Real solutions to equations from geometry

Mathematics is a subject that many students find challenging and sometimes even intimidating. However, cultivating an interest in mathematics can open up a world of possibilities and opportunities. Here are some ways to develop a passion for math:1. Start with RealWorld Applications: Show how math is used in everyday life, from calculating change at the store to understanding the angles in a buildings structure. This can make the subject more relatable and interesting.2. Use Engaging Materials: Books, videos, and online resources that present math in afun and engaging way can help spark interest. Look for materials that tell stories or solve puzzles using mathematical concepts.3. Encourage Curiosity: Encourage questions and exploration. When a student asks Why?, take the time to explain the underlying mathematical principles.4. Make it Competitive: Math competitions can be a fun way to challenge oneself and others. They can also provide a sense of achievement when problems are solved.5. Introduce Different Fields of Math: From geometry to calculus, there are many areas of math to explore. Introducing students to different fields can help them find the area that most interests them.6. Use Technology: Educational apps and software can make learning math interactive and enjoyable. They can also provide immediate feedback, which can be motivating.7. Teach ProblemSolving Skills: Math is all about problemsolving. Teach students that its okay to make mistakes and that the process of finding a solution is just as important as the solution itself.8. Connect with Careers: Show how math skills are essential in various careers, from engineering to economics. This can help students see the practical value of what theyre learning.9. Celebrate Successes: Recognize and celebrate when a student grasps a difficult concept or solves a challenging problem. This positive reinforcement can boost their confidence and interest in math.10. Provide a Supportive Environment: Create a learning environment where its okay to ask for help and where students feel comfortable sharing their ideas.Remember, developing an interest in math is a journey, not a destination. It takes time and patience, but with the right approach, students can come to appreciate the beauty and utility of mathematics.。

Differential equationNot to be confused with Difference equation.Stokes differential equations used to simulate airflow around an obstruction.ClassificationSolutionVisualization of heat transfer in a pump casing, created by solving the heat equation. Heat is being generated internally in the casing and being cooled at the boundary, providing a steady state temperature distribution.A differential equation is amathematical equation that relatessome function of one or more variableswith its derivatives. Differentialequations arise whenever adeterministic relation involving somecontinuously varying quantities(modeled by functions) and their ratesof change in space and/or time(expressed as derivatives) is known orpostulated. Because such relations areextremely common, differentialequations play a prominent role inmany disciplines includingengineering, physics, economics, andbiology.Differential equations aremathematically studied from severaldifferent perspectives, mostlyconcerned with their solutions — the set of functions that satisfy the equation. Only the simplest differential equations are solvable by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form. If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have beendeveloped to determine solutions with a given degree of accuracy.ExampleFor example, in classical mechanics, the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow one (given the position, velocity, acceleration and various forces acting on the body) to express these variables dynamically as a differential equation for the unknown position of the body as a function of time.In some cases, this differential equation (called an equation of motion) may be solved explicitly.An example of modelling a real world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity and air resistance. The ball's acceleration towards the ground is the acceleration due to gravity minus the acceleration due to air resistance. Gravity is considered constant, and air resistance may be modeled as proportional to the ball's velocity. This means that the ball's acceleration, which is a derivative of its velocity, depends on the velocity (and the velocity depends on time). Finding the velocity as a function of time involves solving a differential equation and verifying its validity.Directions of studyThe study of differential equations is a wide field in pure and applied mathematics, physics, and engineering. All of these disciplines are concerned with the properties of differential equations of various types. Pure mathematics focuses on the existence and uniqueness of solutions, while applied mathematics emphasizes the rigorous justification of the methods for approximating solutions. Differential equations play an important role in modelling virtually every physical, technical, or biological process, from celestial motion, to bridge design, to interactions between neurons. Differential equations such as those used to solve real-life problems may not necessarily be directly solvable, i.e. do not have closed form solutions. Instead, solutions can be approximated using numerical methods.Mathematicians also study weak solutions (relying on weak derivatives), which are types of solutions that do not have to be differentiable everywhere. This extension is often necessary for solutions to exist.The study of the stability of solutions of differential equations is known as stability theory.NomenclatureThe theory of differential equations is well developed and the methods used to study them vary significantly with the type of the equation.Ordinary and partial•An ordinary differential equation (ODE) is a differential equation in which the unknown function (also known as the dependent variable) is a function of a single independent variable. In the simplest form, the unknown function is a real or complex valued function, but more generally, it may be vector-valued or matrix-valued: this corresponds to considering a system of ordinary differential equations for a single function.Ordinary differential equations are further classified according to the order of the highest derivative of the dependent variable with respect to the independent variable appearing in the equation. The most important cases for applications are first-order and second-order differential equations. For example, Bessel's differential equation(in which y is the dependent variable) is a second-order differential equation. In the classical literature a distinction is also made between differential equations explicitly solved with respect to the highest derivative and differential equations in an implicit form. Also important is the degree, or (highest) power, of the highest derivative(s) in the equation (cf. : degree of a polynomial). A differential equation is called a nonlinear differential equation if its degree is not one (a sufficient but unnecessary condition).• A partial differential equation (PDE) is a differential equation in which the unknown function is a function of multiple independent variables and the equation involves its partial derivatives. The order is defined similarly to the case of ordinary differential equations, but further classification into elliptic, hyperbolic, and parabolic equations, especially for second-order linear equations, is of utmost importance. Some partial differentialequations do not fall into any of these categories over the whole domain of the independent variables and they are said to be of mixed type.Linear and non-linearBoth ordinary and partial differential equations are broadly classified as linear and nonlinear.• A differential equation is linear if the unknown function and its derivatives appear to the power 1 (products of the unknown function and its derivatives are not allowed) and nonlinear otherwise. The characteristic property of linear equations is that their solutions form an affine subspace of an appropriate function space, which results in much more developed theory of linear differential equations. Homogeneous linear differential equations are a further subclass for which the space of solutions is a linear subspace i.e. the sum of any set of solutions or multiples of solutions is also a solution. The coefficients of the unknown function and its derivatives in a linear differential equation are allowed to be (known) functions of the independent variable or variables; if these coefficients are constants then one speaks of a constant coefficient linear differential equation.•There are very few methods of solving nonlinear differential equations exactly; those that are known typically depend on the equation having particular symmetries. Nonlinear differential equations can exhibit verycomplicated behavior over extended time intervals, characteristic of chaos. Even the fundamental questions of existence, uniqueness, and extendability of solutions for nonlinear differential equations, and well-posedness of initial and boundary value problems for nonlinear PDEs are hard problems and their resolution in special cases is considered to be a significant advance in the mathematical theory (cf. Navier–Stokes existence and smoothness).However, if the differential equation is a correctly formulated representation of a meaningful physical process, then one expects it to have a solution.Linear differential equations frequently appear as approximations to nonlinear equations. These approximations are only valid under restricted conditions. For example, the harmonic oscillator equation is an approximation to the nonlinear pendulum equation that is valid for small amplitude oscillations (see below).ExamplesIn the first group of examples, let u be an unknown function of x, and c and ω are known constants.•Inhomogeneous first-order linear constant coefficient ordinary differential equation:•Homogeneous second-order linear ordinary differential equation:•Homogeneous second-order linear constant coefficient ordinary differential equation describing the harmonic oscillator:•Inhomogeneous first-order nonlinear ordinary differential equation:•Second-order nonlinear (due to sine function) ordinary differential equation describing the motion of a pendulum of length L:In the next group of examples, the unknown function u depends on two variables x and t or x and y.•Homogeneous first-order linear partial differential equation:•Homogeneous second-order linear constant coefficient partial differential equation of elliptic type, the Laplace equation:•Third-order nonlinear partial differential equation, the Korteweg–de Vries equation:Related concepts• A delay differential equation (DDE) is an equation for a function of a single variable, usually called time, in which the derivative of the function at a certain time is given in terms of the values of the function at earlier times.• A stochastic differential equation (SDE) is an equation in which the unknown quantity is a stochastic process and the equation involves some known stochastic processes, for example, the Wiener process in the case of diffusion equations.• A differential algebraic equation (DAE) is a differential equation comprising differential and algebraic terms, given in implicit form.Connection to difference equationsSee also: Time scale calculusThe theory of differential equations is closely related to the theory of difference equations, in which the coordinates assume only discrete values, and the relationship involves values of the unknown function or functions and values at nearby coordinates. Many methods to compute numerical solutions of differential equations or study the properties of differential equations involve approximation of the solution of a differential equation by the solution of a corresponding difference equation.Universality of mathematical descriptionMany fundamental laws of physics and chemistry can be formulated as differential equations. In biology and economics, differential equations are used to model the behavior of complex systems. The mathematical theory of differential equations first developed together with the sciences where the equations had originated and where the results found application. However, diverse problems, sometimes originating in quite distinct scientific fields, may give rise to identical differential equations. Whenever this happens, mathematical theory behind the equations can be viewed as a unifying principle behind diverse phenomena. As an example, consider propagation of light and sound in the atmosphere, and of waves on the surface of a pond. All of them may be described by the same second-order partial differential equation, the wave equation, which allows us to think of light and sound as forms of waves, much like familiar waves in the water. Conduction of heat, the theory of which was developed by Joseph Fourier, is governed by another second-order partial differential equation, the heat equation. It turns out that many diffusion processes, while seemingly different, are described by the same equation; the Black–Scholes equation in finance is, for instance, related to the heat equation.Notable differential equationsPhysics and engineering•Newton's Second Law in dynamics (mechanics)•Euler–Lagrange equation in classical mechanics•Hamilton's equations in classical mechanics•Radioactive decay in nuclear physics•Newton's law of cooling in thermodynamics•The wave equation•Maxwell's equations in electromagnetism•The heat equation in thermodynamics•Laplace's equation, which defines harmonic functions•Poisson's equation•Einstein's field equation in general relativity•The Schrödinger equation in quantum mechanics•The geodesic equation•The Navier–Stokes equations in fluid dynamics•The Diffusion equation in stochastic processes•The Convection–diffusion equation in fluid dynamics•The Cauchy–Riemann equations in complex analysis•The Poisson–Boltzmann equation in molecular dynamics•The shallow water equations•Universal differential equation•The Lorenz equations whose solutions exhibit chaotic flow.Biology•Verhulst equation – biological population growth•von Bertalanffy model – biological individual growth•Lotka–Volterra equations – biological population dynamics•Replicator dynamics – found in theoretical biology•Hodgkin–Huxley model – neural action potentialsEconomics•The Black–Scholes PDE•Exogenous growth model•Malthusian growth model•The Vidale–Wolfe advertising modelReferences•P. Abbott and H. Neill, Teach Yourself Calculus, 2003 pages 266-277•P. Blanchard, R. L. Devaney, G. R. Hall, Differential Equations, Thompson, 2006• E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, 1955• E. L. Ince, Ordinary Differential Equations, Dover Publications, 1956•W. Johnson, A Treatise on Ordinary and Partial Differential Equations[2], John Wiley and Sons, 1913, in University of Michigan Historical Math Collection [3]• A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations (2nd edition), Chapman & Hall/CRC Press, Boca Raton, 2003. ISBN 1-58488-297-2.