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应⽤化学专业英语第⼆版万有志主编版课后答案和课⽂翻译Unit 1 The Roots of ChemistryI. Comprehension.1.C2. B3. D4. C5. BII. Make a sentence out of each item by rearranging the words in brackets.1.The purification of an organic compound is usually a matter of considerable difficulty,and it is necessary to employ various methods for this purpose.2.Science is an ever-increasing body of accumulated and systematized knowledge and isalso an activity by which knowledge is generated.3.Life, after all, is only chemistry, in fact, a small example of chemistry observed on asingle mundane planet.4.People are made of molecules; some of the molecules in people are rather simplewhereas others are highly complex.5.Chemistry is ever present in our lives from birth to death because without chemistrythere is neither life nor death.6.Mathematics appears to be almost as humankind and also permeates all aspects ofhuman life, although many of us are not fully aware of this.III. Translation.1.(a) chemical process (b) natural science (c) the technique of distillation2.It is the atoms that make up iron, water, oxygen and the like/and so on/and soforth/and otherwise.3.Chemistry has a very long history, in fact, human activity in chemistry goes back toprerecorded times/predating recorded times.4.According to/From the evaporation of water, people know/realized that liquids canturn/be/change into gases under certain conditions/circumstance/environment.5.You must know the properties of the material before you use it.IV. Translation化学是三种基础⾃然科学之⼀，另外两种是物理和⽣物。

Mathematical model for heat transfermechanism for particulate systemA.R.Khan *,A.ElkamelDepartment of Chemical Engineering,Faculty of Engineering and Petroleum,Kuwait University,P.O.Box5969,13060Safat,KuwaitAbstractVarious theoretical models for fluidized bed to surface heat transfer have been considered to explain the mechanism of heat transport.The particulate fluidized bed which is the common case for liquid–solid fluidized bed is much simpler and homoge-neous and transport operation can be easily modeled.The heat transfer coefficient in-creases to a maximum and then steadily decreases as the bed void fraction increases from that of a packed bed to unity.The void fraction e max at which the maximum value of heat transfer coefficient occurs is a function of the solid–liquid system properties.An unsteady state thermal conduction model is suggested to describe the heat transfer process.The model consists of strings of the particles with entrained liquid moving parallel to surface,during the time interval heat conduction takes place.These strings are separated by liquid into which the principal mode of transfer is by convec-tion.The model shows a dependence of heat transfer coefficient on void fraction and on physical properties,which is consistent with the results of experimental work.Ó2002Elsevier Science Inc.All rights reserved.1.IntroductionThe high heat transfer coefficient from a hot surface to a fluidized bed fa-cilitates the addition and removal of heat to and from a process efficiently.The role of various parameters and the mechanism of heat transfer in this field have been the subject of extensive investigations during the last three decades.Many Applied Mathematics and Computation 129(2002)295–316*Corresponding author.Tel.:+965-481-7662;fax:+965-483-9498.E-mail address:rehman@.kw (A.R.Khan).0096-3003/02/$-see front matter Ó2002Elsevier Science Inc.All rights reserved.PII:S 0096-3003(01)00039-X296 A.R.Khan,A.Elkamel/put.129(2002)295–316A.R.Khan,A.Elkamel/put.129(2002)295–316297empirical correlations relating bed to surface heat transfer coefficients for a range of operating variables have been proposed.They are of restrictive va-lidity because they cannot make adequate allowance for different geometries of equipment used and varying degree of accuracy of the experimental techniques used.Furthermore,it is difficult to extrapolate outside the experimental range of variables studied.Different models have been proposed to explain the dif-ferent aspects of this complexproblem.There are particularly diverse concepts suggested by different workers regarding the mechanism of heat transfer be-tween afluidized bed and a heat transfer surface.In this paper an attempt has been made to explain the mechanism of heat transfer from bed to surface in liquidfluidized systems.The model results are compared with experimental data[1]and computed values of the existing models.2.Heat transfer models forfluidized bedBoth gas,and to a lesser extent liquidfluidized beds have been employed in chemical engineering practice particularly where the addition or removal of heat from the bed is required.In the case of gasfluidized beds the more important aspects have been collected and presented in detail by Zabrodsky [2].Most of the workers have examined a limited range of experimental variables and presented their results in the form of correlations;the power of any group in the correlations gave some indication of its importance within the experimental range investigated.It is clear that the scale of the equipment in which measurements were made has influenced the results.There is no sufficiently general theory of heat transfer influidized beds,although several different models have been proposed to explain various aspects of this problem.A brief description of some of the models is given in the following section.2.1.The limiting laminar layer modelLeva and Grummer[3]noted that the core of the bed was isothermal and offered negligible thermal resistance while the main resistance limiting the rate of heat transfer between the bed and the heat source lay in afluidfilm near the hot surface.They suggested that particles acted as turbulence promoters,which eroded thefilm reducing its resistive effects.Levenspiel and Walton[4]derived an expression in terms of modified Nusselt and Reynolds numbers for the ef-fectivefluidfilm thickness assuming that thefilm breaks whenever a particle touches the transfer surface.They have to modify the coefficients in the model to account for their own experimental data.Wen and Leva[5]correlated the published heat transfer results on the basis of a scouring action model in which particle movement was assumed to be vertical and parallel to the heat source.Nu¼consC S qSd1:5Pffiffiffig pk g0:4GdPElgR"#0:36:ð1ÞIn this correlation thefluidization parameters are defined as follows:1.E is thefluidization efficiencyðGÀG mfÞ=G mf.2.R is the expansion ration of the bed H=H mf.Richardson and Mitson[6]and Richardson and Smith[7]reported that for liquidfluidized beds the resistance to heat transport lay near the tube wall within the laminar sub-layer where the effective thickness is reduced by the presence of particles for two reasons:1.The particles cause turbulence in thefluid thereby reducing the thickness ofthe laminar boundary layer.2.The particles themselves transport heat as a result of the radial componentof their rapid oscillating motion.Wasan and Ahluwalia[8]proposed that heat transfer through afluidfilm was promoted byfluid eddies beyond thefilm boundary.They assumed that the solid particles were stationary and equally spaced and that heat was transferred through thefilm and then spread byfluid convection into the bulk of the bed.They compared the experimental results of various workers and found deviations of up toÆ44%.These models basically involve a steady-state concept of the heat transfer but Wasan and Ahluwala[8]included some dynamics transfer features for transfer through thefluid into the bed.298 A.R.Khan,A.Elkamel/put.129(2002)295–3162.2.Two resistancefilm modelWasmund and Smith[9]suggested a modified laminar layer model,in which they considered particle convective transfer due to radial motion of particles into the laminar layer.This mechanism contributed50–60%of the total heat transferred and the remainder was fromfluid convective transfer.Tripathi et al.