•R. I. Porter, Further Elementary Analysis, 1978, chapter XIX Differential Equations•Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems[4]. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0.• D. Zwillinger, Handbook of Differential Equations (3rd edition), Academic Press, Boston, 1997.[1]/w/index.php?title=Template:Differential_equations&action=edit[2]/cgi/b/bib/bibperm?q1=abv5010.0001.001[3]/u/umhistmath/[4]http://www.mat.univie.ac.at/~gerald/ftp/book-ode/External links•Lectures on Differential Equations (/courses/mathematics/18-03-differential-equations-spring-2010/video-lectures/) MIT Open CourseWare Videos•Online Notes / Differential Equations (/classes/de/de.aspx) Paul Dawkins, Lamar University•Differential Equations (/diffeq/diffeq.html), S.O.S. Mathematics•Differential Equation Solver (/tools/differential_equation_solver/) Java applet tool used to solve differential equations.•Introduction to modeling via differential equations (/mat/u-u/en/ differential_equations_intro.htm) Introduction to modeling by means of differential equations, with critical remarks.•Mathematical Assistant on Web (http://user.mendelu.cz/marik/maw/index.php?lang=en&form=ode) Symbolic ODE tool, using Maxima•Exact Solutions of Ordinary Differential Equations (http://eqworld.ipmnet.ru/en/solutions/ode.htm)•Collection of ODE and DAE models of physical systems (/research/models.htm) MATLAB models•Notes on Diffy Qs: Differential Equations for Engineers (/diffyqs/) An introductory textbook on differential equations by Jiri Lebl of UIUC•Khan Academy Video playlist on differential equations (/math/ differential-equations) Topics covered in a first year course in differential equations.•MathDiscuss Video playlist on differential equations (/category/courses/ solutions-differential-equations/homogeneous-linear-systems/)Article Sources and Contributors8Article Sources and ContributorsDifferential equation Source: /w/index.php?oldid=610771276 Contributors: 17Drew, After Midnight, Ahoerstemeier, Alarius, Alfred Centauri, Amahoney, AndreiPolyanin, Andres, AndrewHowse, Andycjp, Andytalk, AngryPhillip, Anonymous Dissident, Antoni Barau, Antonius Block, Anupam, Apmonitor, Arcfrk, Asdf39, Asyndeton, Attilios,Babayagagypsies, Baccala@, Baccyak4H, Bejohns6, Bento00, Berland, Bidabadi, Bigusbry, BillyPreset, Bob.v.R, Bolatbek, Brandon, Bryanmcdonald, Btyner, Bygeorge2512,Callumds, Charles Matthews, Christian75, Chtito, Cispyre, Cmprince, Coginsys, ConMan, Cxz111, Cybercobra, DAJF, Danski14, Dbroadwell, Ddxc, Delaszk, DerHexer, Dewritech, Difu Wu, Djordjes, DominiqueNC, Donludwig, Dpv, Dr sarah madden, Drmies, DroEsperanto, Duoduoduo, Dysprosia, EconoPhysicist, Elwikipedista, Epicgenius, EricBright, Erin.Annette.Brown,Estudiarme, F=q(E+v^B), Fintor, Fioravante Patrone, Fioravante Patrone en, Flameturtle, Friend of the Facts, FutureTrillionaire, Gabrielleitao, Gandalf61, Gauss, Genedronek, Geni, Giftlite,GoingBatty, Gombang, Grenavitar, Haham hanuka, Hamiltondaniel, Harry, Haruth, Haseeb Jamal, Heikki m, Holmes1900, Ilya Voyager, Iquseruniv, Iulianu, Izodman2012, J arino, J.delanoy, Ja 62, Jak86, JamesBWatson, Jao, Jarble, Jauhienij, Jayden54, Jeancey, Jersey Devil, Jim Sukwutput, Jim.belk, Jim.henderson, JinJian, Jitse Niesen, JohnOwens, Johndoeisnotmyname, JorisvS,Julesd, K-UNIT, Kayvan45622, KeithJonsn, Kensaii, Khalid Mahmood, Klaas van Aarsen, Kr5t, Krushia, LOL, Lambiam, Lavateraguy, Lethe, LibLord, Linas, Lumos3, Madmath789, Mandarax, Mankarse, MarSch, Martastic, Martynas Patasius, Maschen, Math.geek3.1415926, Matqkks, Mattmnelson, Maurice Carbonaro, Maxis ftw, Mazi, McVities, Mduench, Mets501, Mh, MichaelHardy, Mindspillage, MisterSheik, Mohan1986, Mossaiby, Mpatel, MrOllie, Mtness, Mysidia, Nik-renshaw, Nkayesmith, Norm mit, Okopecz, Oleg Alexandrov, Opelio, Pahio, Parusaro, Paul August, Paul Matthews, Paul Richter, PavelSolin, Pgk, Phoebe, Pine, Pinethicket, Pratyya Ghosh, PseudoSudo, Qwerty Binary, Qzd800, R'n'B, Rama's Arrow, Randomguess, Reallybored999, RexNL, Reyk, RichMorin, Robin S, Romansanders, Rosasco, Ruakh, SDC, SFC9394, SakeUPenn, Salix alba, Sam Staton, Sampathsris, Sardanaphalus, Senoreuchrestud, Silly rabbit, Siroxo,Skakkle, Skypher, SmartPatrol, Snowjeep, Spirits in the Material, Starwiz, Suffusion of Yellow, Sverdrup, Symane, TVBZ28, TYelliot, Tannkrem, Tbhotch, Tbsmith, TexasAndroid, Tgeairn, The Hybrid, The Thing That Should Not Be, Timelesseyes, Tranum1234567890, Tsirel, Tuseroni, User A1, Vanished User 0001, Vishwanathnm, Vthiru, Waffleguy4, Waldir, Waltpohl, Wavelength, Wclxlus, Wihenao, Willtron, Winterheart, Wsears, XJaM, Yafujifide, Zepterfd, ﺪﺟﺎﺳ ﺪﺠﻣﺍ ﺪﺟﺎﺳ, 363 anonymous editsImage Sources, Licenses and ContributorsFile:Airflow-Obstructed-Duct.png Source: /w/index.php?title=File:Airflow-Obstructed-Duct.png License: Public Domain Contributors: Original uploader was User A1 at en.wikipediaFile:Elmer-pump-heatequation.png Source: /w/index.php?title=File:Elmer-pump-heatequation.png License: Creative Commons Attribution-Sharealike 3.0Contributors: Christian1985, Crimerob, Kri, User A1, 2 anonymous editsLicenseCreative Commons Attribution-Share Alike 3.0///licenses/by-sa/3.0/。

Simultaneous Equation MethodIntroductionIn mathematics, simultaneous equations play a crucial role in solving real-world problems and modeling various phenomena. The simultaneous equation method is a powerful technique used to find solutions for a system of equations. This method involves solving multiple equations together to determine the values of unknown variables. In this article, we will explore the simultaneous equation method in detail and discuss its applications.Understanding Simultaneous EquationsDefinitionSimultaneous equations, also known as a system of equations, are a set of equations that share the same variables. The solutions of these equations simultaneously satisfy each equation in the system. The general form of simultaneous equations can be written as:a1x + b1y = c1a2x + b2y = c2Here, x and y are the variables, while a1, a2, b1, b2, c1, and c2 are constants.Types of Simultaneous EquationsSimultaneous equations can be classified into three types based on the number of solutions they have:1.Consistent Equations: These equations have a unique solution,meaning there is a specific set of values for the variables thatsatisfy all the equations in the system.2.Inconsistent Equations: This type of system has no solution. Theequations are contradictory and cannot be satisfied simultaneously.3.Dependent Equations: In this case, the system has infinitely manysolutions. The equations are dependent on each other and represent the same line or plane in geometric terms.To solve simultaneous equations, we employ various methods, with the simultaneous equation method being one of the most commonly used techniques.The Simultaneous Equation MethodThe simultaneous equation method involves manipulating and combining the given equations to eliminate one variable at a time. By eliminating one variable, we can reduce the system to a single equation with one variable, making it easier to find the solution.ProcedureThe general procedure for solving simultaneous equations using the simultaneous equation method is as follows:1.Identify the unknow n variables. Let’s assume we have n variables.2.Write down the given equations.3.Choose two equations and eliminate one variable by employingsuitable techniques such as substitution or elimination.4.Repeat step 3 until you have a single equation with one variable.5.Solve the single equation to determine the value of the variable.6.Substitute the found value back into the other equations to obtainthe values of the remaining variables.7.Verify the solution by substituting the found values into all theoriginal equations. The values should satisfy each equation.If the system is inconsistent or dependent, the simultaneous equation method will also lead to appropriate conclusions.Applications of Simultaneous Equation MethodThe simultaneous equation method finds applications in numerous fields, including:EngineeringSimultaneous equations are widely used in engineering to model and solve various problems. Engineers employ this method to determine unknown quantities in electrical circuits, structural analysis, fluid mechanics, and many other fields.EconomicsIn economics, simultaneous equations help analyze the relationship between different economic variables. These equations assist in studying market equilibrium, economic growth, and other economic phenomena.PhysicsSimultaneous equations are a fundamental tool in physics for solving complex problems involving multiple variables. They are used in areas such as classical mechanics, electromagnetism, and quantum mechanics.OptimizationThe simultaneous equation method is utilized in optimization techniques to find the optimal solution of a system subject to certain constraints. This is applicable in operations research, logistics, and resource allocation problems.ConclusionThe simultaneous equation method is an essential mathematical technique for solving systems of equations. By employing this method, we can find the values of unknown variables and understand the relationships between different equations. The applications of this method span across various fields, making it a valuable tool in problem-solving and modeling real-world situations. So, the simultaneous equation method continues to be akey topic in mathematics and its practical applications in diverse disciplines.。

In the realm of English composition,writing about numbers can be an engaging and informative task.Here are some key points to consider when crafting an essay on this topic:1.Introduction to the Significance of Numbers:Begin your essay by highlighting the importance of numbers in our daily lives,from counting to complex mathematical equations that drive scientific advancements.2.Historical Perspective:Discuss the evolution of numerical systems,such as the transition from Roman numerals to the Arabic numerals we use today.This can provide an interesting historical context.3.Cultural Impact:Numbers hold different significance in various cultures.For example, the number13is considered unlucky in Western cultures,while in some Asian cultures, the number4is associated with death.4.Mathematical Concepts:Delve into fundamental mathematical concepts related to numbers,such as prime numbers,factorials,and Fibonacci sequences.Explain how these concepts are applied in realworld scenarios.5.The Role of Numbers in Science:Explore how numbers are integral to scientific theories and formulas.For instance,the use of piπin calculating the circumference of a circle or the golden ratio in art and architecture.6.Numerical Systems in Technology:Discuss binary and hexadecimal systems used in computer programming and how they differ from the decimal system.7.Statistics and Data Analysis:Numbers play a crucial role in statistics,allowing us to make sense of large data sets,draw conclusions,and make predictions.8.Personal Finance and Economics:Numbers are the backbone of financial systems, from calculating interest rates to understanding economic indicators like GDP and inflation.9.The Beauty of Numbers in Art and Literature:Some authors and artists use numbers creatively in their works,either as themes or structural elements.For example,the use of numerical patterns in poetry or the significance of certain numbers in novels.10.The Future of Numbers:Conclude your essay by speculating on the future of numbers, including potential advancements in mathematics,the role of artificial intelligence innumber theory,and how our understanding and use of numbers may evolve. Remember to use clear examples and explanations to illustrate your points,and ensure that your essay flows logically from one section to the next.Writing about numbers can be both educational and enjoyable,offering a window into the universal language that connects us all.。