[10]used the series model proposed by Ranz[11]for effective transport properties in packed beds.They compared results obtained by Wasmund and Smith[12]using radial velocities of the particles and observed deviation of Æ20%.Brea and Hamilton[13]and Patel and Simpson[14]used a two resis-tancefilm model and emphasized that thefluid eddy convection is the main contributing factor to the heat transfer.Zahavi[15]measured the effective diffusivity of thefluidized beds and also developed a semi-empirical correlation, which represented his results with a maximum deviation ofÆ34%.2.3.Unsteady state heat transferMickley and Trilling[16]suggested that the heat transfer process in a gas-fluidized bed was of an unsteady state ter Mickley and Fairbanks[17] developed a model of heat transfer on the assumption that at any time there is unsteady state heat transfer within thefluidized bed close to heat source;this can be broken down into components due to solid/solid,solid/surface,gas/solid, and gas/surface transfer.Packets of loosely locked particles which are assumed to have uniform thermal properties constitute thefluidized bed.The mean heat transfer coefficient can be calculated for packets of particles moving with a constant speed u passing rapidly along the length of the heat source,thenh¼2ffiffiffip pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik0q0C0Pu0L r:Thermal conductivity k0,density q0and heat capacity C0P for packets can beestimated by use of the Schumann and Voss[18]correlation.The assumption that the thermal properties of the bed are uniform in the neighborhood of the heat source is unrealistic when the source and the bulk of the bed are at considerably different temperatures.Mickley and Fairbanks[17]calculated the residence times of packets from resistancefluctuations recorded for a thin electrically heated platinum strip.The frequency of packets was of the order of two per seconds and the residence time of0.4s.Henwood[19],Catipovic et al.[20],Suarez et al.[21]and George and Smalley[22]used a small heat transfer surface to measure the variations in local heat transfer coefficient within and adjacent to a rising bubble.They concluded that heat transfer took place mainly through thefluid to the particle with the maximum rate in the vicinity of the contact part.A.R.Khan,A.Elkamel/put.129(2002)295–316299300 A.R.Khan,A.Elkamel/put.129(2002)295–3162.4.Simplified modelsThe heterogeneity offluidized beds is an important factor enhancing the value of the heat transfer coefficient up to50–100folds for a gas and5–8folds for liquidfluidized bed.Botterill and Williams[23]have proposed a model for heat transfer in gas-fluidized systems which are based upon the unsteady state conduction of heat to spherical particles adjacent to the transfer surface.The convective transfer through the gas is ignored because the effective diffusivity of thefluidized bed is much higher than the eddy diffusivity of the gas.The particle and gas,whose temperatures were initially the same,approached the surface,the temperature of which remained approximately constant because of its high heat capacity.The Fourier equations for thermal conduction were solved byfinite difference technique.Apart from the axes of symmetry through the particles,there were three space limits to the problem.(A)Close to the heater it was assumed that there was a continuous thin layer ofpurefluid with which a particle was in contact.(B)The temperature of the other end the particle was set at the sink temper-ature,taken as the bulk temperature.(C)Transfer of heat between particles in a direction parallel to the surfacecould be neglected because the temperature difference between particles in adjacent position was very small.The experimental results for heat transfer coefficients for the shortest resi-dence time of metallic particles were far less than the predicted values.This discrepancy was accounted for by assuming that a gasfilm of thickness equals to about10%of the particles around the heat source.Botterill[24]tried both models in which there was triangular and square packing on the particles and a fluidfilm between the particles and the heat transfer surface.The thickness of thefilm was related to the resistance limiting the heat transfer.Different workers made observations but no satisfactory conclusion was reached about the local variation of void fraction in the vicinity of the heat transfer surface.Davies[25]considered the unsteady state heat transfer by conduction be-tween the element at one temperature and spherical particles immersed in a liquid at another temperature.The particles were assumed to enter the thermal boundary layer with a mean radial velocity v r,approached the surface and left it again with the same velocity.The Fourier equations were solved by using the explicitfinite difference method and the same boundary condition as reported by Botterill and Williams[23].The very low values of particle convective heat transfer component predicted by the model indicated that only a very small proportion of the heat transferred from the heated surface to the bed was carried byfluidized particles.The experimental high value was attributable toA.R.Khan,A.Elkamel/put.129(2002)295–316301 the fact that the effective thickness of the thermal boundary layer had been substantially reduced;this was mainly due to the following causes:1.The scouring action of particles.2.The high interstitial velocity of the liquid.2.5.Particle replacement modelGabor[26]has proposed that heat has been absorbed by the particulate bed based on string of spheres of infinite length normal to heat transfer surface. Another simplified approach based on series of alternating gas and solid slabs also provided similar results as the spherical model.Gelperin and Einstein[27]have developed a more refined model taking into account other details of the process involved.They considered that heat is transferred from the heat transfer surface by packets of solid particles by gas bubbles and by gas passing between the packet and the surface.The total heat transfer coefficient is expressed ash¼h PðÀh convÞ1ðÀf0Þþh b f0þh r;ð2Þwhere h P;h conv;h b and h r are the heat transfer coefficients corresponding to packet,convection,bubble and radiation,respectively,and f0is the fraction of time for which the surface is covered by bubbles.They solved the basic equations for their models of bed to surface heat transfer in terms of two heat resistances:R WS the resistance offered by gas entrained by the particles close to the transfer surface and R a the resistance offered by the gas–solid packets.They have tabulated theirfinal equations for instantaneous and mean heat transfer coefficient for different boundaries in their publication[27].For isothermal conditions of heat source,which have already been proposed a simplified solution can be used with little error.Martin[28]has presented a particle convective energy transfer model for wall to bed heat transfer from solid surface immersed in gas-fluidized bed.In his model the following assumptions were applied.1.The contact time is regarded to be proportional to the time taken to coverthe path with the length of one particle diameter in freeflight.2.The wall to particle heat transfer coefficient is calculated by integrating thelocal conduction heatfluxes across the gaseous gap between sphere and plain surface over the whole projected area of the sphere[29].3.The average kinetic energy for the random motion of particles comes from acorresponding potential energy.302 A.R.Khan,A.Elkamel/put.129(2002)295–3163.Development of mathematical modelFor the purpose of establishing a simplified model of a liquidfluidized bed, the system is assumed to consist of strings of particles with liquidfilling the intervening spaces.It is proposed that unsteady state thermal conduction takes place into both the liquid and the solid particles in the string as reported by Gabor[26].Liquid layers into which the principal mode of heat transfer is forced convection as shown in Fig.1to separate the strings.The overall heat transfer coefficient for liquidfluidized bed from immersed surface constitutes solid conductive,liquid conductive and liquidconvective Array Fig.1.Mathematical representation offluidized particle entrained in liquid with initial and boundary conditions.components based on the void fraction determined by bed expansion charac-teristics and particle axial velocity V P;Axial.The convective component is cal-culated for liquid moving with interstitial velocity parallel to the heating surface.The conductive components for solid and liquid are evaluated based on contact time using unsteady state conduction equations for string of par-ticles with entrained liquid.3.1.Heat transfer across incompressible boundary layersThe simulated element is considered as aflat plate located on the axis of the column with the large faces parallel to liquidflow.The liquid with an average velocity u and uniform temperature T B passes over the hotflat plate at a constant temperature T E.At high Prandtl numbers the thermal boundary layer is always confined entirely within the laminar sublayer.This limiting case of forced convection across a turbulent boundary layer can be solved analytically. Kestin and Persen[30]based their analysis on the laminar form of the energy equation and confined their attention to the laminar sublayer only.The other assumption made is that the velocity varies linearly with distance perpendicular to theflat plate.The detailed solution of the energy equation for laminar form assuming linear velocity profile within laminar sublayer is expressed asy s WðxÞlo To xÀy2d s Wd xo To y¼ao2To y2ð3aÞby substitutingg¼y3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðs W=lÞ3q9aR xx0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðs W=lÞpdx;Eq.(3a)can be transformed to ordinary differential equation.d2T d g2þgþ23d Td g¼0;ð3bÞwhereT¼T EÀTT EÀT B¼1;for x¼0and all values of y>0,T¼1,for y¼/and all values of x>0and T¼0,for y¼0and all values of x>0.The solution is given in the form of incomplete c-function asTðgÞ¼c1=3;g ðÞCð1=3Þ:The calculated values of the heat transfer coefficient are presented together with experimental values[1]in Table2.The experimental data are obtainedA.R.Khan,A.Elkamel/put.129(2002)295–316303304 A.R.Khan,A.Elkamel/put.129(2002)295–316from immersed electrically wound heating surface in6-mm glass particleflui-dized bed dimethyl phthalate.The high experimental values are due mainly to the following causes:1.The plane element was not a trueflat plate.2.The edge effects of the element caused a high value.3.The temperature of the element might not be uniform.In the case of afluidized bed the presence of solid particles decreased the free area available forflow and caused an increase in the liquid velocity near the heat source.The contribution of heat transfer due to liquid alone was assessed on the basis of the interstitial velocity which was the factor determining the thickness of the thermal boundary layer.3.2.The contribution offluidized particles3.2.1.Particle velocities influidized bedsBy means of high speed photography several workers have measured the paths taken by a tracer particle in a transparentfluidized bed.Toomey and Johnstone[31]obtained particle velocities near the wall of dense phase gas-fluidized bed.For the particular case of0.376-mm diameter glass spheresflu-idized in air,they reported particle velocities from60to600mm/s.Kondukov et al.[32]used radioactive tracer particles and radiation detectors to measure the particle trajectories in an airfluidized bed.Their results were similar to those of Toomey and Johnstone[31]and gave additional information on particle behavior in the interior of the bed.Handley[33]fluidized1.1-and1.53-mm glass spheres with methyl benzoate in31-and76-mm diameter columns.He reported the radial and axial velocities of the particles for bed voidage ranging from0.67to0.905.He concluded that the velocity of the particles was completely random in a uniformlyfluidized bed.A more extensive study of particle velocities influidized bed was made by Carlos[34]and by Latif[35].Carlosfluidized9mm glass beads with dimethyl phthalate at30°C in a100mm diameter column.He reported the particle velocities(radial,angular,axial,horizontal and total)over the voidage range 0.53–0.7.A set of differential equations governing the mixing process was numerically solved by computer taking into account the effect of radial and axial tif[35]extended the work of Carlos to6-mm glass spheres and developed a simple relationship between particle velocity and axial and radial positions.The constants of these equations are listed in Table1.These equations are used to evaluate particle velocities as a function of bed expansion characteristics.3.2.2.Residence time of particle in the vicinity of hot surfaceDavies [25]assumed that the high heat transfer coefficient for the fluidized bed was attributable to effects arising from the radial velocity of the particles as reported by Figliola and Beasley [36].His model predicted very small values of heat transfer coefficient because the thermal boundary layer thickness was small compared with the diameter of the particle and the residence time in the thin boundary layer was short.However from the average calculated values of radial and axial particle velocity components at the center of the fluidized bed,it was obvious that the axial component of velocity was dominant.On this basis it appeared reason-able to assume,as a first approximation,that particles and fluid both at the bulk temperature of the bed approached the heat transfer surface at a velocity approximately equal to the average axial component of a particle velocity.The string of particles separated by intervening liquid thus traveled parallel to the hot surface and unsteady state heat conduction took place through the solid and liquid in parallel.The residence time for a particle could be given ast ¼L V P ;Axial:3.2.3.Unsteady state thermal conduction for liquid and solidIn the present model in which it is assumed that heat transfer is because of unsteady state thermal conduction in both liquid and solid,the following assumptions are made:1.The temperatures of the bulk of the fluidized bed ðT B Þand the temperature of the element ðT E Þare uniform.2.A string of particles with entrained liquid at a uniform temperature equal to T B arrives quickly in an axial direction at the element.Table 1Coefficients for calculation of axial velocity component [35]V P ;Axial ¼A Ár þB where r is normalized radial coordinate of particle A ¼a 0þa 1z þa 2z 2þa 3e a 4z2B ¼b 0þb 1z þb 2z 2þb 3e b 4z2e a 0a 1a 2a 3a 4b 0b 1b 2b 3b 40.55)0.767.25)8.74)87.4)20.20.42)3.08 3.8331.8)21.90.65)2.749.66)6.92)74.22)10.09 1.17)1.550.3846.5)17.030.75)1.42 4.78)2.84)128.2)15.110.96)3.17 1.9474.56)18.340.850.55)3.3 2.58)162.3)20.0)0.18 3.12)2.9286.7)22.70.95)4.27)17.9822.27)222.0)27.52.2212.04)15.09119.7)29.0A.R.Khan,A.Elkamel /put.129(2002)295–3163053.The particles and entrained liquid absorb heat by unsteady state conductionas they travel along the surface.Immediately the particles leave the vicinity of the element they exchange heat with the surrounding