Pre-requisites:throughout this chapter,the following basic properties and notions are assumed to be known:∙derivatives and antiderivatives;∙usual functions and in particular,their derivatives;∙complex numbers;roots of a second degree polynomial.Contents1General definitions and standard vocabulary2 2First order linear differential equations32.1Exponential functions and their characterization (4)2.2Structure of the set of solutions and sum principle (6)2.3Solutions to the associated homogeneous equation (7)2.4Variation of parameters (8)2.5Solution satisfying a given initial condition:existence and uniqueness (14)3Second order linear differential equations with constant coefficients153.1Definition and structure of the set of solutions (15)3.2Solutions to the associated homogeneous equation (16)3.3Finding a particular solution when the second member is of type exponential-polynomial (20)3.4Variation of parameters (22)3.5Solution satisfying a given initial condition:existence and uniqueness (24)1General definitions and standard vocabularyThroughout this chapter,unless specified otherwise, will be used to state results that are valid with either =ℝor =ℂ.Definition 1.0.1Let be a positive integer.A differential equation of order is an equation in which the unknown is a function (with domain (to be determined)and (at least) times differentiable on )and of the form:( ): ( , , ′,⋅⋅⋅, ( −1), ( ))=0where for each ∈[∣0, ]∣, ( )is the -th derivative of and is a function of ( +2)variables,and is not constant with respect to the last variable.Example 1.0.1You have probably already studied a number of differential equations,in particular during your physics class.Here a few examples of differential equations:∙ ′+2 =0is a (linear)differential equation of order 1(setting ( , , ′)= ′+2 );∙ ′( )− ( )− 2=0is a (linear)differential equation or order 1;∙ ′=1+ 2is a also a differential equation of order 1(but non linear);∙ ′′+ 2 =0is a (linear)differential equation of order 2;∙ (6)− (3)+2 2=cos( )is a (non linear)differential equation of order 6.Remark:the condition “which is not constant with respect to the last variable”is a technical condition.It is there to guarantee that the equation is really of order .For instance,if we set: ( , , , )= + then the differential equation ( , , ′, ′′)=0is + ′=0,which is really of order 1and not 2.Definition 1.0.2We say a function with domain (a non trivial interval)is a solution to ( )if is at least times differentiable on and:∀ ∈ , ( , ( ), ′( ),⋅⋅⋅, ( )( ))=0Therefore,a solution to the differential equation ( )is really a couple ( , ),where is a real interval and a function at least times differentiable on .Example 1.0.2The function with domain [0;1]given by ( )=exp( )for all ∈[0;1]is a solution of ′= .However,it is clearly not the only one as we could define for any interval , ( )=exp( )for ∈ which would also be a solution to the same differential equation,but on .This is why,in our search for solutions,we focus on solutions with maximal domain.Definition 1.0.3To solve the differential equation ( )is to find all maximal solutions,i.e.find all solutions whose domain’s are maximal (that is,there is no solution to ( )with domain such that ⊊ and ∣ = ).Example 1.0.3In our previous example,the maximal solutions of ′= are the functions with domain ℝgiven by ( )= exp( )for all real number .Obviously,there are a number of differential equations we do not know how to solve explicitly.This is where numerical analysis can be very convenient:there are many methods to give approximations of solutions,even though we do not know an exact form.We will study this later in the year.However,there are one kind of differential equations for which,under“reasonable conditions”,we always know how to solve explicitly:namely,linear equations.The object of the next sections is to study these equations,but only when their order is either1or2.Before we do so,we will give a formal definition of what a linear differential equation actually is.Definition1.0.4A differential equation is said to be linear if it is of the form: ( )− =0,where is the unknown, is a function and is a linear map,i.e. satisfies:( 1+ 2)= ( 1)+ ( 2)and ( 1)= ( 1)for all“unknowns” 1and 2and all scalar constants ∈ .In that case,we say that:∙ is the second member of( );∙the differential equation( ): ( )=0is the homogeneous equation associated to( ).Remark:it is often convenient to group both properties in the definition of a linear map.More specifically, is a linear map if and only if,for all unknowns 1, 2and all , ∈ , ( 1+ 2)= 1+ 2.Example1.0.4Setting ( )= ′− ,we have for all unknowns 1and 2,and all , ∈ :( 1+ 2)=( 1+ 2)′−( 1+ 2)= ′1+ ′2− 1− 2= ( ′1− 1)+ ( ′2− 2)= ( 1)+ ( 2)Therefore, ′− =0is indeed a linear differential equation.Example1.0.5We stated earlier that ′=1+ 2is not a linear differential equation.Indeed,we have here: ′− 2=1 so that ( )= ′− 2and =1(constant function).However, (2 )=2 ′−4 2=2( ′− 2)if =0.Therefore, is not a linear map(the condition must be verified for all unknowns).2First order linear differential equationsDefinition2.0.5A linear differential equation of order1is a differential equation of the form:( ) ′( )+ ( ) ( )= ( )where , and are three functions defined on a real interval .We will only work withfirst order linear differential equations which are“resolved”in ′,i.e.of the form:′( )+ ( ) ( )= ( )Remark:often,we write ′+ ( ) = ( )which theoretically is abusive.The proper forms would either be ′( )+ ( ) ( )= ( )for all in a certain interval or ′+ = (equality amongst functions).However,this abuse is currently accepted and we will always write the equations in that form.Proposition2.0.1Afirst order linear differential equation is indeed a linear differential equation.Hence,the name.Proof:To begin,we observe that the equation can be written in the form: ( )= ,where =and ( )= ( ) ′+ ( ) (again,we should write ( ): −→ ( ) ′( )+ ( ) ( )).Now,let 1and 2be two unknowns and let , be two scalars in .We then have:( 1+ 2)= ( )( 1+ 2)′+ ( )( 1+ 2)= ( )( ′1+ ′2)+ ( )( 1+ 2)= ( ( ) ′1+ ( ) 1)+ ( ( ) ′2+ ( ) 2)= ( 1)+ ( 2)Thus, is a linear map and the equation is indeed linear.⊠2.1Exponential functions and their characterizationTheorem2.1.1Let ∈ .Then the function :ℝ−→−→exp( )is the only(maximal)solution of ′=that satisfies the initial condition (0)=1.Proof:Suppose that is a solution of ′= that verifies (0)=1and let ( )= ( )exp(− ).Then is differentiable onℝ(product of two functions that are)and for all real number ,′( )= ′( )exp(− )− ( )exp(− )=exp(− )( ′( )− ( ))=0Hence, is constant onℝ.Furthermore, (0)= (0)exp(0)=1×1=1so that for all ∈ℝ,( )exp(− )=1.Multiplying both sides by exp( )=0,wefind that: ( )=exp( )forall real number .This proves that if is a solution of ′= that satisfies (0)=1then= .Conversely,one can easily check that satisfies ′ = and (0)=1.⊠Remark:this is one way of characterizing the exponential functions(real or complex)as the only solutions of ′= with initial condition (0)=1.Theorem2.1.2(Second characterization)The only functions differentiable onℝthat verify:∀ , ∈ℝ, ( + )= ( )× ( )(∗)are either exponential functions with ∈ℂor identically zero.Proof:⇐=The identically zero function and exponential functions obviously are differentiable onℝand verify(∗).=⇒Let be a function which is differentiable onℝand verifies the functional equation(∗).Next,let ∈ℝand consider::ℝ−→−→ ( + )− ( ) ( )As is differentiable on ℝ, is also.Furthermore,by hypothesis, and hence, ′are identically zero.Therefore,by differentiating:∀ ∈ℝ, ′( + )= ′( ) ( )Setting =0,we have: ′( )= ′(0) ( ),i.e. ′= with = ′(0).We now consider two possibilities:∙either is identically zero (and the result follows);∙else there is at least one real number 0such that ( 0)=0.Setting = 0and =0in (∗),we have: ( 0)= ( 0+0)= (0)× ( 0).However,by hypothesis, ( 0)=0;hence, (0)=1.We have therefore shown that satisfies ′= and (0)=1.According to our last theorem, = for some ∈ℂ.⊠Remark:one can actually show that the conditions we imposed are a bit strong.Indeed,we need not suppose differentiable.The hypothesis continuous on ℝis sufficient,and actually,even continuous at one point is enough.Exercise 2.1.1Let be a function that verifies (∗).Prove that:(1) is identically zero or does not vanish at all and (0)=1;(2)if is continuous at (with ∈ℝ),then is continuous on ℝ;(3)if is not identically zero,then there exists >0such that∫( )d =0and that therefore, is differentiableon ℝ.Remark:this second characterization of exponential functions is very useful.For instance,this is how one can prove that continuous random variable without memory: ( > + ∣ > )= ( > )for all , ⩾0has necessarily an exponential distribution.Remark:we can also mention a few applications of these theorems to Physics:∙radioactive decay:take for instance plutonium isotope Pu-239.Physics laws state that if ( )is the number ofradioactive atoms (or otherwise the mass (multiplying by the molar mass)),then the activity -given by =−dd-is proportional to .This leads to the relation −d d = .Here,ln 2is what we call the half-life,i.e.thetime required for half the atoms in a sample of radioactive material to decay.∙Newton’s law of cooling states that the rate of change in the temperature of an object is proportional to the difference between the object’s temperature and the temperature of the surrounding medium,i.e.dd=ℎ( 0− )where is the temperature of the object, 0the temperature of the surrounding medium,andℎis what we call the Newton coefficient.2.2Structure of the set of solutions and sum principleTheorem2.2.1Let( )be a linear differential equation,( ): ( )= with a linear map.Assume that is a particular solution of( ).Then for any differentiable function , is a solution of( )if and only if − is a solution of( ).In other words,setting (respectively( ))the set of solutions of( )(respectively( )),one has:= +Proof:Let be any differentiable function.Then,is a solution of( )⇐⇒ ( )=⇐⇒ ( )= ( )(because is a solution of(E))⇐⇒ ( )− ( )=0⇐⇒ ( − )=0(because is linear)⇐⇒ − is a solution of( )⊠this fundamental theorem has a very important practical signification.In order to solve alinear differential equation,one mustfind one solution(what we call a particular solution)and then add all the solutions to the homogeneous equation.Remark:this is why in the next paragraphs,we will focus on how to solve the homogeneous equation and then how to find a particular solution.Remark:for your information,we will say that is an affine space-that is the sum of“a point”( )and a“vector space”( ).Affine spaces and vector spaces are fundamental algebraic structures which we will study next year. Proposition2.2.1(Sum principle)Let be a linear map, 1be a solution of ( )= 1and 2a solution of ( )= 2,where 1and 2are two functions.Then:(i) 1+ 2is a solution of ( )= 1+ 2;(ii)for all ∈ , 1is a solution of ( )= 1.Proof:All these properties are direct consequences of the fact that is linear.Indeed,( 1+ 2)= ( 1)+ ( 2)= 1+ 2and( 1)= ( 1)= 1⊠Example2.2.1Say we want to solve the differential equation( ): ′− =cos( )+3 2 .We then proceed in three steps:∙Step1:wefind a particular solution of ′− =cos( ).∙Step2:wefind a particular solution of ′− = 2 .∙Step3:we solve the homogeneous equation(here the solutions are −→ )We can therefore conclude that the solutions of( )are the functions:= 1+3 2particular solutionby the sum principle +solutions of( )There are now two questions that arise naturally:1.How do we solve the homogeneous equation?2.How do wefind a particular solution if there is no evident one?That’s what we will now focus on.2.3Solutions to the associated homogeneous equationReminder(which we have already reviewed in the chapter on usual functions)Definition2.3.1Let be a function.We say that a function is an antiderivative of on the real interval if is differentiable on and ′( )= ( )for all in .Theorem2.3.1Any continuous function on an interval has antiderivatives.Furthermore,if ∈ ,then −→∫( )d is the only antiderivative of on which vanishes at .Proposition2.3.1Let be a continuous function on the real interval (with values in )and consider the differential equation( ): ′+ ( ) =0.Let be any antiderivative of on .Then the solutions of( )are the functions with domain given by:∀ ∈ , ( )= exp(− ( )),with ∈Proof:First,we notice that since we assumed to be continuous on , does indeed have an-tiderivatives.Therefore, exists.Next,let be any function and set = exp( ).As theexponential function(real or complex)does not vanish,we have:= exp( )⇐⇒ = exp(− )Hence, is differentiable on if and only if is.We then have:∈ ⇐⇒∀ ∈ , ′( )+ ( ) ( )=0⇐⇒∀ ∈ , ′( )exp(− ( ))− ( ) ( )exp−( ( ))+ ( ) ( )exp(− ( ))=0⇐⇒∀ ∈ , ′( )exp(− ( ))=0⇐⇒∀ ∈ , ′( )=0⇐⇒∃ ∈ ,∀ ∈ , ( )=⇐⇒∃ ∈ ,∀ ∈ , ( )= exp(− ( ))⊠Remark:please note that this result holds for =ℂas well;and also,that this proposition states that the maximal solutions are all defined on !Example2.3.1Consider the following differential equations:∙ ′= where ∈ℂ.We already know from the characterization of exponential functions that the solutions are ( )= (0)exp( ).Setting this aside,( )can be written as ′− =0and if we apply the previous proposition,we know that −→− is continuous onℝand −→ is an antiderivative onℝ.Therefore,the solutions are of the form:( )= exp( ),with ∈We do indeedfind the same solutions,which is reassuring!∙( ): ′+11+ 2=0. −→11+ 2is continuous onℝand has −→arctan( )for antiderivative.Therefore,the solutions of( )are the functions defined on by:∀ ∈ , ( )= exp(−arctan( )), ∈Remark:solving a homogeneous differential equation of order one basically boils down tofinding an antiderivative of a given function.Proposition2.3.2With the previous notations,(i)solutions of( ),other than zero,do not vanish on ;(ii) ={ − , ∈ }is a“one-dimensional vector space”or a“line”:all elements of are“proportional”(“collinear”)to − .Proof:(ii)is simply a formal way of writing the set of solutions.As for( ),let be a solution of( ).Then by the previous proposition,there is a constant ∈ such that = exp(− ).Moreover,we know that the exponential never vanishes;therefore,if ( 0)=0for some0∈ ,then =0and =0(identically zero).⊠2.4Variation of parametersNow we consider the full equation( ): ′+ ( ) = ( ).Let 0be a non-zero solution of( )(( 0)is a basis of ).From the previous paragraph,we know that the solutions of the homogeneous equation( )are of the form 0,with ∈ .We are therefore going to try tofind a particular solution of the same form,but by“variation of the parameter ”,i.e.of the form:( )= ( ) 0( )By( )in the previous proposition,we know that for all ∈ , 0( )=0;hence,= 0⇐⇒ = 0and therefore, is differentiable on if and only if is.We then deduce that for any differentiable function : = 0; ′= ′ 0+ ′0et ′+ = ′ 0+( ′+ )0( 0solution of( ))= ′ 0Hence,is a solution of( )⇐⇒ ′ 0= ⇐⇒ ′= 0We can now state our result formally:Theorem 2.4.1Consider the differential equation ( ): ′+ ( ) = ( ),where and are two continuous functions on .Let be an antiderivative of and an antiderivative of → ( ) ( ).Then a particular solution of ( )is given by:∀ ∈ , ( )= ( ) − ( )In particular,if 0∈ ,then the solutions of ( )aregiven by:∀ ∈ , ( )=(∫( ) ( )d ) − ( )+ − ( )Proof:We choose 0=exp(− ).The previous calculations show that:is a solution of ( )⇐⇒ ′= exp( )⇐⇒= +Substituting in = exp(− )yields the results.⊠△!Caution:beware!All the theorems we stated apply for equations which are “resolved”in ′.If they are not,i.e.of the form ( ) ′+ ( ) = ( ),then one most solve the equation on intervals on which does not vanish,wherewe can write ′+ ( ) ( ) = ( )( ).Example 2.4.1Consider a resistor and a capacity mounted in series with a generator delivering a constant tension .We know that the tension (or voltage)then satisfies the following differential equation:( 1):dd+ = As = =0,( 1)is equivalent tod d + =.We now apply our method.∙Solutions to the associated homogeneous equation:−→1 is continuous on ℝand has −→for antiderivative.Therefore,the solutions to ( 1, )are thefunctions given by:∀ ∈ℝ, ( )= exp(−), ∈ℝ∙Search for a particular solution:we could simply apply the variation of parameters method.However,it is much simpler here to search for a trivial solution.The second member is constant:we therefore try to find a constant solution,i.e.such that dd =0.This yields: = .∙ConclusionThis proves that the tension is of the form:∀ ∈[0;+∞[, ( )= + −.We can then determine the value of by using the initial condition.Example2.4.2Consider the equation( 2): ′− =cos( ).∙Solutions to the associated homogeneous equation:It is clear that the solutions to the homogeneous equation are given by: ( )= for all ∈ℝ,with ∈ℝ.∙Search for a particular solution:–First method:variation of parametersApplying the variation of parameters,we set for all inℝ, ( )= ( ) where is a differentiable function onℝ.Then:is a solution of( 2)⇐⇒∀ ∈ℝ, ′ ( ) =cos( )⇐⇒∀ ∈ℝ, ( )= − cos( )We must therefore determine an antiderivative onℝof −→ − cos( ).To do this,we can,for example, use integration by parts twice:∫ 0 − cos( )d =[− − cos( )]−∫(− − )(−sin( ))d=− − cos( )+1−[− − sin( )]−∫− cos( )dThus,2∫− cos( )d =− − cos( )+ − sin( )+1So that −→ −2(sin( )−cos( ))is an antiderivative of −→ − cos( ).Hence,one particular solution of( 2)is given by:∀ ∈ℝ, ( )=12(sin( )−cos( ))–Second method:using complex numbersConsider the new equation:( ): ′− = .If wefind a particular solution ,thenℜ ( )will be a particular solution of( 2)1.Next,we try tofind a solution of the form = with ∈ℂ(for more details,see section3.3).We then have the following equivalences:solution of( )⇐⇒∀ ∈ℝ, − =⇐⇒( −1) =1⇐⇒ =1 −1⇐⇒ =− −12Thus,a particular solution of( 2)is given by:for all ∈ℝ,( )=ℜ (−1+2)=−12cos( )+12sin( )–Third method:trying tofind a trivial solutionGiven the second member,it might seem reasonable to look for a solution of the form: −→ cos( )+ sin( ).∙ConclusionWe can now conclude that the solutions to( 2)are given by:∀ ∈ℝ, ( )= −2sin( )− −2cos( )+ − , ∈ℝ1Careful!This method works here because the coefficients are all real!Otherwise,we could not say thatℜ ( ′− )=ℜ ( )′−ℜ ( ).Example2.4.3Consider( 3): ′+1+ 2=11+ 2.∙Step1:solutions to the associated homogeneous equation−→1+ 2is continuous onℝand has −→12ln(1+ 2)for antiderivative.Thus, ={ℝ−→ℝ−→ exp(−12ln(1+ 2)), ∈ℝ}={−→√1+ 2, ∈ℝ}∙Step2:search for a particular solutionApplying the previous theorem,we know that a particular solution is of the form: −→( )√1+ 2where is anantiderivative of −→ ( ) − ( )=1√1+ 2.Here,we can choose =Argsh which leads to:∀ ∈ℝ, ( )=Argsh( )√1+ 2∙ConclusionThis shows that the solutions to the equation( 3)are the functions given by:∀ ∈ℝ, ( )=Argsh( )+√1+ 2,with ∈ℝExercise2.4.1Solve the differential equation: ′+ = 3.Finally,to conclude this paragraph,we will study one example of equation which is not“resolved”in ′and see how one goes about determining maximal solutions.Example2.4.4We wish to solve the differential equation:( ):(1− 2) ′−2 =sin( ).First,notice that1− 2=0if and only if =±1.Therefore,even though( )is defined onℝ,we cannot simply apply our theorems onℝ.We must solve the equation on each interval where1− 2does not vanish,i.e.on 1=]−∞;−1[, 2=]−1;1[and 3=]1;+∞[,and then determine whether or nor,there are any solutions onℝ.∙Step1:Solutions to( )on each interval where1− 2does not vanishLet ∈{1,2,3}.We then have:solution of( )on ⇐⇒ ′+22−1=sin( )1− 2–Solutions to the associated homogeneous equation−→22−1is continuous on and −→ln∣ 2−1∣is an antiderivative of that function on .Therefore,the solutions to( )(the homogeneous equation)are given by:∀ ∈ , ( )= exp (−ln∣ 2−1∣)=∣ 2−1∣, ∈ℝFurthermore,on the interval , 2−1has constant sign.We can thus remove the absolute value and incorporate the sign of 2−1in the sign of the constant .In other words,the solutions to( )are ofthe form:∀ ∈ , ( )=1− 2, ∈ℝ△!Caution:it is extremely important to realize that the parameter that wefind solving the equation depends of the interval on which we are solving the equation.Thus,the notation to indicate this fact.–Search for a particular solutionApplying the variation of parameter method,we set ( )= ( )1−for in ,with a differentiablefunction on .Then,is a solution of( )on ⇐⇒∀ ∈ , ′( )1−=sin( )1−⇐⇒∀ ∈ , ′ ( )=sin( ) Hence,a particular solution of( )on is given by:∀ ∈ , ( )=−cos( ) 1− 2–ConclusionThis proves that the solution of( )on are the functions of the form:∀ ∈ , ( )= −cos( )1− 2,with ∈ℝRemark:we actually could have solved( )much faster by noticing that(1− 2) ′−2 =((1− 2) )′so that( )is equivalent to(1− 2) =−cos( )+ .∙Solving( )completely–Necessary conditionsSuppose is a solution of( )(i.e.a solution onℝ).Then,its restriction to each interval (for ∈{1,2,3}) is a solution of( )on .According to what we have just shown,there are three real constants 1, 2and3such that:( )=⎧⎨⎩1−cos( )1− 2if ∈]−∞;−1[ 2−cos( )1−if ∈]−1;1[ 3−cos( )1− 2if ∈]1;+∞[Also,setting =±1in the equation,we see that (±1)=−sin(1)2so that in fact,if is solution of( )onℝ,then necessarily:( )=⎧⎨⎩1−cos( )1−if ∈]−∞;−1[−sin(1)2if =−12−cos( )1− 2if ∈]−1;1[−sin(1)2if =13−cos( )1− 2if ∈]1;+∞[Furthermore,if is a solution of( ),we know that necessarily, is differentiable at±1,hence continuous.Moreover,lim→−1<−1(1− 2)=0−and lim→−1<−1( 1−cos( ))= 1−cos(1).Therefore,for to be continuousfrom the left at−1,it is necessary that 1−cos(1)=0,i.e. 1=cos(1).Proceeding in the same way,one shows that for to be continuous(period)at−1and1,one must have 1= 2= 3=cos(1).Therefore, we have shown that if a solution to( )exists,it is necessarily given by:( )=⎧⎨⎩cos(1)−cos( )1− 2if =±1−sin(1)2if =±1–Search for sufficient conditionsAssume that is defined onℝby the previous piecewise expression.We wish to determine whether or not is differentiable onℝand solution of( ).First,it is clear that is differentiable onℝ∖{−1,1}and solution of( )on each interval]−∞;−1[,]−1;1[ and]1;+∞[.We must therefore study the differentiability at±1:by construction,if is differentiable at ±1,then will automatically satisfy the equation for =±1.Next,let be any real number such that∣ ∣=1.Then,( )=cos(1)−cos( )(1− )(1+ )=−12×sin(−12)−12×sin(+12)+12Setting ( )=⎧⎨⎩sin( )if =01if =0,we see that in fact,for all real number (including =±1):( )=−12(−12)× (+12)Thus,to prove the differentiability of at ±1,it is sufficient to prove that is differentiable on ℝ.I leave that fact as an exercise for you to do 2.∙ConclusionWe have proved that the differential equation ( )has one and only one solution on ℝwhich is given by:∀ ∈ℝ, ( )=⎧ ⎨ ⎩cos(1)−cos( )1− 2if =±1−sin(1)2if =±1Important facts to remember from this example:(1)when solving a first order differentiable equation which isn’t “resolved”in ′,we cannot directly apply ourtheorems (notice that we found only one solution as opposed to an infinite number of solutions);(2)in this case,it is important to remember to solve on each interval where the function (coefficient in frontof ′)does not vanish,and that the constants (or parameters)we find depend on the interval;(3)then comes the big piece of work:determining necessary conditions on the constants so that a function canbe a solution on the full interval,and then,conversely,checking that such a function is indeed differentiable and a solution to the differentiable equation.(4)note that these equations require a whole lot more work then the other form....2.5Solution satisfying a given initial condition:existence and uniquenessTheorem 2.5.1Consider the differential equation ( ): ′+ ( ) = ( ),where and are two continuous functions on .Let 0∈ and let 0be any scalar in .Then the equation ( )has one and only one solution satisfying the initial condition ( 0)= 0.Furthermore,we can give an exact expression for this solution:∀ ∈ , ( )=( 0+∫ 0( ) ∫ 0 ( )d d )−∫ 0 ( )d Proof:Let be a solution of ( ).Consider : −→ given by ( )=∫( )d so that isthe only antiderivative of on that vanishes at 0.By theorem 2.4.1,we know that thereexists a scalar ∈ such that:∀ ∈ , ( )=(∫ 0 ( ) ( )d ) − ( )+ − ( )Thus,( 0)= 0⇐⇒(∫ 0 0 ( ) ( )d ) − ( 0)+ − ( 0)= 0⇐⇒= 0⊠2hint:start by proving that for any non-negative , − 36⩽sin( )⩽ .3Second order linear differential equations with constant coefficients3.1Definition and structure of the set of solutionsDefinition3.1.1A second order linear differential equation with constant coefficients is by definition a differential equation of the form:( ): ′′+ ′+ = ( )where , , are three scalars in such that =0and a function.Proposition3.1.1A second order linear differential equation with constant coefficients is indeed a linear differ-ential equation.Proof:Set ( )= ′′+ ′+ ;then( )is equivalent to ( )= .Now we show that is alinear map.Let 1, 2be two functions twice differentiable on a real interval and let ,[]be two scalars in .Then:( 1+ 2)= ( 1+ 2)′′+ ( 1+ 2)′+ ( 1+ 2)= ( ′′1+ ′′2)+ ( ′1+ ′2)+ ( 1+ 2)= ( ′′1+ ′1+ 1)+ ( ′′2+ ′2+ 2)= ( 1)+ ( 2)This proves that is a linear map and therefore,that( )is a linear differential equation.⊠As for the structure of the set of solutions,if you read the proof of theorem2.2.1and proposition2.2.1,you can easily see that they do not depend on the order of the differential equation,but solely on the fact that it is linear.Hence, they also apply for second order linear differential equations:Theorem3.1.1Let( )be a linear differential equation,( ): ( )= with a linear map.Assume that is a particular solution of( ).Then for any differentiable function , is a solution of( )if and only if − is a solution of( ).In other words,setting (respectively( ))the set of solutions of( )(respectively( )),one has:= +Proposition3.1.2(Sum principle)Let be a linear map, 1be a solution of ( )= 1and 2a solution of ( )= 2,where 1and 2are two functions.Then:(i) 1+ 2is a solution of ( )= 1+ 2;(ii)for all ∈ , 1is a solution of ( )= 1.Once again,we are now left with two problems:finding the solutions to the homogeneous equation andfinding a particular solution.3.2Solutions to the associated homogeneous equationWe now focus on the associated homogeneous equation:( ): ′′+ ′+ =0We know that for linear differential equations of order 1with constant coefficients,the solutions of the homogeneous equation are exponentials.Also,we know that when differentiating an exponential −→exp( )(for ∈ ),we get a function which is proportional to the same exponential.Therefore,it is perfectly logical to try to find solutions to the homogeneous equation of the form: ( )= with ∈ .Let ∈ .The function is at least twice differentiable on ℝ, ′ = and ′′ = 2 .Thus,the followingpropositions are equivalent:is a solution of ( )⇐⇒′′ + ′ + =0⇐⇒2 + + =0⇐⇒ ×( 2+ + )=0Considering the fact that the exponential function is never zero (or does not vanish),we find that:is a solution of ( )⇐⇒ 2+ + =0Definition 3.2.1The characteristic polynomial associated to ( )(or to ( ))is the second degree polynomial:( )= 2+ +One also says that 2+ + =0is the characteristic equation of ( )or of ( ).Our previous results show that is a solution of ( )if and only if ( )=0.Therefore,the number of different exponentials which will be solutions depends of the number of roots that has in .Hence,one must distinguish two possibilities.∙First case: has at least one root inLet 1be a root of in .Relations between roots and coefficients show that has another root 2(eventually 2= 1is a root with multiplicity 2)that verifies 1+ 2=−.Next,let be any function twice differentiable on ℝ.Copying our variation of parameter,we set: ( )= ( ) − 1 ,or equivalently, ( )= ( ) 1 .Then is twice differentiable on ℝas well and for all real number ,′( )= ′( ) 1( )+ ( ) 1 1( )′′( )= ′′( ) 1( )+2 1 ′( ) 1( )+ 21 ( ) 1( )Consequently,collecting terms and replacing in the equation,we find that:is a solution of ( )⇐⇒ 1×( ′′+2 1 ′+ 21+ ′+ 1 + )=0⇐⇒ ′′+(2 1+ ) ′+( 21+ 1+ ) =0Furthermore,by definition, 1is a root of so that ( 1)=0,i.e 21+ 1+ =0.Also,substituting =− ( 1+ 2)we find that 2 1+ = ( 1− 2)and thus:is solution of ( )⇐⇒∀ ∈ℝ, ′′+ ( 1− 2) ′=0⇐⇒∀ ∈ℝ, ′′+( 1− 2) ′=0⇐⇒∀ ∈ℝ, ′( )= ( 2− 1) ,for some ∈We must now find ,i.e.find antiderivatives of −→ ( 2− 1) .which again introduces two separate cases:First sub-case: 2= 1(i.e. has two distinct roots in )。