liquid.The model explains the way in which thefluidized particles contribute to-wards the transfer of heat between the hot surface and the bed.At any time the heat contents of both the liquid and the particles may be obtained from a knowledge of the temperature distribution within the particle and liquid.The temperature distribution in the solid and liquid in contact with the surface may be obtained by the heat conduction equation for both the liquid and solid over the residence time for which they are present at the hot surface.The heat conduction equation in spherical coordinates necessitates the use of three space dimensions;this may cause complications in solving the equation with its ap-propriate boundary conditions.The situation may be simplified by defining the system in terms of Cartesian coordinates and assuming that the particles may be replaced by cubes,the length of the side of each of which is equal to the diameter of the particle.Each cube moves with one face in contact with the surface and liquid occupies the intervening spaces.Symmetry is assumed along the plane perpendicular to the surface.The length of the liquid slug between the particles and the thickness of the liquid layer separating the strings will be calculated as follows: In Fig.1a cube of dimensionðX SþX LÞis considered and the void fraction in the vicinity of the surface assumed to be the same as in the bulk where X S and X L are dimensions of particle and liquid slug,respectively.The volume ofthe particle is X3S and of the liquid slug X2SX L and void fraction is expressed ase¼ðX SþX LÞ3ÀX3S ðX SþX LÞ:On rearranginge¼1ÀX SX SþX L 3orX L¼X S11Àe1=3"À1#:The Fourier equations for unsteady state thermal conduction within the two homogeneous phases of the system areFor solid phaseo2T P o x2þo2T Po y2¼1a Po T Po t:ð4aÞ306 A.R.Khan,A.Elkamel/put.129(2002)295–316For liquid phaseo2T f o x2þo2T fo y2¼1a fo T fo t:ð4bÞThese equations are put in dimensionless form by definingT P¼T EÀT PT EÀT B;T f¼T EÀT fT EÀT Band s¼tÁa P;s¼tÁa fo2T P o x2þo2T Po y2¼o T Po s;ð4cÞo2T f o x2þo2T fo y2¼o T fo s:ð4dÞInitial and boundary conditions,at t¼0,the particle and liquid slug both are divided to give a mesh nÂn and all points within the particle and liquid slug are at the bulk temperatureT P¼T f¼1;T P¼T f¼T B:ð5aÞThe temperature at the face of the liquid slug at x¼0and at all distances in the y-direction perpendicular to the surface is considered to be at the bulk temperature.The temperature of the similar face of the solid particle is taken as the computed values of the liquid temperature at the end of thefirst time step at x¼X Lat y¼0and x¼0;T f¼1;T f¼T B;T P¼T f and T P¼T fðAfter first time step at x¼X LÞ:ð5bÞAt t P0,the faces of both particle and liquid which are in contact with the surface are at all times at the surface temperature.At y¼0;T P¼T f¼0;T P¼T f¼T E:ð5cÞEqs.(4c)and(4d)are solved simultaneously usingfinite difference techniques for afixed and for a variable boundary.For t>0thefixed boundary is ex-pressed by Eq.(5c)and the variable boundary is aty¼0at x¼0for liquidT f¼T P and T f¼T PðAfter previous time step at x¼X SÞand for particleA.R.Khan,A.Elkamel/put.129(2002)295–316307T P¼T f and T P¼T fðAfter current time step at x¼X LÞ:ð5dÞThe explicit method used by Botterill and Williams[23]and by Davies[25] restricts the time and space increments to ensure stability according to the equationD s61ðD XÞÀ2þðD YÞÀ2 h i:Their difference equation isTði;jÞkþ1ÀTði;jÞkD s ¼TðiÀ1;jÞÀ2Tði;jÞþTðiþ1;jÞD X2kþTði;jÀ1ÞÀ2Tði;jÞþTði;jþ1ÞD Y2k:ð6aÞThe above-mentioned method is very sensitive to the value of the operator k h which is given ask hX¼a hD tD X2and k hY¼a hD tD Y2:For the sake of simplicity equal increments are taken in the X-and Y-directions,i.e.D X¼D Y.An implicit method can make the equations inde-pendent of the operator value as well as of space and time increments.The difference equation is thenTði;jÞkþ1ÀTði;jÞkD s ¼TðiÀ1;jÞÀ2Tði;jÞþTðiþ1;jÞD X2þTði;jÀ1ÞÀ2Tði;jÞþTði;jþ1ÞD Y2kþ1:ð6bÞOn rearranging one gets a pentadiagonal matrixÀTðiÀ1;jÞÀTðiþ1;jÞþ1k hþ4Tði;jÞÀTði;jÀ1ÞÀTði;jþ1Þkþ1¼1k hTði;jÞk;ð6cÞwherek h¼a h D t D X2;which can be solved by either the Gaussian elimination method or the Gauss-Seidel iterative method to givefive unknowns.The implicit alternating direction method discussed by Carnhan et al.[37], which avoids all the disadvantages discussed above,is thought to be most suited for this type of problem.The two difference equations are used in turn over successive time steps,each of duration D s=2.Thefirst Eq.(7a)is implicit 308 A.R.Khan,A.Elkamel/put.129(2002)295–316。

f¨u r Mathematikin den NaturwissenschaftenLeipzigRandom perturbations of spiking activity in apair of coupled neuronsbyBoris Gutkin,J¨u rgen Jost,and Henry TuckwellPreprint no.:492007Random perturbations of spiking activity in apair of coupled neuronsBoris Gutkin∗,J¨u rgen Jost and Henry C.Tuckwell†May14,2007AbstractWe examine the effects of stochastic input currents on thefiring be-haviour of two coupled Type1or Type2neurons.In Hodgkin-Huxleymodel neurons with standard parameters,which are Type2,in the bistableregime,synaptic transmission can initiate oscillatory joint spiking,butwhite noise can terminate it.In Type1cells(models),typified by aquadratic integrate andfire model,synaptic coupling can cause oscilla-tory behaviour in excitatory cells,but Gaussian white noise can againterminate it.We locally determine an approximate basin of attraction,A,of the periodic orbit and explain thefiring behaviour in terms of theeffects of noise on the probability of escape of trajectories from A.1IntroductionHodgkin(1948)found that various squid axon preparations responded in quali-tatively different ways to applied currents.Some preparations gave a frequency offiring which rose smoothly from zero as the current increased whereas oth-ers manifested the sudden appearance of a train of spikes at a particular input current.Cells that responded in thefirst manner were called Class1(which we refer to as Type1)whereas cells with a discontinuous frequency-current curve were called Class2(Type2).Mathematical explanations for the two types are found in the bifurcation which accompanies the transition from rest state to a periodicfiring mode.For Type1behaviour,a resting potential vanishes via a saddle-node bifurcation whereas for Type2behaviour the instability of the rest point is due to an Andronov-Hopf bifurcation,see Rinzel and Ermentrout (1989).Stochastic effects in thefiring behaviour of neurons have been widely reported, discussed and analyzed since their discovery in the1940’s.One of thefirst reports for the central nervous system was by Frank and Fuortes(1955)for catX1X3X2X4X1X2TIMEFigure1:On the left are shown the solutions of(1)-(4)for two coupled QIF model neurons with the standard parameters.X1and X2are the potential variables of neurons1and2and X3and X4are the inputs to neurons1and2, respectively.On the right is shown the periodic orbit in the(x1,x2)-plane.The square marked P was explored in detail in reference to the extent of the basin of attraction of the periodic orbit.spinal neurons.Although there have been many single neuron studies,the effect of noise on systems of coupled neurons have not been extensively investigated. Some preliminary studies are those of Gutkin,Hely and Jost(2004)and Casado and Baltan´a s(2003).2The quadratic integrate andfire modelA relatively simple neural model which exhibits Type1firing behaviour is the quadratic integrate andfire(QIF)model.We couple two model neurons in the following manner(Gutkin,Hely and Jost,2004).Let{X1(t),X2(t),t≥0}be the depolarizations of neurons1and2,where t is the time index.Then the model equations are,for subthreshold states of two identical neurons,dX1=[(X1−x R)2+β+g s X3]dt+σdW1(1)dX2=[(X2−x R)2+β+g s X4]dt+σdW2(2)dX3=−X3τ+F(X1)(4)2where X3is the synaptic input to neuron1from neuron2and X4is the synaptic input to neuron2from neuron1.The quantity x R is a resting value.g s is the coupling strength.