自然哲学的数学原理万有引力公式怎么得到质量相乘The mathematical principle in natural philosophy that describes the universal law of gravitation was formulated by Sir Isaac Newton in his work Principia Mathematica. 这个数学原理描述了质量之间的引力关系，是由艾萨克·牛顿爵士在他的《数学原理》中提出的。

This principle states that every particle of matter in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. 这个原则表明，宇宙中的每一个物质粒子都以与其质量乘积成正比、与它们中心之间的距离的平方成反比的力量互相吸引。

The equation that represents this principle is known as the universal law of gravitation formula: F = (G m1 m2) / r2. 这个等式表示的即为万有引力公式：F = (G m1 m2) / r^2。

In this formula, F represents the force of gravity between two objects, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between their centers. 在这个公式中，F代表两个物体之间的引力，G是万有引力常数，m1和m2分别是两个物体的质量，r是它们中心之间的距离。

一般力学类：分析力学 analytical mechanics拉格朗日乘子 Lagrange multiplier拉格朗日[量] Lagrangian拉格朗日括号 Lagrange bracket循环坐标 cyclic coordinate循环积分 cyclic integral哈密顿[量] Hamiltonian哈密顿函数 Hamiltonian function正则方程 canonical equation正则摄动 canonical perturbation正则变换 canonical transformation正则变量 canonical variable哈密顿原理 Hamilton principle作用量积分 action integral哈密顿-雅可比方程 Hamilton-Jacobi equation作用--角度变量 action-angle variables阿佩尔方程 Appell equation劳斯方程 Routh equation拉格朗日函数 Lagrangian function诺特定理 Noether theorem泊松括号 poisson bracket边界积分法 boundary integral method并矢 dyad运动稳定性 stability of motion轨道稳定性 orbital stability李雅普诺夫函数 Lyapunov function渐近稳定性 asymptotic stability结构稳定性 structural stability久期不稳定性 secular instability弗洛凯定理 Floquet theorem倾覆力矩 capsizing moment自由振动 free vibration固有振动 natural vibration暂态 transient state环境振动 ambient vibration反共振 anti-resonance衰减 attenuation库仑阻尼 Coulomb damping同相分量 in-phase component非同相分量 out-of -phase component超调量 overshoot 参量[激励]振动 parametric vibration模糊振动 fuzzy vibration临界转速 critical speed of rotation阻尼器 damper半峰宽度 half-peak width集总参量系统 lumped parameter system 相平面法 phase plane method相轨迹 phase trajectory等倾线法 isocline method跳跃现象 jump phenomenon负阻尼 negative damping达芬方程 Duffing equation希尔方程 Hill equationKBM方法 KBM method, Krylov-Bogoliu- bov-Mitropol'skii method马蒂厄方程 Mathieu equation平均法 averaging method组合音调 combination tone解谐 detuning耗散函数 dissipative function硬激励 hard excitation硬弹簧 hard spring, hardening spring谐波平衡法harmonic balance method久期项 secular term自激振动 self-excited vibration分界线 separatrix亚谐波 subharmonic软弹簧 soft spring ,softening spring软激励 soft excitation邓克利公式 Dunkerley formula瑞利定理 Rayleigh theorem分布参量系统 distributed parameter system优势频率 dominant frequency模态分析 modal analysis固有模态natural mode of vibration同步 synchronization超谐波 ultraharmonic范德波尔方程 van der pol equation频谱 frequency spectrum基频 fundamental frequencyWKB方法 WKB methodWKB方法Wentzel-Kramers-Brillouin method缓冲器 buffer风激振动 aeolian vibration嗡鸣 buzz倒谱cepstrum颤动 chatter蛇行 hunting阻抗匹配 impedance matching机械导纳 mechanical admittance机械效率 mechanical efficiency机械阻抗 mechanical impedance随机振动 stochastic vibration, random vibration隔振 vibration isolation减振 vibration reduction应力过冲 stress overshoot喘振surge摆振shimmy起伏运动 phugoid motion起伏振荡 phugoid oscillation驰振 galloping陀螺动力学 gyrodynamics陀螺摆 gyropendulum陀螺平台 gyroplatform陀螺力矩 gyroscoopic torque陀螺稳定器 gyrostabilizer陀螺体 gyrostat惯性导航 inertial guidance 姿态角 attitude angle方位角 azimuthal angle舒勒周期 Schuler period机器人动力学 robot dynamics多体系统 multibody system多刚体系统 multi-rigid-body system机动性 maneuverability凯恩方法Kane method转子[系统]动力学 rotor dynamics转子[一支承一基础]系统 rotor-support- foundation system静平衡 static balancing动平衡 dynamic balancing静不平衡 static unbalance动不平衡 dynamic unbalance现场平衡 field balancing不平衡 unbalance不平衡量 unbalance互耦力 cross force挠性转子 flexible rotor分频进动 fractional frequency precession半频进动half frequency precession油膜振荡 oil whip转子临界转速 rotor critical speed自动定心 self-alignment亚临界转速 subcritical speed涡动 whirl固体力学类：弹性力学 elasticity弹性理论 theory of elasticity均匀应力状态 homogeneous state of stress 应力不变量 stress invariant应变不变量 strain invariant应变椭球 strain ellipsoid均匀应变状态 homogeneous state of strain应变协调方程 equation of strain compatibility拉梅常量 Lame constants各向同性弹性 isotropic elasticity旋转圆盘 rotating circular disk 楔wedge开尔文问题 Kelvin problem布西内斯克问题 Boussinesq problem艾里应力函数 Airy stress function克罗索夫--穆斯赫利什维利法 Kolosoff- Muskhelishvili method基尔霍夫假设 Kirchhoff hypothesis板 Plate矩形板 Rectangular plate圆板 Circular plate环板 Annular plate波纹板 Corrugated plate加劲板 Stiffened plate,reinforcedPlate中厚板 Plate of moderate thickness弯[曲]应力函数 Stress function of bending 壳Shell扁壳 Shallow shell旋转壳 Revolutionary shell球壳 Spherical shell[圆]柱壳 Cylindrical shell锥壳Conical shell环壳 Toroidal shell封闭壳 Closed shell波纹壳 Corrugated shell扭[转]应力函数 Stress function of torsion 翘曲函数 Warping function半逆解法 semi-inverse method瑞利--里茨法 Rayleigh-Ritz method松弛法 Relaxation method莱维法 Levy method松弛 Relaxation量纲分析 Dimensional analysis自相似[性] self-similarity影响面 Influence surface接触应力 Contact stress赫兹理论 Hertz theory协调接触 Conforming contact滑动接触 Sliding contact滚动接触 Rolling contact压入 Indentation各向异性弹性 Anisotropic elasticity颗粒材料 Granular material散体力学 Mechanics of granular media热弹性 Thermoelasticity超弹性 Hyperelasticity粘弹性 Viscoelasticity对应原理 Correspondence principle褶皱Wrinkle塑性全量理论 Total theory of plasticity滑动 Sliding微滑Microslip粗糙度 Roughness非线性弹性 Nonlinear elasticity大挠度 Large deflection突弹跳变 snap-through有限变形 Finite deformation 格林应变 Green strain阿尔曼西应变 Almansi strain弹性动力学 Dynamic elasticity运动方程 Equation of motion准静态的Quasi-static气动弹性 Aeroelasticity水弹性 Hydroelasticity颤振Flutter弹性波Elastic wave简单波Simple wave柱面波 Cylindrical wave水平剪切波 Horizontal shear wave竖直剪切波Vertical shear wave体波 body wave无旋波 Irrotational wave畸变波 Distortion wave膨胀波 Dilatation wave瑞利波 Rayleigh wave等容波 Equivoluminal wave勒夫波Love wave界面波 Interfacial wave边缘效应 edge effect塑性力学 Plasticity可成形性 Formability金属成形 Metal forming耐撞性 Crashworthiness结构抗撞毁性 Structural crashworthiness 拉拔Drawing破坏机构 Collapse mechanism回弹 Springback挤压 Extrusion冲压 Stamping穿透Perforation层裂Spalling塑性理论 Theory of plasticity安定[性]理论 Shake-down theory运动安定定理 kinematic shake-down theorem静力安定定理 Static shake-down theorem 率相关理论 rate dependent theorem载荷因子load factor加载准则 Loading criterion加载函数 Loading function加载面 Loading surface塑性加载 Plastic loading塑性加载波 Plastic loading wave简单加载 Simple loading比例加载 Proportional loading卸载 Unloading卸载波 Unloading wave冲击载荷 Impulsive load阶跃载荷step load脉冲载荷 pulse load极限载荷 limit load中性变载 nentral loading拉抻失稳 instability in tension加速度波 acceleration wave本构方程 constitutive equation完全解 complete solution名义应力 nominal stress过应力 over-stress真应力 true stress等效应力 equivalent stress流动应力 flow stress应力间断 stress discontinuity应力空间 stress space主应力空间 principal stress space静水应力状态hydrostatic state of stress对数应变 logarithmic strain工程应变 engineering strain等效应变 equivalent strain应变局部化 strain localization应变率 strain rate应变率敏感性 strain rate sensitivity应变空间 strain space有限应变 finite strain塑性应变增量 plastic strain increment 累积塑性应变 accumulated plastic strain 永久变形 permanent deformation内变量 internal variable应变软化 strain-softening理想刚塑性材料 rigid-perfectly plastic Material刚塑性材料 rigid-plastic material理想塑性材料 perfectl plastic material 材料稳定性stability of material应变偏张量deviatoric tensor of strain应力偏张量deviatori tensor of stress 应变球张量spherical tensor of strain应力球张量spherical tensor of stress路径相关性 path-dependency线性强化 linear strain-hardening应变强化 strain-hardening随动强化 kinematic hardening各向同性强化 isotropic hardening强化模量 strain-hardening modulus幂强化 power hardening塑性极限弯矩 plastic limit bending Moment塑性极限扭矩 plastic limit torque弹塑性弯曲 elastic-plastic bending弹塑性交界面 elastic-plastic interface弹塑性扭转 elastic-plastic torsion粘塑性 Viscoplasticity非弹性 Inelasticity理想弹塑性材料 elastic-perfectly plastic Material极限分析 limit analysis极限设计 limit design极限面limit surface上限定理 upper bound theorem上屈服点upper yield point下限定理 lower bound theorem下屈服点 lower yield point界限定理 bound theorem初始屈服面initial yield surface后继屈服面 subsequent yield surface屈服面[的]外凸性 convexity of yield surface截面形状因子 shape factor of cross-section 沙堆比拟 sand heap analogy屈服Yield屈服条件 yield condition屈服准则 yield criterion屈服函数 yield function屈服面 yield surface塑性势 plastic potential能量吸收装置 energy absorbing device能量耗散率 energy absorbing device塑性动力学 dynamic plasticity塑性动力屈曲 dynamic plastic buckling塑性动力响应 dynamic plastic response塑性波 plastic wave运动容许场 kinematically admissible Field静力容许场 statically admissibleField流动法则 flow rule速度间断 velocity discontinuity滑移线 slip-lines滑移线场 slip-lines field移行塑性铰 travelling plastic hinge塑性增量理论 incremental theory ofPlasticity米泽斯屈服准则 Mises yield criterion普朗特--罗伊斯关系 prandtl- Reuss relation特雷斯卡屈服准则 Tresca yield criterion洛德应力参数 Lode stress parameter莱维--米泽斯关系 Levy-Mises relation亨基应力方程 Hencky stress equation赫艾--韦斯特加德应力空间Haigh-Westergaard stress space洛德应变参数 Lode strain parameter德鲁克公设 Drucker postulate盖林格速度方程Geiringer velocity Equation结构力学 structural mechanics结构分析 structural analysis结构动力学 structural dynamics拱 Arch三铰拱 three-hinged arch抛物线拱 parabolic arch圆拱 circular arch穹顶Dome空间结构 space structure空间桁架 space truss雪载[荷] snow load风载[荷] wind load土压力 earth pressure地震载荷 earthquake loading弹簧支座 spring support支座位移 support displacement支座沉降 support settlement超静定次数 degree of indeterminacy机动分析 kinematic analysis 结点法 method of joints截面法 method of sections结点力 joint forces共轭位移 conjugate displacement影响线 influence line三弯矩方程 three-moment equation单位虚力 unit virtual force刚度系数 stiffness coefficient柔度系数 flexibility coefficient力矩分配 moment distribution力矩分配法moment distribution method力矩再分配 moment redistribution分配系数 distribution factor矩阵位移法matri displacement method单元刚度矩阵 element stiffness matrix单元应变矩阵 element strain matrix总体坐标 global coordinates贝蒂定理 Betti theorem高斯--若尔当消去法 Gauss-Jordan elimination Method屈曲模态 buckling mode复合材料力学 mechanics of composites 复合材料composite material纤维复合材料 fibrous composite单向复合材料 unidirectional composite泡沫复合材料foamed composite颗粒复合材料 particulate composite层板Laminate夹层板 sandwich panel正交层板 cross-ply laminate斜交层板 angle-ply laminate层片Ply多胞固体 cellular solid膨胀 Expansion压实Debulk劣化 Degradation脱层 Delamination脱粘 Debond纤维应力 fiber stress层应力 ply stress层应变ply strain层间应力 interlaminar stress比强度 specific strength强度折减系数 strength reduction factor强度应力比 strength -stress ratio横向剪切模量 transverse shear modulus 横观各向同性 transverse isotropy正交各向异 Orthotropy剪滞分析 shear lag analysis短纤维 chopped fiber长纤维 continuous fiber纤维方向 fiber direction纤维断裂 fiber break纤维拔脱 fiber pull-out纤维增强 fiber reinforcement致密化 Densification最小重量设计 optimum weight design网格分析法 netting analysis混合律 rule of mixture失效准则 failure criterion蔡--吴失效准则 Tsai-W u failure criterion 达格代尔模型 Dugdale model断裂力学 fracture mechanics概率断裂力学 probabilistic fracture Mechanics格里菲思理论 Griffith theory线弹性断裂力学 linear elastic fracturemechanics, LEFM弹塑性断裂力学 elastic-plastic fracture mecha-nics, EPFM断裂 Fracture脆性断裂 brittle fracture解理断裂 cleavage fracture蠕变断裂 creep fracture延性断裂 ductile fracture晶间断裂 inter-granular fracture准解理断裂 quasi-cleavage fracture穿晶断裂 trans-granular fracture裂纹Crack裂缝Flaw缺陷Defect割缝Slit微裂纹Microcrack折裂Kink椭圆裂纹 elliptical crack深埋裂纹 embedded crack[钱]币状裂纹 penny-shape crack预制裂纹 Precrack 短裂纹 short crack表面裂纹 surface crack裂纹钝化 crack blunting裂纹分叉 crack branching裂纹闭合 crack closure裂纹前缘 crack front裂纹嘴 crack mouth裂纹张开角crack opening angle,COA裂纹张开位移 crack opening displacement, COD裂纹阻力 crack resistance裂纹面 crack surface裂纹尖端 crack tip裂尖张角 crack tip opening angle,CTOA裂尖张开位移 crack tip openingdisplacement, CTOD裂尖奇异场crack tip singularity Field裂纹扩展速率 crack growth rate稳定裂纹扩展 stable crack growth定常裂纹扩展 steady crack growth亚临界裂纹扩展 subcritical crack growth 裂纹[扩展]减速 crack retardation止裂crack arrest止裂韧度 arrest toughness断裂类型 fracture mode滑开型 sliding mode张开型 opening mode撕开型 tearing mode复合型 mixed mode撕裂 Tearing撕裂模量 tearing modulus断裂准则 fracture criterionJ积分 J-integralJ阻力曲线 J-resistance curve断裂韧度 fracture toughness应力强度因子 stress intensity factorHRR场 Hutchinson-Rice-Rosengren Field守恒积分 conservation integral有效应力张量 effective stress tensor应变能密度strain energy density能量释放率 energy release rate内聚区 cohesive zone塑性区 plastic zone张拉区 stretched zone热影响区heat affected zone, HAZ延脆转变温度 brittle-ductile transitiontemperature剪切带shear band剪切唇shear lip无损检测 non-destructive inspection双边缺口试件double edge notchedspecimen, DEN specimen单边缺口试件 single edge notchedspecimen, SEN specimen三点弯曲试件 three point bendingspecimen, TPB specimen中心裂纹拉伸试件 center cracked tension specimen, CCT specimen中心裂纹板试件 center cracked panelspecimen, CCP specimen紧凑拉伸试件 compact tension specimen, CT specimen大范围屈服large scale yielding小范围攻屈服 small scale yielding韦布尔分布 Weibull distribution帕里斯公式 paris formula空穴化 Cavitation应力腐蚀 stress corrosion概率风险判定 probabilistic riskassessment, PRA损伤力学 damage mechanics损伤Damage连续介质损伤力学 continuum damage mechanics细观损伤力学 microscopic damage mechanics累积损伤 accumulated damage脆性损伤 brittle damage延性损伤 ductile damage宏观损伤 macroscopic damage细观损伤 microscopic damage微观损伤 microscopic damage损伤准则 damage criterion损伤演化方程 damage evolution equation 损伤软化 damage softening损伤强化 damage strengthening 损伤张量 damage tensor损伤阈值 damage threshold损伤变量 damage variable损伤矢量 damage vector损伤区 damage zone疲劳Fatigue低周疲劳 low cycle fatigue应力疲劳 stress fatigue随机疲劳 random fatigue蠕变疲劳 creep fatigue腐蚀疲劳 corrosion fatigue疲劳损伤 fatigue damage疲劳失效 fatigue failure疲劳断裂 fatigue fracture疲劳裂纹 fatigue crack疲劳寿命 fatigue life疲劳破坏 fatigue rupture疲劳强度 fatigue strength疲劳辉纹 fatigue striations疲劳阈值 fatigue threshold交变载荷 alternating load交变应力 alternating stress应力幅值 stress amplitude应变疲劳 strain fatigue应力循环 stress cycle应力比 stress ratio安全寿命 safe life过载效应 overloading effect循环硬化 cyclic hardening循环软化 cyclic softening环境效应 environmental effect裂纹片crack gage裂纹扩展 crack growth, crack Propagation裂纹萌生 crack initiation循环比 cycle ratio实验应力分析 experimental stressAnalysis工作[应变]片 active[strain] gage基底材料 backing material应力计stress gage零[点]飘移zero shift, zero drift应变测量 strain measurement应变计strain gage应变指示器 strain indicator应变花 strain rosette应变灵敏度 strain sensitivity机械式应变仪 mechanical strain gage 直角应变花 rectangular rosette引伸仪 Extensometer应变遥测 telemetering of strain横向灵敏系数 transverse gage factor 横向灵敏度 transverse sensitivity焊接式应变计 weldable strain gage 平衡电桥 balanced bridge粘贴式应变计 bonded strain gage粘贴箔式应变计bonded foiled gage粘贴丝式应变计 bonded wire gage 桥路平衡 bridge balancing电容应变计 capacitance strain gage 补偿片 compensation technique补偿技术 compensation technique基准电桥 reference bridge电阻应变计 resistance strain gage温度自补偿应变计 self-temperature compensating gage半导体应变计 semiconductor strain Gage集流器slip ring应变放大镜 strain amplifier疲劳寿命计 fatigue life gage电感应变计 inductance [strain] gage 光[测]力学 Photomechanics光弹性 Photoelasticity光塑性 Photoplasticity杨氏条纹 Young fringe双折射效应 birefrigent effect等位移线 contour of equalDisplacement暗条纹 dark fringe条纹倍增 fringe multiplication干涉条纹 interference fringe等差线 Isochromatic等倾线 Isoclinic等和线 isopachic应力光学定律 stress- optic law主应力迹线 Isostatic亮条纹 light fringe 光程差optical path difference热光弹性 photo-thermo -elasticity光弹性贴片法 photoelastic coating Method光弹性夹片法 photoelastic sandwich Method动态光弹性 dynamic photo-elasticity空间滤波 spatial filtering空间频率 spatial frequency起偏镜 Polarizer反射式光弹性仪 reflection polariscope残余双折射效应 residual birefringent Effect应变条纹值 strain fringe value应变光学灵敏度 strain-optic sensitivity 应力冻结效应 stress freezing effect应力条纹值 stress fringe value应力光图 stress-optic pattern暂时双折射效应 temporary birefringent Effect脉冲全息法 pulsed holography透射式光弹性仪 transmission polariscope 实时全息干涉法 real-time holographicinterfero - metry网格法 grid method全息光弹性法 holo-photoelasticity全息图Hologram全息照相 Holograph全息干涉法 holographic interferometry 全息云纹法 holographic moire technique 全息术 Holography全场分析法 whole-field analysis散斑干涉法 speckle interferometry散斑Speckle错位散斑干涉法 speckle-shearinginterferometry, shearography散斑图Specklegram白光散斑法white-light speckle method云纹干涉法 moire interferometry[叠栅]云纹 moire fringe[叠栅]云纹法 moire method云纹图 moire pattern离面云纹法 off-plane moire method参考栅 reference grating试件栅 specimen grating分析栅 analyzer grating面内云纹法 in-plane moire method脆性涂层法 brittle-coating method条带法 strip coating method坐标变换 transformation ofCoordinates计算结构力学 computational structuralmecha-nics加权残量法weighted residual method有限差分法 finite difference method有限[单]元法 finite element method配点法 point collocation里茨法 Ritz method广义变分原理 generalized variational Principle最小二乘法 least square method胡[海昌]一鹫津原理 Hu-Washizu principle 赫林格-赖斯纳原理 Hellinger-Reissner Principle修正变分原理 modified variational Principle约束变分原理 constrained variational Principle混合法 mixed method杂交法 hybrid method边界解法boundary solution method有限条法 finite strip method半解析法 semi-analytical method协调元 conforming element非协调元 non-conforming element混合元 mixed element杂交元 hybrid element边界元 boundary element强迫边界条件 forced boundary condition 自然边界条件 natural boundary condition 离散化 Discretization离散系统 discrete system连续问题 continuous problem广义位移 generalized displacement广义载荷 generalized load广义应变 generalized strain广义应力 generalized stress界面变量 interface variable 节点 node, nodal point[单]元 Element角节点 corner node边节点 mid-side node内节点 internal node无节点变量 nodeless variable杆元 bar element桁架杆元 truss element梁元 beam element二维元 two-dimensional element一维元 one-dimensional element三维元 three-dimensional element轴对称元 axisymmetric element板元 plate element壳元 shell element厚板元 thick plate element三角形元 triangular element四边形元 quadrilateral element四面体元 tetrahedral element曲线元 curved element二次元 quadratic element线性元 linear element三次元 cubic element四次元 quartic element等参[数]元 isoparametric element超参数元 super-parametric element亚参数元 sub-parametric element节点数可变元 variable-number-node element拉格朗日元 Lagrange element拉格朗日族 Lagrange family巧凑边点元 serendipity element巧凑边点族 serendipity family无限元 infinite element单元分析 element analysis单元特性 element characteristics刚度矩阵 stiffness matrix几何矩阵 geometric matrix等效节点力 equivalent nodal force节点位移 nodal displacement节点载荷 nodal load位移矢量 displacement vector载荷矢量 load vector质量矩阵 mass matrix集总质量矩阵 lumped mass matrix相容质量矩阵 consistent mass matrix阻尼矩阵 damping matrix瑞利阻尼 Rayleigh damping刚度矩阵的组集 assembly of stiffnessMatrices载荷矢量的组集 consistent mass matrix质量矩阵的组集 assembly of mass matrices 单元的组集 assembly of elements局部坐标系 local coordinate system局部坐标 local coordinate面积坐标 area coordinates体积坐标 volume coordinates曲线坐标 curvilinear coordinates静凝聚 static condensation合同变换 contragradient transformation形状函数 shape function试探函数 trial function检验函数test function权函数 weight function样条函数 spline function代用函数 substitute function降阶积分 reduced integration零能模式 zero-energy modeP收敛 p-convergenceH收敛 h-convergence掺混插值 blended interpolation等参数映射 isoparametric mapping双线性插值 bilinear interpolation小块检验 patch test非协调模式 incompatible mode 节点号 node number单元号 element number带宽 band width带状矩阵 banded matrix变带状矩阵 profile matrix带宽最小化minimization of band width波前法 frontal method子空间迭代法 subspace iteration method 行列式搜索法determinant search method逐步法 step-by-step method纽马克法Newmark威尔逊法 Wilson拟牛顿法 quasi-Newton method牛顿-拉弗森法 Newton-Raphson method 增量法 incremental method初应变 initial strain初应力 initial stress切线刚度矩阵 tangent stiffness matrix割线刚度矩阵 secant stiffness matrix模态叠加法mode superposition method平衡迭代 equilibrium iteration子结构 Substructure子结构法 substructure technique超单元 super-element网格生成 mesh generation结构分析程序 structural analysis program 前处理 pre-processing后处理 post-processing网格细化 mesh refinement应力光顺 stress smoothing组合结构 composite structure流体动力学类：流体动力学 fluid dynamics连续介质力学 mechanics of continuous media介质medium流体质点 fluid particle无粘性流体 nonviscous fluid, inviscid fluid连续介质假设 continuous medium hypothesis流体运动学 fluid kinematics水静力学 hydrostatics 液体静力学 hydrostatics支配方程 governing equation伯努利方程 Bernoulli equation伯努利定理 Bernonlli theorem毕奥-萨伐尔定律 Biot-Savart law欧拉方程Euler equation亥姆霍兹定理 Helmholtz theorem开尔文定理 Kelvin theorem涡片 vortex sheet库塔-茹可夫斯基条件 Kutta-Zhoukowskicondition布拉休斯解 Blasius solution达朗贝尔佯廖 d'Alembert paradox 雷诺数 Reynolds number施特鲁哈尔数 Strouhal number随体导数 material derivative不可压缩流体 incompressible fluid 质量守恒 conservation of mass动量守恒 conservation of momentum 能量守恒 conservation of energy动量方程 momentum equation能量方程 energy equation控制体积 control volume液体静压 hydrostatic pressure涡量拟能 enstrophy压差 differential pressure流[动] flow流线stream line流面 stream surface流管stream tube迹线path, path line流场 flow field流态 flow regime流动参量 flow parameter流量 flow rate, flow discharge涡旋 vortex涡量 vorticity涡丝 vortex filament涡线 vortex line涡面 vortex surface涡层 vortex layer涡环 vortex ring涡对 vortex pair涡管 vortex tube涡街 vortex street卡门涡街 Karman vortex street马蹄涡 horseshoe vortex对流涡胞 convective cell卷筒涡胞 roll cell涡 eddy涡粘性 eddy viscosity环流 circulation环量 circulation速度环量 velocity circulation 偶极子 doublet, dipole驻点 stagnation point总压[力] total pressure总压头 total head静压头 static head总焓 total enthalpy能量输运 energy transport速度剖面 velocity profile库埃特流 Couette flow单相流 single phase flow单组份流 single-component flow均匀流 uniform flow非均匀流 nonuniform flow二维流 two-dimensional flow三维流 three-dimensional flow准定常流 quasi-steady flow非定常流unsteady flow, non-steady flow 暂态流transient flow周期流 periodic flow振荡流 oscillatory flow分层流 stratified flow无旋流 irrotational flow有旋流 rotational flow轴对称流 axisymmetric flow不可压缩性 incompressibility不可压缩流[动] incompressible flow 浮体 floating body定倾中心metacenter阻力 drag, resistance减阻 drag reduction表面力 surface force表面张力 surface tension毛细[管]作用 capillarity来流 incoming flow自由流 free stream自由流线 free stream line外流 external flow进口 entrance, inlet出口exit, outlet扰动 disturbance, perturbation分布 distribution传播 propagation色散 dispersion弥散 dispersion附加质量added mass ,associated mass收缩 contraction镜象法 image method无量纲参数 dimensionless parameter几何相似 geometric similarity运动相似 kinematic similarity动力相似[性] dynamic similarity平面流 plane flow势 potential势流 potential flow速度势 velocity potential复势 complex potential复速度 complex velocity流函数 stream function源source汇sink速度[水]头 velocity head拐角流 corner flow空泡流cavity flow超空泡 supercavity超空泡流 supercavity flow空气动力学 aerodynamics低速空气动力学 low-speed aerodynamics 高速空气动力学 high-speed aerodynamics 气动热力学 aerothermodynamics亚声速流[动] subsonic flow跨声速流[动] transonic flow超声速流[动] supersonic flow锥形流 conical flow楔流wedge flow叶栅流 cascade flow非平衡流[动] non-equilibrium flow细长体 slender body细长度 slenderness钝头体 bluff body钝体 blunt body翼型 airfoil翼弦 chord薄翼理论 thin-airfoil theory构型 configuration后缘 trailing edge迎角 angle of attack失速stall脱体激波detached shock wave 波阻wave drag诱导阻力 induced drag诱导速度 induced velocity临界雷诺数critical Reynolds number前缘涡 leading edge vortex附着涡 bound vortex约束涡 confined vortex气动中心 aerodynamic center气动力 aerodynamic force气动噪声 aerodynamic noise气动加热 aerodynamic heating离解 dissociation地面效应 ground effect气体动力学 gas dynamics稀疏波 rarefaction wave热状态方程thermal equation of state喷管Nozzle普朗特-迈耶流 Prandtl-Meyer flow瑞利流 Rayleigh flow可压缩流[动] compressible flow可压缩流体 compressible fluid绝热流 adiabatic flow非绝热流 diabatic flow未扰动流 undisturbed flow等熵流 isentropic flow匀熵流 homoentropic flow兰金-于戈尼奥条件 Rankine-Hugoniot condition状态方程 equation of state量热状态方程 caloric equation of state完全气体 perfect gas拉瓦尔喷管 Laval nozzle马赫角 Mach angle马赫锥 Mach cone马赫线Mach line马赫数Mach number马赫波Mach wave当地马赫数 local Mach number冲击波 shock wave激波 shock wave正激波normal shock wave斜激波oblique shock wave头波 bow wave附体激波 attached shock wave激波阵面 shock front激波层 shock layer压缩波 compression wave反射 reflection折射 refraction散射scattering衍射 diffraction绕射 diffraction出口压力 exit pressure超压[强] over pressure反压 back pressure爆炸 explosion爆轰 detonation缓燃 deflagration水动力学 hydrodynamics液体动力学 hydrodynamics泰勒不稳定性 Taylor instability 盖斯特纳波 Gerstner wave斯托克斯波 Stokes wave瑞利数 Rayleigh number自由面 free surface波速 wave speed, wave velocity 波高 wave height波列wave train波群 wave group波能wave energy表面波 surface wave表面张力波 capillary wave规则波 regular wave不规则波 irregular wave浅水波 shallow water wave深水波deep water wave重力波 gravity wave椭圆余弦波 cnoidal wave潮波tidal wave涌波surge wave破碎波 breaking wave船波ship wave非线性波 nonlinear wave孤立子 soliton水动[力]噪声 hydrodynamic noise 水击 water hammer空化 cavitation空化数 cavitation number 空蚀 cavitation damage超空化流 supercavitating flow水翼 hydrofoil水力学 hydraulics洪水波 flood wave涟漪ripple消能 energy dissipation海洋水动力学 marine hydrodynamics谢齐公式 Chezy formula欧拉数 Euler number弗劳德数 Froude number水力半径 hydraulic radius水力坡度 hvdraulic slope高度水头 elevating head水头损失 head loss水位 water level水跃 hydraulic jump含水层 aquifer排水 drainage排放量 discharge壅水曲线back water curve压[强水]头 pressure head过水断面 flow cross-section明槽流open channel flow孔流 orifice flow无压流 free surface flow有压流 pressure flow缓流 subcritical flow急流 supercritical flow渐变流gradually varied flow急变流 rapidly varied flow临界流 critical flow异重流density current, gravity flow堰流weir flow掺气流 aerated flow含沙流 sediment-laden stream降水曲线 dropdown curve沉积物 sediment, deposit沉[降堆]积 sedimentation, deposition沉降速度 settling velocity流动稳定性 flow stability不稳定性 instability奥尔-索末菲方程 Orr-Sommerfeld equation 涡量方程 vorticity equation泊肃叶流 Poiseuille flow奥辛流 Oseen flow剪切流 shear flow粘性流[动] viscous flow层流 laminar flow分离流 separated flow二次流 secondary flow近场流near field flow远场流 far field flow滞止流 stagnation flow尾流 wake [flow]回流 back flow反流 reverse flow射流 jet自由射流 free jet管流pipe flow, tube flow内流 internal flow拟序结构 coherent structure 猝发过程 bursting process表观粘度 apparent viscosity 运动粘性 kinematic viscosity 动力粘性 dynamic viscosity 泊 poise厘泊 centipoise厘沱 centistoke剪切层 shear layer次层 sublayer流动分离 flow separation层流分离 laminar separation 湍流分离 turbulent separation 分离点 separation point附着点 attachment point再附 reattachment再层流化 relaminarization起动涡starting vortex驻涡 standing vortex涡旋破碎 vortex breakdown 涡旋脱落 vortex shedding压[力]降 pressure drop压差阻力 pressure drag压力能 pressure energy型阻 profile drag滑移速度 slip velocity无滑移条件 non-slip condition 壁剪应力 skin friction, frictional drag壁剪切速度 friction velocity磨擦损失 friction loss磨擦因子 friction factor耗散 dissipation滞后lag相似性解 similar solution局域相似 local similarity气体润滑 gas lubrication液体动力润滑 hydrodynamic lubrication 浆体 slurry泰勒数 Taylor number纳维-斯托克斯方程 Navier-Stokes equation 牛顿流体 Newtonian fluid边界层理论boundary later theory边界层方程boundary layer equation边界层 boundary layer附面层 boundary layer层流边界层laminar boundary layer湍流边界层turbulent boundary layer温度边界层thermal boundary layer边界层转捩boundary layer transition边界层分离boundary layer separation边界层厚度boundary layer thickness位移厚度 displacement thickness动量厚度 momentum thickness能量厚度 energy thickness焓厚度 enthalpy thickness注入 injection吸出suction泰勒涡 Taylor vortex速度亏损律 velocity defect law形状因子 shape factor测速法 anemometry粘度测定法 visco[si] metry流动显示 flow visualization油烟显示 oil smoke visualization孔板流量计 orifice meter频率响应 frequency response油膜显示oil film visualization阴影法 shadow method纹影法 schlieren method烟丝法smoke wire method丝线法 tuft method。