βis the mean background input.W1and W2are independent standard Wiener processes which enter with strengthσ.This term may model variations in nonspecific inputs to the circuit as well as possibly intrinsic membrane and channel noise.By construction,we take this term to be much weaker than the mutual coupling between the cells in our circuit.The function F is given byF(x)=1+tanh(α(x−θ))whereθcharacterizes the threshold effect of synaptic activation.Since when a QIF neuron is excited and it receives no inhibition,its potential reaches an infinite value in afinite time,for numerical simulations a cutoffvalue x max is introduced so that the above model equations for the potential apply only if X1 or X2are below x max.To complete a“spike”in any neuron,taken as occurring when its potential reaches x max,its potential is instantaneously reset to some value x reset which may be taken as−x max.At the bifurcation point g s=g∗s, two heteroclinic orbits between unstable rest points turn into a periodic orbit of antiphase oscillations.3Results and theoryIn the numerical work,the following constants are employed throughout.x R= 0,x max=20,θ=10,α=1,β=−1,g s=100andτ=0.25.The initial values of the neural potentials are X1(0)=1.1,X2(0)=0and the initial values of the synaptic variables are X3(0)=X4(0)=0.When there is no noise,σ=0,the results of Figure1are obtained.The spike trains of the two coupled neurons and their synaptic inputs are shown on the left.Thefiring settles down to be quite regular and the periodic orbit,S,is shown on the right.The patch marked P is the location of the region explored in detail below.The effects of a small amount of noise are shown in Figure2.The neural excitation variables are shown on the left and the corresponding trajectories in the(x1,x2)-plane are shown on the right.In the top portion an example of the trajectory forσ=0.1is shown.Here three spikes arise in neuron1and two in neuron2,but the time between spikes increases and eventually the orbit collapses away from the periodic orbit.In the example(lower part)forσ=0.2 there are no spikes in either neuron.In10trials,the average numbers of spikes obtained for the pair of neurons were(2.5,2.2)forσ=0.1,(1.4,1.1)forσ=0.2 and(1.3,0.9)forσ=0.3;these may be compared with(5,5)for zero noise. 3.1Exit-time and orbit stabilityIf a basin of attraction for a periodic orbit can be found,then the probabil-ity that the process with noise escapes from the region of attraction gives the probability,in the present context,that spiking will cease.Since the system3TIMEX1X21 X2Figure2:On the left are shown examples of the neuronal potentials for neurons 1and2(QIF model)for two values of the noise,σ=0.1andσ=0.2.On the right are shown the trajectories corresponding to the results on the left,showing how noise pushes or keeps the trajectories out of the basin of attraction of the periodic orbit.(1)-(4)is Markovian,we may apply standardfirst-exit time theory(Tuckwell, 1989).Letting A be a set in R4and letting x=(x1,x2,x3,x4)∈A be a values of X1,X2,X3,X4)at some given time,the probability p(x1,x2,x3,x4)that the process ever escapes from A is given byL p≡σ2∂x21+σ2∂x22(5)+[(x1−x R)2+β+g s x3]∂p∂x2+ F(x2)−x3∂x3+ F(x1)−x4∂x4=0,x∈Awith boundary condition that p=1on the boundary of A(since the process is continuous).If one also adds an arbitrarily small amount of noise for X3and X4(or considers those solutions of(5)that arise from the limit of vanishing noise for X3,X4),the solution of the linear elliptic partial differential equation (5)is unique and≡1,that is,the process will eventually excape from A with probability1.Hence,the expected time f(x)of exit of the process from A satisfies L f=−1,x∈A with boundary condition f=0on the boundary of A.In fact,for small noise,the logarithm of the expected exit time from A,that4is,the time at whichfiring stops,behaves like the inverse of the square of the noise amplitude(Freidlin and Wentzell,1998).These linear partial differential equations can be solved numerically,for example by Monte-Carlo techniques.The basin of attraction A must be found in order to identify the domain of(5).We have done this approximately for the square P in Figure1.The effects of perturbations of the periodic orbit S within P on the spiking activity were found by solving(1)-(4)with various initial conditions in the absence of noise.The values of x1were from−0.43to1.57in steps of0.2and the values of x2were from-4to2also in steps of0.2.For this particular region, as expected from geometrical considerations,the system responded sensitively to to variations in x1but not x2.For example,to the left of S there tended to be no spiking activity whereas just to the right there was a full complement of spikes and further to the right(but still inside P)one spike.4Coupled Hodgkin-Huxley neuronsAs an example of a Type2neuron,we use the standard Hodgkin-Huxley(HH) model augmented with synaptic input variables as in the model for coupled QIF neurons given by equations(3)and(4),but with different parameter values. It has been long known that additive noise has a facilitative effect on single HH neurons(Yu and Lewis,1989).Coupled pairs of HH neurons have been employed with a different approach using conductance noise in order to analyze synchronization properties(e.g.Casado and Balt´a nas,2003).For the present approach,with X1and X2as the depolarizations of the two cells,we putdX1=1g K n4(V K−X1)+it was found that transient synchronization can terminate sustained activity. For Type2neurons,we have investigated coupled Hodgkin-Huxley neurons and found that in the bistable regime,noise can again terminate sustained spiking activity initiated by synaptic connections.We have investigated a minimal cir-cuit model of sustained neural activity.Such sustained activity in the prefrontal cortex has been proposed as a neural correlate of working memory(Fuster and Alexander,1973).ReferencesCasado,J.M.,Balt´a nas,J.P.(2003).Phase switching in a system of two noisy Hodgkin-Huxley neurons coupled by a diffusive interaction.Phys.Rev.E68,061917,Frank,K.,Fuortes,M.G.(1955).Potentials recorded from the spinal cord with microelectrodes,J.Physiol.130,625-654.Freidlin,M.I.,Wentzell,A.D.(1998),Random Perturbations of Dynamical Sys-tems,2nd ed.,Springer,New York Fuster,J.M.and Alexander,G.E.(1971),Neuron activity related to short-term memory.Science652-654 Gutkin,B.,Ermentrout,G.B.(1998).Dynamics of membrane excitability de-termine interval variability:a link between spike generation mechanismsand cortical spike train statistics.Neural Comp.10,1047-1065. Gutkin,B.S.et al.(2001)Turning on and offwith p.Neurosc.11:2,121-134Gutkin,B.,Hely,T.,Jost,J.(2004).Noise delays onset of sustainedfiring in a minimal model of persistent activity.Neurocomputing58-60,753-760. Hodgkin,A.L.(1948).The local changes associated with repetitive action in a non-medullated axon.J.Physiol.107,165-181.Rinzel,J.,Ermentrout,G.B.(1989).Analysis of neural excitability and oscilla-tions;in:Koch C.&Segev I.,eds.MIT Press.Tateno,T.,Harsch,A.,Robinson,H.P.C.(2004).Thresholdfiring frequency-current relationships of neurons in rat somatosensory cortex:Type1and Type2dynamics.J.Neurophysiol.92,2283-2294.Tuckwell,H.C.(1989).Stochastic Processes in the Neurosciences.SIAM,Philadel-phia.Yu,X.,Lewis,E.R.(1989).Studies with spike initiators:linearization by noise allows continuous signal modulation in neural networks.IEEE Trans.Biomed.Eng.36,36-43.6。

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A COmmOn tungsten mineral is Scheelite, CaICiUm tungstate, CdWO4.(a)DraW a StrUctUre ShOWing the bonding in the WθΛ ion and give the O-W-O bond angle.Treatment Of SCheellte Wlth aqueous SOdiUm CarbOnate gives a SOlutKXI Of SOdiUm tungstate and a White insoluble salt. AdditiOn Of hydrochloric add to aqueous SOdiUm tungstate PrOdUCeS tungstic acid. WhiCh On heating gives Iungsten(VI) oxide. ThiS may be reduced Wrth hydrogen to give PUre UJngSten metal.(b)(i) GiVe the equation for the reaction between SCheeIite and aqueous SodiUmCart)Onate.(ii) GiVe the equation for the reactiOn Of aqueous SOdiUm tungstate Wrth hydrochloric acid and SUggeSt a StrUctUre for tungstic acid・(Iii) GiVe the equation for the formation Of tungsten(VI) OXKje from tungstic aαd.(iv) GiVe the equation for the (OrmatiOn tungsten metal.In electrochromic Windows. a VOltage is applied between a WanSParent Iayer Of Iungsten(VI) OXkje and a SOUrCe Of ions SUCh as a IrthIUm SaIt・and the following reacbo∩ takes PIaCe during wħich SOme Of the IrthiUm ions are in∞rporated into the StrUCtUre Of the oxide:WO3÷ xU* ÷ Xe- ---------- ► LiXWo3The PrOdUct t Li1WO3. is known as a tungsten bronze, and its ColoUr depends On the ValUe Of x. The VaIUe Of X Can Vary from O to 1： a typical ValUe giving a blue-black COlOUr is When X = 0.3.(C) (i) CaIcUlate the OXidatiOn State Of the tungsten When X= 1 ・(ii) CakXIIate the average OXidatiOn State Of the tungsten When X = 0.3.BOth elemental SUlfUr and tungsten react Wrth fluorir»e gas to form the hexafluorides. SFe and WF6are both gases at r∞m temperature and PreSSUre. Wrth WF6being the most dense gas known Underthese ∞nditio∩s. WF6is extremely toxic due to its rapid reaction With Water to form twoSUbStances. In COntrast, SF6is inert in Water and non-toxic・(d)By assuming air to be made UP Of nItrOgen gas OnM CaICUlate the densities Of SF e and WF6relative to air.(e)CaleUlate the actual density Of WFβ(g) in g cm"3 at 298 K and Standard PreSSUre.(f)SUggeSt an equation for the reaction Of WF e Wfth water.2017年考题ThiS question is about the Green POOl Of RiOWhen the Water in the diving PoOI at the RiO OlymPiC GameS turned green, it WaS SUggeSted that Ihe growth Of algae WaS to blame; this WaS StrOngM COnteSted by the organisers. EVen after Ihe OffiCial report WaS PUbliShed t there is StiIl much SPeCUIation as to theactual reason behind the ∞tour Change-One Of the most COmmOnfy USed ComPOUndS in thechlorination Of SWimming POoIS is SOdiUmhypochlorite・ NaCIO.(a)Determine the OXidatiOn State Of ChiOrine inSOdiUm hypochlorite, NaelO.OnCe dissolved, an equilibrium is established between CIO" and its COnjUgate acid.(b)GrVe an equation for this equilibrium.ThiS equilibrium is highly PH dependent and Un der acidic co nd tons ChIOrine is produced.(C) GiVe an equation for the generation Of cħlorine from hypochlorite and HCLThe Organisers eventually explained the green COlOUr in the POOI as being due to the growth Of algae after the i∩advertent addition Of a Iarge quantity Of hydrogen PerOXide. WhiCh destroyed thehypochk>rrte and formed ChlOride ions.(d)GiVe an equation for the reaction between hydrogen PeroXkJe and hypochlorite.KyPoChlOrrteS also have a tendency to react Wrth ammonia and ammonia-like ComPOUndS to form ∞mpou∩ds COntaining nitrogen and ChlOnne. One SUCh COmPOUnd is nitrogen trichloride. WhiCh Can CaUSe eyeirritation and Ihe distinctive SmeIl Of Swimmlng p∞ls(e)(i) GiVe an equation for the formation Of nitrogen trichloride.(ii) DraW a StrUCtUre for nitrogen trichloride, ShOWing its shape, and State the approximate CkN-Cl bond angle.DePending On reacting ratios, another POSSibIe OUtCOme Of Ihe reaction between ammonia and hypoChlorrte is the formation Of hydrazine. H2N-NH2. and ChlOride ions.(O GrVe an equation for this reaction.COPPer(II) SUtfate is SOmetimeS added to SWimming POOIS and this WaS also suggested as the CaUSe Of the green ∞lour. COPPer(II) ions Wefe also blamed for the green t int given to Ihe bleached hair Of the AmeriCan SWimmer Ryan LoChte・ The COPPer(Il) ions precipitate On the hair due to the high PH OfCertain sħamp∞s・(g)SUggeSt a formula for the blue PreCiPitate On Ryan LOChte,s h.2018年考题4. ThiS question is about COUgh SUPPreSSantSIn SePtember 2017. the UK Prime MiniSter l ThereSa May, had a bad ∞υgh during her SPeeCh at the ConSerVatiVe Party COnferenCe- The COUgh suppressant drugOextromethorphan, WhiCh is PreSent in ∞ugh remedies SUCh as BenyIin®. CoUld have helped her out. ThiS questiOn is about the SyntheSlS Of dextromethorphan. The SyntheSiS involves the formation Of SOme Str(Jng b∞ds and SOme StabIe CarbOCations.DeXtromethorphan is Onen administered as the hydrobromide monohydrate salt.(a) In the anSWer b∞k. Ci rCle the atom in dextromethorphan that is PrOtOnated in the salt. (b) Determine the molecular formula Of dextromethorphan and hence CaIaJlate Ihe molar mass Of thedextromethorphan hydrobromkie monohydrate salt. The SyntheSiS Of dextRxnethorphan takes a number Of StePS. PteaSe note that in the SChemeS describing the Synthesis, by-products Of the reactions are not always shown.The SyntheSiS Of dextromethorphan begins With the SyntheSiS Of COmPoUnd F.1 MgS∞h ICO 2 SOCh ------------------------ COmPOUnd A ■ COmPOUnd B 3 M ZMzOIR 3100 Cm t (v9rybΦM1716 Cm ' (3hΛφ)A COmPOUnd D c ∙ - W 2 HZH 2O♦ CoIOUrIeSS gas X π⅞o∕∙ mass of 44 01 g moi 1COnIPOUnd C♦COmPOUnd ECOmPOUnd FDraW the StUCtUreS Of COmPOUndS A. B. C. D. E and g‘COmPOUnd CCOmPOUnd EO δ2019年考题4・ ThiS question is about CarbOn dioxideThe food and drink industries USe a IOt OfCarbon dioxide DUring SUmmer 2018, a globalShOrtage Ied to SUPermarketS Iimiting frozenfood deliveries and ratbning beer. ThiS isironic COnSidering the do∞mented rise Ofatmospheric CO2 IeVeIs.(a) (i) DraW dot and CrOSS diagrams forCarbon dioxide and CartXXI monoxide.(Ii) CalaJlate the difference in the OXIdatiOn State between the CarbOnS in CarbOn dioxide and in CarbOn monoxide・The EngIiSh ChemiSt WiIliam Henry StUdied the equilibria When a gas dissolves in a IiqUid He PrOPOSed Ihat Ihe ConCentratiOn Of a gas dissolved in a IlqUld IS ProPOrt)Onal to the gas' PartiaI PreSSUre When in the gas phase. The PrOPOrti∞alrty fac tor is CaIIed the Henry l S IaW ∞nstant. The Henry S IaW ConStant for 82 is 3.3 × 1(Γ2 mol dm'3 atm"1.SeaIed COntainerS Of fizzy drinks COntain dissolved CQ. ThiS dissolved CO： is in equilibrium With a SmaIl quantity Of gase∞s CQ at the top Of the COn tain er.(b) (i) The Partial PreSSUre Of CO2gas in a 250 αn3 Can Of fizzy drink is 3.0 atm at25 0C. What is the COnCentratiOn Of CO2 in the fizzy drink?(H) What mass Of CO2 is ClISSOIVed in a 250 cm3 Can Of fizzy drink?(III)If the Can c∞ta∣ned Onty the mass Of C O? CaICUIated in Part (O) as a gas. CaIaJlate the PreSSUre in Ihe Can When it is StOred at 25 °C.(IV)Under What ∞nditio∩s WoUId CO2 gas be most SOlUble jn water?TiCk the COrreet OPtiOn in the anSWer b∞klet:•high PreSSUre and IOW temperature•high PreSSUre and high temperature•IoW PreSSUre and IOW temperature•IOW PreSSUre and high temperature2020年考题2. ThiS question is about hydrogen as a fuelCart)On dioxide emissions from fossil fuelsare a major factor in Climate change.Hydrogen is a Potenual alternative tofossil fuels. PrOViding ,dean energy* WrthOnly Water as a byproduct. The UK govemmentis in vestIgatlng COnVeftIng the naturalgas ghd to Carry hydrogen instead.FOr this question・ assume all PrOCeSSeStake PIaCe at 298 K.EnthaIPy Change Of formation Of CH4(g). ∆∕^f = -74.8 kJ moΓ'EnthaIPy Change Of formation Of CO?(g). Δ∕Λ = -393.5 kJ moΓ1EnthaIPy Change Of (Ofmation Of H2O(∣> ∆∕^f ≡ -285 8 kJ mol*1EntroPy Change Of formation Of H2O(I). ∆^e f = -163.0 J K*1moΓ1One IOW COSt method for PrOdUCing hydrogen is reforming methane. ThoUgh this PrOdUCeS CO2, this Can be easily captured. The reforming ProCeSS Can be represented by the OVeraH reaction:CH4(g) ÷ 2H2O(I) -→ CO2(g) + 4H2(g)(a) CalCUIate the enthalpy Change for this reaction.EleCtr OIySiS Of Water is another method Of producing hydrogen On a Iarge SCaIe .it OJrrently ∞sts more than reforming methane.In POIymer electrolyte membrane electrolysts, PfOlOnS are transferred thr∞gh a membrane between Ihe two electrodes. The two half reactions are:1.2H2O(l) → O2(g) + 4Hφ(aq) + 4eβ2.2H∙(aq) ÷ 2e" → H2(g)WhiCh Of these half reactions OCeUrS at the cathode?。