&RS\ULJKW E\ 6SDWLDO $XWRPDWLRQ /DERUDWRU\6$/$ 0HVKIUHH 0HWKRG IRU ,QFRPSUHVVLEOH )OXLG '\QDPLFV 3UREOHPV, 7VXNDQRY 9 6KDSLUR 6 =KDQJA Meshfree Method for Incompressible Fluid Dynamics ProblemsI.Tsukanov a∗,V.Shapiro a,S.Zhang ba Spatial Automation LaboratoryUniversity of Wisconsin-Madison1513University AvenueMadison,WI53706,U.S.A.b General Motors R&D CenterWarren,MI48090,U.S.A.Accepted for publication in Int.Journal forNumerical Methods in EngineeringAbstractWe show that meshfree variational methods may be utilized for solution of incompressiblefluid dynamics prob-lems using the R-function method(RFM).The proposed approach constructs an approximate solution that satisfiesall prescribed boundary conditions exactly using approximate distancefields for portions of the boundary,transfiniteinterpolation,and computations on a non-conforming spatial grid.We give detailed implementation of the methodfor two common formulations of the incompressiblefluid dynamics problem:first using scalar stream function for-mulation and then using vector formulation of the Navier-Stokes problem with artificial compressibility approach.Extensive numerical comparisons with commercial solvers and experimental data for the benchmark back-facing stepchannel problem reveal strengths and weaknesses of the proposed meshfree method.Keywords:meshfree method,distancefield,solution structure,Navier-Stokes problem,stream function,artificialcompressibility approach1Introduction1.1Towards meshfree solution of computationalfluid dynamics problemsModeling of the incompressiblefluidflow involves solution of the Navier-Stokes equations inside a geometric domain. The interaction between thefluid and the boundary of the geometric domain,in terms of the mathematical model is described by boundary conditions,formulated for viscousfluid as known velocity profile at the inlet and zero velocity at the walls.The nature of this problem makes its treatment difficult:the solution algorithm needs to incorporate two distinct types of information—(1)analytical information that describes the Navier-Stokes equations and func-tions given as boundary conditions;and(2)geometric information about boundaries where the boundary conditions are prescribed.Conventional methods of engineering analysis solve this problem by employingfirst,the spatial dis-cretization of the geometric domain(a mesh that conforms to the boundary of the geometric domain),and second,the discretization of the Navier-Stokes equations and the boundary conditions over the discretized geometry domain.The resulting approximation,therefore,unifies both functional and geometric information.Such approach,despite its wide usage,has some drawbacks.For example,it is well known that the construction of a“good”mesh is a difficult and time consuming task.In engineering practice design iterations require efficient feedback from the analysis results to the geometric model.However,employment of conforming meshes for solution of engineering problems is not quite suitable for design purposes,because the spatial grid restricts changes of the parameters of the geometric model such that it is difficult or even impossible to change the shape of the model without remeshing.∗Corresponding author.E-mail:igor@These difficulties in the conventional approaches led to the development of methods which use non-conforming1 meshes or no meshes at all.These new meshfree(sometimes they are also called meshless)methods represent a solu-tion of the problem by linear combination of basis functions which may be constructed over meshes not conforming to the shape of the geometric model[3,23,4,17,8,21,25,18,7,9].However,the employment of non-conforming spatial discretization makes the treatment of boundary conditions more difficult.Proposed remedies include the combination of Element Free Galerkin Method(EFG)[4]withfinite element shape functions near the boundary[17],the use of modified variational principle[20],window or correction functions that vanish on the boundary[9],and Lagrange multipliers.Although these techniques appear promising,they often contradict the meshfree nature of the approxi-mation near the boundary,introduce additional constraints on solutions,or lead to systems with an increased number of unknowns[13].Several promising transformation-based approaches to satisfying essential boundary conditions at desired nodal locations have been recently proposed and compared by J.-S.Chen[8].The meshing problem can be substantially simplified by employment of the Cartesian grid methods[42,1,11,5]. These methods represent the geometric model by a hierarchical set of cubical/rectangular cells that simplify computa-tion of the partial derivatives using afinite difference scheme.Instead of requiring that cells conform to the boundaries of the domain,the geometric model of the domain is approximated by quad/octtree spatial decompositions to any prescribed accuracy.This approach is accompanied by introduction of additional sources of errors and potentially exponential(in the subdivision depth)increase in computational cost.In contrast to Cartesian grid methods,immersed boundary methods[24,14,15]solve the problem using a uniform non-conforming grid of points that cover the geometric model.Influence of the boundaries and boundary conditions is accounted for by modification of the differential equation of the problem,based on special case analysis.In this paper,we describe a method that also relies on a non-conforming uniform rectangular grid,but goes sub-stantially further.All prescribed boundary conditions are satisfied exactly by transfinitely interpolating individual boundary conditions inversely proportional to the approximate distance to each boundary portion.The technique can be applied systematically to any and all boundary value problems using the theory of R-functions[30],and the result-ing interpolant can be combined with just about any standard numerical solution method.The method is demonstrated with variational methods applied to the solution of incompressiblefluid dynamics problems:first using scalar stream function formulation,and then using vector formulation of the Navier-Stokes problem with artificial compressibility approach.1.2Brief History of the MethodKantorovich showed that Dirichlet boundary conditions could be satisfied exactly using functions vanishing on the boundary of a geometric object[16].He proposed to represent a solution satisfying Dirichlet boundary conditionu|∂Ω=ϕin the following form:u=ωNi=1C iχi+ϕ,(1)whereωis a function taking on zero value on the boundary of the domain;{χi}N i=1is a system of linearly independent basis functions;{C i}N i=1is a vector of unknown coefficients andϕis a function given as a boundary condition. Different sets of the coefficients{C i}N i=1give different functions u but all of them satisfy the prescribed boundary condition.Numerical values of the unknown coefficients can be obtained via variational or projectional methods. Application of Kantorovich’s method was limited to very simple geometric domains,because at that time it was unclear how to construct functionωfor arbitrary geometric domains.Several years later,Rvachev proposed that functions taking on zero value on the boundary of a geometric domain can be constructed for virtually any geometric object using R-functions[27,28,34].Informally,R-functions serve as a construction toolkit transforming a set-theoretic description of the boundary of a geometric object into a real valued function whose zero set coincides with the boundary.Detailed discussion on R-functions and construction techniques is outside of the scope of this paper,but it can be found in numerous references[28,34,35,26,29]and will be illustrated in section2.2.Functions constructed using R-functions are differentiable everywhere except a finite number of points[28,35]and behave as distances to the boundaries near the boundary points.We will refer 1This should not be confused with the another commonly used terminology of“conforming/non-conformingfinite element”.In this paper the non-conformance of the spatial grid to the shape of the geometric domain means that the grid is extended beyond,and unconstrained by the boundary of the geometric domain.to such functions as approximate distancefields.Besides techniques based on the theory of R-functions[28],other methods may also be applied for construction of approximate distancefields.For example,the level set method [33,32]results in a distance-like functions,albeit defined at a discrete set of points and usually implicitly.In contrast, the approximate distancefields constructed using R-functions are explicitly defined at all points of the space.The successful employment of the level set method to model holes and inclusions was discussed in[38].Similar technique was used to model crack development and propagation in[37].Approximate distancefields can be used for interpolation of the functions and their derivatives prescribed on the boundary pieces of a geometric object[31].Representation of boundaries of a geometric object by approximate distancefields made possible the extension of the Kantorovich’s method into the R-function method(RFM).The RFM allows the satisfaction of many types of boundary conditions exactly by employing solution structures that incorporate boundary conditions,approximate distancefields,and basis functions with unknown coefficients[30]. RFM is essentially a meshfree method because it places no restriction on the choice of the basis functions:they can be constructed over conforming or non-conforming mesh.For example,finite elements can be used as basis functions;in this case,RFM can be viewed as an enhancedfinite element method that treats all given boundary conditions exactly. But in this paper,all computations were performed over a uniform rectangular grid of B-splines and performed within the SAGE system developed by authors[41].In[36],we showed that the method is particularly effective in dealing with moving and deforming boundaryconditions.Figure1:Parametrization of the geometry of a back facing step channel1.3Scope and outlineThis paper serves two purposes.First,the application of the RFM to Computational Fluid Dynamics(CFD)is illus-trated through two different formulations;second,the numerical properties(accuracy and convergence)of the RFM, applied to these formulations,are investigated.The paper will show that the RFM approach to CFD provides a unique andflexible method to link both the functional and geometric information in a unified manner.It will also be shown that the artificial compressibility approach gives good results for low Reynolds numberflow(Re<400).For higher Reynolds numbersflows the RFM needs to be implemented either with Finite V olume(FV)/Finite Difference(FD) numerical schemes or using the regularization approach described in[10].Throughout the paper,we assume thatflow is laminar,fluid is Newtonian,and we focus on two-dimensional problems.In this case incompressible two dimensional viscousflow is described by Navier-Stokes equations and the continuity equation[19]:u ∂u∂x+v∂u∂y−1Re∇2u=−Eu∂p∂x;u ∂v∂x+v∂v∂y−1Re∇2v=−Eu∂p∂y∂u ∂x +∂v∂y=0,(2)where variables u and v are the velocity components in the x and y coordinate directions respectively,p is thepressure variable,Re=23u max2h inletνand Eu=Pρu2maxare Reynolds and Euler numbers respectively.We explorethe accuracy of the RFM and its convergence properties,solving a standard textbook benchmark problem:an incom-pressible viscousfluidflow in a two-dimensional back-facing step channel,whose parametrization is shown in Figure 1.For this problem experimental data[2],as well as the computer simulation results given by the conventionalfluiddynamics systems,are available.For concreteness we let the geometric parameters take on the following numericalvalues:L inlet=5,L channel=12,h inlet=0.5and s=0.471.In order to simplify the comparison of the RFM modeling results with experimental data,we use the same ratio between h inlet and s as in[2].Boundary conditions areformulated as a parabolic velocity profile with u max=1.5at the inlet and zero velocity at the walls of the channel.Most meshfree methods employ some variational principle in order to solve the problem,and the RFM is no ex-ception.Since RFM treats the given boundary conditions exactly,the variational principle is applied to the differentialequation(s)of the problem only.Because viscousfluidflows do not conserve energy,we employ a least squares method.In the paper we discuss the application of the RFM to two different formulation of the incompressiblefluid dynamics problem:stream function and artificial compressibility formulations.The stream function formulation discussed in Section2substantially simplifies the initial problem reducing the system of the Navier-Stokes and continuity equations to a single equation.We use the stream function formulation as an introductory example in order to explain the concept of the RFM solution structure and the RFM solution procedure. The velocity profiles given by the RFM are in good agreement with the experimental data for Reynolds number100. For higher Reynolds numbers the RFM overestimates the position of the reattachment point.The accuracy of the modeling results can be improved by applying the RFM to the primitive variables of the Navier-Stokes equations via an artificial compressibility approach,detailed in Section3.In contrast to the stream function formulation,the artificial compressibility formulation allows modeling offluidflows in channels with arbitrary geo-metric shape including multiple connected channels.Further,it can be easily extended to model three-dimensional and turbulentflows.Since the artificial compressibility formulation leads to solution of a vector problem,application of the RFM to this formulation of incompressiblefluid dynamics problem results in vector solution structures whose construction we explain in Section3.2.Section2.5and Section3.4contain the analysis of the RFM modeling results and their comparison with exper-imental data and numerical results obtained using the commercialfluid analysis system Fluent.Distributions of the velocity components and the pressurefield given by the RFM are in good agreement with experimental data,however the employment of variational methods appears to raise some issues.These observations and possible ways to improve effectiveness of the method are discussed in Section4.2RFM with stream function formulation2.1Stream function formulation and solutionIntroduction of a stream functionψsuch that u=∂ψ∂y and v=−∂ψ∂xallows us to satisfy the continuity equation andto exclude the pressure p from the momentum equations.Substitution of the stream functionψinstead of derivatives of velocity components gives a differential equation for the stream function[19]:1 Re ∇4ψ−∂ψ∂y∂∂x∇2ψ+∂ψ∂x∂∂y∇2ψ=0.