收稿日期：2020-11-25基金项目：福建省自然科学基金(2016J01032)作者简介：程金发(1966-)，男，江西省乐平市人，博士，教授，博士生导师.*通信作者.E-mail ：***************.cn非一致格子上离散分数阶差分与分数阶和分程金发*（厦门大学数学科学学院福建厦门，361005）摘要：众所周知，一致格子上分数阶和分与分数阶差分的思想概念也是最近几年才兴起的，并且在该邻域得到了很大的发展.但是在非一致格子x ()z =c 1z 2+c 2z +c 3或者x ()z =c 1q z +c 2q -z +c 3上，分数阶和分与分数阶差分的定义是什么，这是一个十分复杂和有趣的问题.本文首次提出非一致格子上分数阶和分与Riemann-Liouville 分数阶差分、Caputo 分数阶差分的定义以及非一致格子上广义Abel 积分方程的求解等基础性结果.关键词：超几何差分方程；非一致格子；分数阶和分；分数阶差分；特殊函数中图分类号：33C45；33D45；26A33；34K37文献标志码：A文章编号：2095-7122（2021）01-0001-013On the fractional sum and fractional difference on nonuniform latticesCHENG Jinfa *(School of Mathematical Sciences,Xiamen University,Xiamen,Fujian 361005,China )Abstract:As is well known,the idea of a fractional sum and difference on uniform lattice is more current,and gets a lot of development in this field.But the definitions of fractional sum and fractional difference of f ()z on nonuniform lattices x ()z =c 1z 2+c 2z +c 3or x ()z =c 1q z +c 2q -z +c 3seem much more complicated andinteresting.In this article,for the first time we propose the definitions of the fractional sum and fractional difference on nonuniform lattices.The solution of the generalized Abel equation is obtained etc.Key words:special function;orthogonal polynomials;adjoint difference equation;difference equation of hy-pergeometric type;nonuniform lattice第34卷第1期2021年3月闽南师范大学学报（自然科学版）Journal of Minnan Normal University （Natural Science ）Vol.34Ｎｏ．1Mar.20211背景回顾及问题提出正如我们在本文序言指出的，分数阶微积分的概念几乎与经典微积分同时起步，可以回溯到Euler 和Leibniz 时期.经过几代数学家的努力，特别是近几十年来，分数阶微积分已经取得了惊人的发展和广阔的应用，有关分数阶微积分的著作层出不穷，例如文献[1-4],但是在一致格子x ()z =z 和x ()z =q z 或者q -z ,z ∈C 上关于离散分数阶微积分的思想，仍然是最近才兴起的.虽然关于一致格子x ()z =z 和x ()z =q z 的离散分数微积分出现和建立相对较晚,但是该领域目前已经做出了大量的工作，且取得了很大的发展[5-8].在最近十年的学术著作中，程金发[9],Goodrich 和Peterson [10]相继出版了两本有关离散分数阶方程理论、离散分数微积分的著作，其中全面系统地介绍了离散分数微积分的基本定义和基本定理，以及最新的参考资料.有关q -分数阶微积分方面的著作可参见Annaby 和Mansour [11].非一致格子的定义回溯到超几何型微分方程[12-13]:σ()z y ′′()z +τ()z y ′()z +λy ()z =0，(1)的逼近，这里σ()z 和τ()z 分别是至多二阶和一阶多项式，λ是常数.Nikiforov 等[14-15]将式(1)推广到如下最一般的复超几何差分方程σˉ[]x ()s ΔΔx ()s -12éëêùûú∇y ()s ∇x ()s +12τˉ[]x ()s éëêùûúΔy ()s Δx ()s +∇y ()s ∇x ()s +λy ()s =0,(2)这里σˉ()x 和τˉ()x 分别是关于x ()s 的至多二阶和一阶多项式，λ是常数，Δy ()s =y ()s +1-y ()s ,∇y ()s =y ()s -y ()s -1，并且x ()s 必须是以下非一致格子.定义1[16-17]两类格子函数x ()s 称之为非一致格子，如果它们满足x ()s =-c 1s 2+-c 2s +-c 3,(3)x ()s =c 1q s +c 2q -s +c 3,(4)这里c i ,-c i 是任意常数，且c 1c 2≠0,-c 1-c 2≠0.当c 1=1,c 2=c 3=0，或c 2=1,c 1=c 3=0或者-c 2=1,-c 1=-c 3=0时，这两种格子函数x ()s ：x ()s =s ，(5)x ()s =q s 或x ()s =q -s(6)称之为一致格子.给定函数F ()s ，定义关于x γ()s 的差分或差商算子为∇γF ()s =∇F ()s ∇x γ()s ,且∇k γF ()z =∇∇x γ()z ()∇∇x γ+1()z ⋯()∇()F ()z ∇x γ+k -1()z .()k =1,2,⋯关于差商算子，命题1是常用的.命题1给定两个复函数f ()s ,g ()s ，成立恒等式Δυ()f ()s g ()s =f ()s +1Δυg ()s +g ()s Δυf ()s =g ()s +1Δυf ()s +f ()s Δυg ()s ,Δυ()f ()s g ()s =g ()s +1Δυf ()s -f ()s +1Δυg ()s g ()s g ()s +1=g ()s Δυf ()s -f ()s Δυg ()s g ()s g ()s +1,Δυ()f ()s g ()s =f ()s -1Δυg ()s +g ()s Δυf ()s =g ()s -1Δυf ()s +f ()s Δυg ()s ,(7)Δυ()f ()s g ()s =g ()s -1Δυf ()s -f ()s -1Δυg ()s g ()s g ()s -1=闽南师范大学学报（自然科学版）2021年2g ()s Δυf ()s -f ()s Δυg ()s g ()s g ()s -1.我们必须指出,在非一致格子式(3)或者式(4),即使当n ∈N ，如何建立非一致格子的n -差商公式，也是一件很不平凡的工作，因为它是十分复杂的，也是难度很大的.事实上,在文献[14-15]中,Nikiforov 等利用插值方法得到了如下n -阶差商∇()n 1[]f ()s 公式：定义2[12-13]对于非一致格子式(3)或式(4)，让n ∈N +，那么∇()n 1[]f ()s =∑k =0n ()-1n -k[]Γ()n +1q[]Γ()k +1q[]Γ()n -k +1q×∏l =0n∇x []s +k -()n -12∇x []s +()k -l +12f ()s -n +k =∑k =0n()-1n -k[]Γ()n +1q[]Γ()k +1q[]Γ()n -k +1q×∏l =0n ∇x n +1()s -k ∇x []s +()n -k -l +12f ()s -k ,(8)这里[]Γ()s q 是修正的q -Gamma 函数，它的定义是[]Γ()s q=q -()s -1()s -24Γq ()s ，并且函数Γq ()s 被称为q -Gamma 函数;它是经典Euler Gamma 函数Γ()s 的推广.其定义是Γq ()s =ìíîïïïï∏k =0∞(1-q k +1)()1-q s -1∏k =0∞(1-q s +k),当||q <1;q -()s -1()s -22Γ1q ()s ,当||q >1.(9)经过进一步化简后，Nikiforov 等在文献[14]中将n 阶差分∇()n 1[]f ()s 的公式重写成下列形式:定义3[14]对于非一致格子式(3)或式(4)，让n ∈N +，那么∇()n 1[]f ()s =∑k =0n ()[]-n qk[]k q ![]Γ()2s -k +c q[]Γ()2s -k +n +1+c qf ()s -k ∇x n +1()s -k ,这里[]μq=γ()μ=ìíîïïïïq u2-q -u 2q 12-q -12如果x ()s =c 1q s +c 2q -s +c 3;μ,如果x ()s =-c 1s 2+-c 2s +c 3,(10)且c =ìíîïïïïïïïïlog c 2c 1log q ,当x ()s =c 1q s +c 2q -s +c 3,-c 2-c 1,当x ()s =-c 1s 2+-c 2s +c 3.程金发：非一致格子上离散分数阶差分与分数阶和分第1期3现在存在两个十分重要且具有挑战性的问题需要进一步深入探讨:1）对于非一致格子上超几何差分方程式（2），在特定条件下存在关于x ()s 多项式形式的解，如果用Rodrigues 公式表示的话，它含有整数阶高阶差商.一个新的问题是：若该特定条件不满足，那么非一致格子上超几何差分方程式（2）的解就不存在关于x ()s 的多项式形式，这样高阶整数阶差商就不再起作用了.此时非一致格子超几何方程的解的表达形式是什么呢？这就需要我们引入一种非一致格子上分数阶差商的新概念和新理论.因此,关于非一致格子上α-阶分数阶差分及α-阶分数阶和分的定义是一个十分有趣和重要的问题.显而易见，它们肯定是比整数高阶差商更为难以处理的困难问题，自专著[14-15]出版以来，Nikiforov 等并没有给出有关α-阶分数阶差分及α-阶分数阶和分的定义，我们能够合理给出非一致格子上分数阶差分与分数阶和分的定义吗?2)另外，我们认为作为非一致格子上最一般性的离散分数微积分，它们也会有独立的意义，并可以导致许多有意义的结果和新理论.本文的目的是探讨非一致格子上离散分数阶和差分.受文章篇幅所限，本文我们仅合理给出非一致格上分数阶和分与分数阶差分的基本定义，其它更多结果例如：非一致格子离散分数阶微积分的一些基本定理，如:Euler Beta 公式，Cauchy Beta 积分公式，Taylor 公式、Leibniz 公式在非一致格子上的模拟形式，非一致格子上广义Abel 方程的解，以及非一致格子上中心分数差分方程的求解，离散分数阶差和分与非一致格子超几何方程之间联系等内容，请参见笔者新专著[16].2非一致格子上的整数和分与整数差分设x ()s 是非一致格子，这里s ∈ℂ.对任意实数γ，x γ()s =x ()s +γ2也是一个非一致格子.让∇γF ()s =f ()s .那么F ()s -F ()s -1=f ()s []x γ()s -x γ()s -1.选取z ,a ∈ℂ，和z -a ∈N .从s =a +1到z ，则有F ()z -F ()a =∑s =a +1zf ()s ∇x r()s .因此，我们定义∫a +1z f ()s d ∇x γ()s =∑s =a +1zf ()s ∇xγ()s .容易直接验证下列式子成立.命题2给定两个复变函数F ()z ,f ()z ，这里复变量z ,a ∈C 以及z -a ∈N ，那么成立1）∇γéëêùûú∫a +1zf ()s d ∇x γ()s =f ()z ;2）∫a +1z∇γF ()s d ∇x γ()s =F ()z -F ()a .现在让我们定义非一致格子上的广义n -阶幂函数[]x ()s -x ()z ()n 为[]x ()s -x ()z ()n =∏k =0n -1[]x ()s -x ()z -k ,()n ∈N +,当n 不是正整数时，需要将广义幂函数加以进一步推广，它的性质和作用是非常重要的，非一致格子上广义幂函数[]x γ()s -x γ()z ()α的定义如下：闽南师范大学学报（自然科学版）2021年4定义4[17-18]设α∈C ，广义幂函数[]x γ()s -x γ()z ()α定义为[]x γ()s -x γ()z ()α=ìíîïïïïïïïïïïïïïïïïïïïïΓ()s -z +a Γ()s -z ,如果x ()s =s ,Γ()s -z +a Γ()s +z +γ+1Γ()s -z Γ()s +z +γ-α+1,如果x ()s =s 2,()q -1αq α()γ-α+12Γq ()s -z +αΓq ()s -z ,如果x ()s =q s ,12α()q -12αq -α()s +γ2Γq ()s -z +αΓq ()s +z +γ+1Γq ()s -z Γq ()s +z +γ-α+1,如果x ()s =q s +q -s 2.(11)对于形如式(4)的二次格子，记c =-c 2-c 1，定义[]x γ()s -x γ()z ()α=-c 1αΓ()s -z +a Γ()s +z +γ+c +1Γ()s -z Γ()s +z +γ-α+c +1；(12)对于形如式(3)的二次格子，记c =logc 2c 1log q，定义[]xγ()s -x γ()z ()α=éëùûc 1()1-q 2αq -α()s +γ2Γq()s -z +a Γq()s +z +γ+c +1Γq()s -z Γq()s +z +γ-α+c +1,(13)这里Γ()s 是Euler Gamma 函数，且Γq ()s 是Euler q -Gamma 函数，其定义如式(9).命题3[17-18]对于x ()s =c 1q s +c 2q -s +c 3或者x ()s =-c 1s 2+-c 2s +-c 3，广义幂[]x γ()s -x γ()z ()α满足下列性质：[]x γ()s -x γ()z []x γ()s -x γ()z -1()μ=[]x γ()s -x γ()z ()μ[]xγ()s -x γ()z -μ=(14)[]xγ()s -x γ()z ()μ+1；(15)[]xγ-1()s +1-x γ-1()z ()μ[]xγ-μ()s -x γ-μ()z =[]x γ-μ()s +μ-x γ-μ()z []x γ-1()s -x γ-1()z ()μ=[]x γ()s -x γ()z ()μ+1；(16)ΔzΔx γ-μ+1()z []xγ()s -x γ()z ()μ=-∇s∇x γ+1()s []x γ+1()s -x γ+1()z ()μ=(17)-[]μq []x γ()s -x γ()z ()μ-1；(18)∇z∇x γ-μ+1()z {}1[]xγ()s -x γ()z ()μ=-ΔsΔx γ-1()s ìíîïïüýþïï1[]x γ-1()s -x γ-1()z ()μ=(19)[]μq[]xγ()s -x γ()z ()μ+1(20)这里[]μq 定义如式（10）.程金发：非一致格子上离散分数阶差分与分数阶和分第1期5现在让我们详细给出非一致格子x γ()s 上整数阶和分的定义，这对于我们进一步给出非一致格子x γ()s 上分数阶和分的定义是十分有帮助的.设γ∈R ，对于非一致格子x γ()s ，数集{}a +1,a +2,⋯,z 中f ()z 的1-阶和分定义为y 1()z =∇-1γf ()z =∫a +1z f ()s d ∇x γ()s ，（21）这里y 1()z =∇-1γf ()z 定义在数集{}a +1,mod ()1中.那么由命题2，我们有∇1γ∇-1γf ()z =∇y 1()z ∇x γ()z =f ()z ，（22）并且对于非一致格子x γ()s ，数集{}a +1,a +2,⋯,z 中f ()z 的2-阶和分定义为y 2()z =∇-2γf ()z =∇-1γ+1[]∇-1γf ()z =∫a +1z y 1()s d ∇x γ+1()s =∫a +1z d ∇x γ+1()s ∫a +1s f ()t d ∇x γ()t =∫a +1z f ()t d ∇x γ()t ∫tz d ∇x γ+1()s =∫a +1z []x γ+1()z -x γ+1()t -1f ()s d ∇x γ()s .