(3)Boundary conditions for the stream function at the inlet can be derived from the velocity profile at the inlet which is usually known:ψ|inlet=sV(x,y)cos(n,V)dS,(4)where V=u i+v j is the velocity vector and n is the normal vector to the inlet section.Assuming a parabolic profile for the u component of the velocity vector,as shown in Figure1,with u max=1.5and v=0at the inlet,we obtain the boundary condition for stream function as:ψ|inlet=2h2inlety3+3h inlety2;∂ψ∂n|inlet =0.(5)On the walls of the channel the stream function should satisfy the following boundary conditions:ψ|lower wall =0;ψ|upper wall =h inlet .(6)Since we are dealing with viscous flow,velocity is assumed to be zero on all walls.This condition is expressed in terms of homogeneous Neumann boundary condition for the stream function:∂ψ∂n |walls =0.(7)Boundary conditions for the stream function at the outlet can be derived from two conditions:(a)the total discharge at the outlet should be the same as at the inlet,and (b)the velocity components at the outlet should respectively be:v =0,and u should possess a parabolic profile as is shown in Figure 1.After simplification we get:ψ|outlet =m −y 33+h inlet −s 2y 2+h inlet sy +k ;m =6h inlet (h inlet +s )3;k =m 4 3s 2h inlet +s 3 ;∂ψ∂n |outlet=0.(8)The geometry of the channel,equation (3)together with boundary conditions (4–8)constitutes a complete mathe-matical formulation of the problem.The first steps of solving this problem with RFM are construction of approximate distance fields for boundary pieces and the RFM solution structure for the problem.The solution structure interpolates all given boundary conditions over the specified geometry,but also includes a set of basis functions with undetermined coefficients.The subsequent solution procedure will determine the coefficients that best approximate the govern-ing differential equation in some sense.The solution structure,as well as approximate distance fields,are usually constructed automatically without user’s intervention,but below we show all the construction details manually and explicitly.2.2Theory of R -functions and approximate distance fieldsThe theory of R -functions was originally developed in Ukraine by V .L.Rvachev and his students [28,26,29].A complete list of references through 2001can be found in [22].A brief English summary of the theory of R -functions written by Shapiro in 1988[34]is available as a technical report.An R -function is real-valued function whose sign is completely determined by the signs of its arguments.For example,the function xyz can be negative only when the number of its negative arguments is odd.A similar property is possessed by functions x +y + xy +x 2+y 2and xy +z +|z −yx |,and so on.Such functions ‘encode’Boolean logic functions and are called R -functions .Every Boolean function is a companion to infinitely many R -functions,which form a branch of the set of R -functions.For example,it is well known that min(x 1,x 2)is an R -function whose companion Boolean function is logical “and”(∧),and max(x 1,x 2)is an R -function whose companion Boolean function is logical “or”(∨).But the same branches of R -functions contain many other functions,e.g.x 1∧αx 2≡11+α x 1+x 2− x 21+x 22−2αx 1x 2 ;x 1∨αx 2≡11+α x 1+x 2+x 21+x 22−2αx 1x 2 ,(9)where α(x 1,x 2)is an arbitrary symmetric function such that −1<α(x 1,x 2)≤1.The precise value of αmay or may not matter,and often it can be set to a constant.For example,setting α=1yields the min and max respectively,but setting α=0results in much nicer functions ∨0and ∧0that are analytic everywhere except when x 1=x 2=0.Similarly,R -functionsx 1∧m αx 2≡(x 1∧αx 2)(x 21+x 22)m 2;x 1∨m αx 2≡(x 1∨αx 2)(x 21+x 22)m 2(10)(a)(b)Figure2:(a)Halfspaces that constitute a CSG representation of the channel;(b)the corresponding approximate distancefield(a)(b)Figure3:Approximate distancefields for the portions of the boundary where non-slip boundary conditions are pre-scribed:(a)for the car presented in Figure23(b);(b)for the car presented in Figure23(c)are analytic everywhere except the origin(x1=x2=0),where they are m times differentiable.Many other systems of R-functions are studied in[28].The choice of an appropriate system of R-functions is dictated by many considerations, including simplicity,continuity,differential properties,and computational convenience.Just as Boolean functions,R-functions are closed under ing R-functions,any object defined by a predicate on“primitive”geometric regions(e.g.regions defined by a system of inequalities)can now also be represented by a single inequality,or equation.The latter can be evaluated,differentiated,and possesses many other useful properties.In particular:•the functions are constructed in a‘logical’fashion and can be controlled through intuitive user-defined parame-ters;•functions can be normalized,in which case they behave as distance functions near the boundary of the object and can be differentiated everywhere[28,35];•functions can also be constructed for individual cells and cells complexes,given prescribed values for the func-tions and their gradients;•the functions can be used to define time-varying geometry and used for modeling various complex physical phenomena.Theory of R-functions provides the connection between logical and set operations on geometric primitives and analytic constructions.For every logical or set-theoretic construction,there is a corresponding approximate distance function with the above properties.Furthermore,the translation from logical and set-theoretic description is a matter of simple syntactic substitution that does not require expensive symbolic computations.For example,the geometric domain of the channel in Figure2(a)can be defined as a Boolean(Constructive Solid Geometry)combination of six primitives:Ω=(f1∪f2)∩f3∩f4∩f5∩f6,where x denotes the regularized complement of x,and individual primitives f1through f6are defined by the following inequalities:f1=y≥0;f2=x−L inlet≥0;f3=y−s≥0;f4=L inlet+L channel−x≥0;f5=h inlet−y≥0;f6=x≥0Naturally,all numeric constants can be viewed as specific values for some parameters(size,position,etc.).The constructed Boolean representation can be translated into the approximate distancefield shown in Figure2(b)using R-functions:ω=(f1∨0f2)∧0f3∧0f4∧0f5∧0f6,(11) which is also parameterized by h inlet,s,L inlet and L channel.This example clearly shows that any Boolean representation may be translated into the corresponding approximate distancefield.Similarly,boundary representation of a solid is a union of solid’s faces,each face is a subset of some surface bounded by edges,and so on.This logical description can also be directly translated into a function such that is zero for every point on the boundary and positive elsewhere. Our recent results[35,41]indicate that such functions can be constructed directly from the commercially available solid modeling representations,as well as from a variety of other geometric data structures,such as cell complexes. For example,Figures3(a)and(b)show approximate distancefields for the portions of the boundary where non-slip boundary conditions are prescribed(compare to the car shapes in Figures23(b)and(c)respectively).In the next Section we explain the usage of approximate distancefields for construction of the RFM solution structures and transfinite interpolation of the prescribed boundary conditions.2.3RFM solution structure for stream functionA solution structure is a function that satisfies exactly all prescribed boundary conditions.In general,any RFM solution structure can be represented as a sum of two functions:ψ=ψ0+ψ1(12) whereψ0satisfies homogeneous boundary conditions and contains necessary degrees of freedom in order to approxi-mate the differential equation of the problem;functionψ1interpolates the functions given as boundary conditions(5),Figure4:The RFM solution structure that satisfies boundary conditions(4-8)exactly(6)and(8).The interpolation term is constructed using the transfinite interpolation method[31]which is a generaliza-tion of the inverse distance weighting technique;it matches all specified boundary conditions and extends them inside the domain by some arbitrary but well behaved function.For the problem considered here,it takes:ψ1=ψoutletω2outlet+ψinletω2inlet+ψupper wallω2upper wall+ψlower wallω2lower wall1ω2outlet+1ω2inlet+1ω2upper wall+1ω2lower wall,(13)whereωoutlet,ωinlet,ωupper wall andωlower wall are approximate distancefields that describe outlet,inlet and walls of the channel as it is shown in Figure4.Rasing these functions to the second power assures that boundary condition∂ψ∂n|whole boundary =0is satisfied.Functionψ0in the solution structure(12)serves for approximation of the differential equation of the problem.In our case it can be represented as a product of the second power of an approximate distance to the boundary of the channelωand unknown functionΦwhose sole purpose is the approximation of the differential equation of theproblem:ψ0=ω2Φ.Sinceωtakes on zero value on boundary of the geometric domain,ψ0vanishes on the boundary together with itsfirst normal derivative.Therefore,regardless of the chosen functionΦ,functionψ0satisfies thehomogeneous boundary conditions exactly.In many practical situations functionΦcannot be determined exactly,which is why it is usually represented by a linear combination of basis functions{χi}N i=1:Φ=Ni=1C iχi.(14)The basis functions{χi}N i=1have to be smooth enough in order to approximate the differential equation of the problem. Thus,the RFM solution structure(12)may be rewritten as follows:ψ=ψoutletω2outlet+ψinletω2inlet+ψupper wallω2upper wall+ψlower wallω2lower wall1ω2outlet+1ω2inlet+1ω2upper wall+1ω2lower wall+ω2Ni=1C iχi.(15)This solution structure corresponds to the space that contains functions satisfying the prescribed boundary conditions and is sufficiently complete in the sense of being able to approximate the exact solution with an arbitrary degree of accuracy[30].Employment of the RFM solution structures to represent a solution of a physical problem offers several advantages. In particular:an RFM solution structure treats the prescribed boundary conditions exactly;an RFM solution structure contains no information about the differential equation of the problem which means that the same solution structure can be used to represent solutions of different physical problems with similar types of boundary conditions;basis functions in the solution structure can be constructed over a mesh conforming or non-conforming to a geometric model;solution structure can be easily adjusted to a new geometric model—only approximate distancefields have to be reconstructed in order to represent the boundary pieces of new geometric model;an RFM solution structure can be evaluated and differentiated at any point inside the computational domain;finally,an RFM solution structure can be integrated over the geometric model using adaptive numerical procedures[41].2.4Computation of the coefficients in the solution structureSince the RFM solution structure satisfies the given boundary conditions exactly,to solve the problem we need to find the set of the unknown coefficients{C i}N i=1in the RFM solution structure that gives the best approximation to the differential equation of the boundary value problem.Numerical values of these coefficients can be determined via variational or projectional methods.The differential equation(3)for the stream function contains non-linear terms that have to be linearized before the solution method is applied.After substitution of solution structure(12)into differentialequation (3)and application of Newton-Kantorovich linearization scheme we obtain:1Re ∇4ψn +10− ∂ψn +10∂y ∂∇2ψn 0∂x +∂ψn 0∂y ∂∇2ψn +10∂x −∂ψn +10∂x ∂∇2ψn 0∂y −∂ψn 0∂x ∂∇2ψn +10∂y−∂ψn +10∂y ∂∇2ψ1∂x −∂ψ1∂y ∂∇2ψn +10∂x +∂ψn +10∂x ∂∇2ψ1∂y +∂ψ1∂x ∂∇2ψn +10∂y=−1Re ∇4ψ1+∂ψ1∂y ∂∇2ψ1∂x −∂ψ1∂x ∂∇2ψ1∂y −∂ψn 0∂y ∂∇2ψn 0∂x +∂ψn 0∂x ∂∇2ψn 0∂y .(16)This equation is formulated for the function ψ0satisfying the homogeneous boundary conditions ψ0|∂Ω=0,∂ψ0∂n |∂Ω=0.Equation (16)is solved by an iterative algorithm,and the superscripts n +1and n in the equation denote solutions at the current and previous iterations respectively.The iterative process finishes as soon as the difference between twoconsecutive solutions becomes sufficiently small.At each iteration the least squares method is applied to equation (16)minimizing the residual of the equation:F = Ω1Re ∇4ψn +10− ∂ψn +10∂y ∂∇2ψn 0∂x +∂ψn 0∂y ∂∇2ψn +10∂x −∂ψn +10∂x ∂∇2ψn 0∂y −∂ψn 0∂x ∂∇2ψn +10∂y −∂ψn +10∂y ∂∇2ψ1∂x −∂ψ1∂y ∂∇2ψn +10∂x +∂ψn +10∂x ∂∇2ψ1∂y +∂ψ1∂x ∂∇2ψn +10∂y+1Re ∇4ψ1−∂ψ1∂y ∂∇2ψ1∂x +∂ψ1∂x ∂∇2ψ1∂y +∂ψn 0∂y ∂∇2ψn 0∂x −∂ψn 0∂x ∂∇2ψn 0∂y 2d Ω→min.(17)From the necessary condition of the existence of minimum ∂F ∂C i =0,i =1,...,N we obtain a system of linear equations AC =B whose solution gives the numerical values of the unknown coefficients in the solution structure.Elements of the matrix A and vector B are defined as follows:a ij = Ω 1Re ∇4 ω2χi − ∂∂y ω2χi ∂∇2ψn 0∂x +∂ψn 0∂y ∂∂x ∇2 ω2χi −∂∂x ω2χi ∂∇2ψn 0∂y −∂ψn 0∂x ∂∂y ∇2 ω2χi −∂∂y ω2χi ∂∇2ψ1∂x −∂ψ1∂y ∂∂x ∇2 ω2χi +∂∂x ω2χi ∂∇2ψ1∂y +∂ψ1∂x ∂∂y ∇2(ωχi ) 1Re ∇4 ω2χj − ∂∂y ω2χj ∂∇2ψn 0∂x +∂ψn 0∂y ∂∂x ∇2 ω2χj −∂∂x ω2χj ∂∇2ψn 0∂y −∂ψn 0∂x ∂∂y ∇2 ω2χj −∂∂y ω2χj ∂∇2ψ1∂x −∂ψ1∂y ∂∂x ∇2 ω2χj +∂∂x ω2χj ∂∇2ψ1∂y +∂ψ1∂x ∂∂y∇2 ω2χj d Ω;(18)b i = Ω 1Re ∇4 ω2χi − ∂∂y ω2χi ∂∇2ψn 0∂x +∂ψn 0∂y ∂∂x ∇2 ω2χi −∂∂x ω2χi ∂∇2ψn 0∂y −∂ψn 0∂x ∂∂y ∇2 ω2χi −∂∂y ω2χi ∂∇2ψ1∂x −∂ψ1∂y ∂∂x ∇2 ω2χi +∂∂x ω2χi ∂∇2ψ1∂y +∂ψ1∂x ∂∂y ∇2 ω2χi −1Re ∇4ψ1+∂ψ1∂y ∂∇2ψ1∂x −∂ψ1∂x ∂∇2ψ1∂y −∂ψn 0∂y ∂∇2ψn 0∂x +∂ψn 0∂x ∂∇2ψn 0∂yd Ω(19)Integrals (18)and (19)are computed using adaptive integration algorithm based on the Gauss-Legendre quadrature rule in conjunction with hierarchical space decomposition technique [41].。