(23)这里y 2()z =∇-2γf ()z 定义在数集{}a +1,mod ()1中.同时，可得∇1γ+1∇1γ-1y 1()z =∇y 2()z ∇x γ+1()z =y 1()z ，∇2γ∇-2γf ()z =∇∇x γ()z ()∇y 2()z ∇x γ+1()z =∇y 1()z ∇x γ()z =f ()z .（24）更一般地，由数学归纳法，对于非一致格子x γ()s ，数集{}a +1,a +2,⋯,z 中函数f ()z ，我们可以给出函数f ()z 的n -阶和分定义为y k ()z =∇-kγf ()z =∇-1γ+k -1[]∇-()k -1γf ()z =∫a +1z y k -1()s d ∇x γ+k -1()s =1[]Γ()k q∫a +1z []xγ+k -1()z -x γ+k -1()t -1()k -1f ()t d ∇x γ()t ,()k =1,2,⋯(25)这里[]Γ()k q=ìíîïïq -()k -1()k -2Γq ()k ,如果x ()s =c 1q s +c 2q -s +c 3;Γ()α,如果x ()s =-c 1s 2+-c 2s +c 3,这满足下式[]Γ()k +1q=[]k q []Γ()k q ,[]Γ()2q =[]1q []Γ()1q =1.那么成立∇kγ∇-k γf ()z =∇∇x γ()z ()∇∇x γ+1()z ⋯()∇y k ()z ∇x γ+k -1()z =f ()z .()k =1,2,⋯（26）需要指出的是，当k ∈C 时，式（25）右边仍然是有意义的，因此自然地，我们就可以对非一致格子x γ()s 闽南师范大学学报（自然科学版）2021年6给出函数f ()z 的分数阶和分定义如下：定义5（非一致格子分数阶和分）对任意Re α∈R +，对于非一致格子式（3）和式（4），数集{}a +1,a +2,⋯,z 中的函数f ()z ，我们定义它的α-阶分数阶和分为∇-αγf ()z =1[]Γ()αq∫a +1z []xγ+α-1()z -x γ+α-1()t -1()α-1f ()s d ∇x γ()s ，（27）这里[]Γ()αq=ìíîïïq -()s -1()s -2Γq ()α,如果x ()s =c 1q s +c 2q -s +c 3;Γ()α,如果x ()s =-c 1s 2+-c 2s +c 3,这满足下式[]Γ()α+1q=[]αq []Γ()αq .3非一致格子上的Abel 方程及分数阶差分非一致格子x γ()s 上f ()z 的分数阶差分定义相对似乎更困难和复杂一些.我们的思想是起源于非一致格子上广义Abel 方程的求解.具体来说，一个重要的问题是：让m -1<Re α≤m ，定义在数集{}a +1,a +2,⋯,z 的f ()z 是一给定函数，定义在数集{}a +1,a +2,⋯,z 的g ()z 是一未知函数，它们满足以下广义Abel 方程∇-αγg ()z =∫a +1z []x γ+α-1()z -x γ+α-1()t -1()α-1[]Γ()αqg ()t d ∇x γ()t =f ()t ，（28）怎样求解该广义Abel 方程式（28）？为了求解方程式（28），我们需要利用重要的Euler Beta 公式在非一致格子下的基本模拟.定理1[16]（非一致格子上Euler Beta 公式）对于任何α,β∈C ，那么对非一致格子x ()s ，我们有∫a +1z []x β()z -x β()t -1()β-1[]Γ()βq[]x ()t -x ()αα[]Γ()α+1qd ∇x 1()t =[]x β()z -x β()α()α+β[]Γ()α+β+1q.（29）定理2（Abel 方程的解）设定义在数集{}a +1,mod ()1中的函数f ()z 和函数g ()z 满足∇-αγg ()z =f ()z ,0<m -1<Re α≤m ，那么g ()z =∇m γ∇-m +αγ+αf ()z （30）成立.证明我们仅需证明∇-m γg ()z =∇-m +αγ+αf ()z ，即∇-()m -αγ+αf ()z =∇-()m -αγ+α∇-αγg ()z =∇-m γg ()z .事实上，由定义5可得程金发：非一致格子上离散分数阶差分与分数阶和分第1期7∇-()m -αγ+af ()z =∫a +1z []xγ+m -1()z -x γ+m -1()t -1()m -α-1[]Γ()m -αqf ()t d ∇x γ+α()t =∫a +1z []x γ+m -1()z -x γ+m -1()t -1()m -α-1[]Γ()m -αqd ∇x γ+α()t ⋅∫a +1z []xγ+α-1()t -x γ+α-1()s -1()α-1[]Γ()αqg ()s d ∇x γ()s =∫a +1zg ()s ∇x γ()s ∫sz []xγ+m -1()z -x γ+m -1()t -1()m -α-1[]Γ()m -αq⋅[]xγ+α-1()t -x γ+α-1()s -1()α-1[]Γ()αqd ∇x γ+α()t .在定理1中，将α+1替换成s ；α替换成α-1；β替换成m -α，且将x ()t 替换成x γ+α-1()t ，那么x β()t 替换成x γ+m -1()t ，则我们能够得出下面的等式∫sz []xγ+m -1()z -x γ+m -1()t -1()m -α-1[]Γ()m -αq[]xγ+α-1()t -x γ+α-1()s -1()α-1[]Γ()αqd ∇x γ+α()t =[]xγ+m -1()z -x γ+m -1()s -1()-m -1[]Γ()m q,因此，我们有∇-()m -αγ+af ()z =∫a +1z []x γ+m -1()z -x γ+m -1()s -1()-m -1[]Γ()m qg ()s d ∇x γ()s =∇-mγg ()z ,这样就有∇m γ∇-()m -αγ+a f ()z =∇m γ∇-m γg ()z =g ()z .由定理2得到启示，很自然地我们给出关于f ()z 的Riemann-Liouville 型α-阶()0<m -1<Re α≤m 分数阶差分的定义如下：定义6（Riemann-Liouville 分数阶差分）让m 是超过Re α的最小正整数，对于非一致格子x γ()s ，数集{}α,mod ()1中f ()z 的Riemann-Liouville 型α-阶分数阶差分定义为∇αγf ()z =∇m γ()∇α-mγ+αf ()z .（31）形式上来说，在定义5中，如果α替换成-α，那么式（27）的右边将变为∫a +1z []xγ-α-1()z -x γ-α-1()t -1()-α-1[]Γ()-αqf ()t d ∇x γ()t =∇∇x γ-α()t ()∇∇x γ-α+1()t ⋯()∇∇x γ-α+n -1()t ⋅∫a +1z[]xγ+n -α-1()z -x γ+n -α-1()t -1()n -α-1[]Γ()n -αqf ()t d ∇x γ()t =∇n γ-α∇-n +αγf ()z =∇αγ-αf ()z .（33）闽南师范大学学报（自然科学版）2021年8从式（33），我们也可以得到f ()z 的Riemann-Liouville 型α-阶分数阶差分如下:定义7（Riemann-Liouville 型分数阶差分2）对任意Re α>0，对于非一致格子x γ()s ，数集{}a +1,a +2,⋯,z 中f ()z 的Riemann-Liouville 型α-阶分数阶差分定义为∇αγ-αf ()z =∫a +1z x γ-α-1()z -x γ-α-1()t -1()-α-1[]Γ()-αqf ()t d ∇x γ()t ，（34）将∇γ-α()t 替换成∇γ()t ，那么∇αγf ()z =∫a +1z []x γ-1()z -x γ-1()t -1()-α-1[]Γ()-αqf ()t d ∇x γ+α()t ,（35）这里假定[]Γ()-αq ≠0.4非一致格子上Caputo 型分数阶差分在本节，我们将给出非一致格子上Caputo 型分数阶差分的合理定义.定理3（分部求和公式）给定两个复变函数f (s )，g (s )，那么∫a +1z g (s )∇γf (s )d ∇x γ(s )=f (z )g (z )-f (a )g (a )-∫a +1z f (s -1)∇γg (s )d ∇x γ(s )，这里z ,a ∈C ，且假定z -a ∈N .证明应用命题1，可得g (s )∇γf (s )=∇γ[f (z )g (z )]-f (s -1)∇γg (s )，这样就有g (s )∇r f (s )=∇r [f (z )g (z )]-f (s -1)∇r g (s ).关于变量s ，从a +1到z 求和，那么可得∫a +1z g (s )∇γf (s )d ∇x γ(s )=∫a +1z ∇γ[f (z )g (z )]∇x γ(s )-∫a +1z f (s -1)∇γg (s )d ∇x γ(s )=f (z )g (z )-f (a )g (a )-∫a +1z f (s -1)∇γg (s )d ∇x γ(s ).与非一致格子上Riemann-Liouville 型分数阶差分定义的思想来源一样，对于非一致格子上Caputo 型分数阶差分定义思想，也是受启发于非一致格子上广义Abel 方程式（28）的解.在本文第3节，借助于非一致格子上的Euler Beta 公式，我们已经求出广义Abel 方程∇-αγg (z )=f (z )，0<m -1<α≤m ，是g (z )=∇αγf (z )=∇m γ∇-m +αγ+αf (z ).（36）现在我们将用分部求和公式，给出式（36）的另一种新的表达式.事实上，我们有∇a γf (z )=∇m γ∇-m +aγ+a f (z )=∇mγ∫a +1z[x γ+m -1(z )-x γ+m -1(s -1)](m -α-1)[Γ(m -α)]qf (s )d ∇x γ+α(s ).（37）应用恒等式∇(s )[x γ+m -1(z )-x γ+m -1(s )](m -α)∇x γ+α(s )=∇(s )[x γ+m -1(z )-x γ+m -1(s -1)](m -α)∇x γ+α(s -1)=-[m -α]q [x γ+m -1(z )-x γ+m -1(s -1)](m -α-1)，那么以下表达式程金发：非一致格子上离散分数阶差分与分数阶和分第1期9∫a +1z[x γ+m -1(z )-x γ+m -1(s -1)](m -α-1)[Γ(m -α)]qf (s )d ∇x γ+α(s )，可被改写成∫a +1zf (s )∇(s ){-[x γ+m -1(z )-x γ+m -1(s )](m -α)[Γ(m -α+1)]q}d ∇s =∫a +1z f (s )∇γ+α-1{-[x γ+m -1(z )-x γ+m -1(s )](m -α)[Γ(m -α+1)]q}d ∇x γ+α-1(s ).应用分部求和公式，可得∫a +1zf (s )∇γ+α-1{-[x γ+m -1(z )-x γ+m -1(s )](m -α)[Γ(m -α+1)]q}d ∇x γ+α-1(s )=f (a )[x γ+m -1(z )-x γ+m -1(a )](m -α)[Γ(m -α+1)]q+∫a +1z[x γ+m -1(z )-x γ+m -1(s -1)](m -α)[Γ(m -α+1)]q∇γ+α-1[f (s )]d ∇x γ+α-1(s ).因此，这可导出∫a +1z[x γ+m -1(z )-x γ+m -1(s -1)](m -α-1)[Γ(m -α)]q}f (s )d ∇x γ+α(s )=f (a )[x γ+m -1(z )-x γ+m -1(a )](m -α)[Γ(m -α+1)]q+∫a +1z[x γ+m -1(z )-x γ+m -1(s -1)](m -α)[Γ(m -α+1)]q∇γ+α-1[f (s )]d ∇x γ+α-1(s ).(38)进一步，考虑∫a +1z[x γ+m -1(z )-x γ+m -1(s -1)](m -α)[Γ(m -α+1)]q∇γ+α-1[f (s )]d ∇x γ+α-1(s )，(39)利用恒等式∇(s )[x γ+m -1(z )-x γ+m -1(s )](m -α+1)∇x γ+α-1(s )=∇(s )[x γ+m -1(z )-x γ+m -1(s -1)](m -α+1)∇x γ+α-1(s -1)=-[m -α+1]q [x γ+m -1(z )-x γ+m -1(s -1)](m -α)，表达式(39)能被改写成∫a +1z∇γ+α-1[f (s )]∇(s ){-[x γ+m -1(z )-x γ+m -1(s -1)](m -α+1)[Γ(m -α+2)]q}d ∇s =∫a +1z∇γ+α-1[f (s )]∇γ+α-2{-[x γ+m -1(z )-x γ+m -1(s -1)](m -α+1)[Γ(m -α+2)]q}d ∇x γ+α-2(s ).由分部求和公式，我们有∫a +1z ∇γ+α-1[f (s )]∇γ+α-2{-[x γ+m -1(z )-x γ+m -1(s -1)](m -α+1)[Γ(m -α+2)]q}d ∇x γ+α-2(s )=∇γ+α-1f (a )[x γ+m -1(z )-x γ+m -1(a )](m -α+1)[Γ(m -α+2)]q +∫a +1z[x γ+m -1(z )-x γ+m -1(s -1)](m -α+1)[Γ(m -α+2)]q[∇γ+α-2∇γ+α-1]f (s )d ∇x γ+α-2(s )=闽南师范大学学报（自然科学版）2021年10∇γ+α-1f (a )[x γ+m -1(z )-x γ+m -1(a )](m -α+1)[Γ(m -α+2)]q+∫a +1z[x γ+m -1(z )-x γ+m -1(s -1)](m -α+1)[Γ(m -α+2)]q∇2γ+α-2f (s )d ∇x γ+α-2(s ).因此，我们得到∫a +1z[x γ+m -1(z )-x γ+m -1(s -1)](m -α)[Γ(m -α+1)]q∇γ+α-1[f (s )]d ∇x γ+α-1(s )=∇γ+α-1f (a )[x γ+m -1(z )-x γ+m -1(a )](m -α+1)[Γ(m -α+2)]q+∫a +1z[x γ+m -1(z )-x γ+m -1(s -1)](m -α+1)[Γ(m -α+2)]q∇2γ+α-2f (s )d ∇x γ+α-2(s ).(40)同理，用数学归纳法，我们可得∫a +1z[x γ+m -1(z )-x γ+m -1(s -1)](m -α+k -1)[Γ(m -α+k )]q∇kγ+α-k [f (s )]d ∇x γ+α-k (s )=∇kγ+α-kf (a )[x γ+m -1(z )-x γ+m -1(a )](m -α+k )[Γ(m -α+k +1)]q+∫a +1z[x γ+m -1(z )-x γ+m -1(s -1)](m -α+k )[Γ(m -α+k +1)]q∇k +1γ+α-(k +1)f (s )d ∇x γ+α-(k +1)(s ).(k =0,1,⋯,m -1)(41)将式（38）,（40）和（41）代入式（37），则有∇αγf ()z =∇m γìíîïïf ()a []x γ+m -1()z -x γ+m -1()a ()m -α[]Γ()m -α+1q +∇γ+α-1f ()a []xγ+m -1()z -x γ+m -1()a ()m -α+1[]Γ()m -α+2q+∇kγ+α-kf ()a []x γ+m -1()z -x γ+m -1()a ()m -α+k []Γ()m -α+k +1q+⋯+∇m -1γ+α-()m -1f ()a []x γ+m -1()z -x γ+m -1()a ()2m -α-1[]Γ()2m -αq+üýþïï∫a +1z []xγ+m -1()z -x γ+m -1()s -1()2m -α-1[]Γ()2m -αq∇m γ+α-mf ()s d ∇x γ+α-m ()s =∇m γ{}∑k =0m -1∇kγ+α-kf ()a []x γ+m -1()z -x γ+m -1()a ()m -α+k []Γ()m -α+k +1q+∇α-2m γ+α-m ∇mγ+α-m f ()z =∑k =0m -1∇kγ+α-kf ()a []x γ-1()z -x γ-1()a ()-α+k []Γ()-α+k +1q+∇α-m γ+α-m ∇mγ+α-m f ()z .总之，我们有下面的程金发：非一致格子上离散分数阶差分与分数阶和分第1期11定理4（广义Abel 方程解2）假设定义在数集{}a +1,a +2,⋯,z 上的函数f ()z 和g ()z 满足∇-αγg ()z =f ()z ,0<m -1<Re α≤m ,那么g ()z =∑k =0m -1∇k γ+α-kf ()a []xγ-1()z -x γ-1()a ()-α+k []Γ()-α+k +1q+∇α-m γ+α-m ∇mγ+α-m f ()z .受到定理4的启示，我们很自然地给出函数f ()z 的α-阶()0<m -1<Re α≤m Caputo 分数阶差分如下：定义8（Caputo 分数阶差分）让m 是Re α超过的最小整数，非一致格子上定义在数集{}a +1,a +2,⋯,z 函数f ()z 的α-阶Caputo 分数阶差分定义为C∇αγf ()z =∇α-m γ+α-m ∇mγ+α-m f ()z .最后，本文再强调指出：对于非一致格子上超几何差分方程式（2），在特定条件下存在关于x ()s 多项式形式的解，如果用Rodrigues 公式表示的话，它含有整数阶高阶差分.一个重要的问题是：若该特定条件不满足，那么非一致格子超几何差分方程的解就不存在关于x ()s 的多项式形式，这样高阶整数阶差分将不再起作用了，这就迫切需要我们引入一种非一致格子上分数阶差分的新概念和新理论.因此，关于非一致格子上阶分数阶差分及阶分数阶和分的定义是一个十分有趣和重要的问题.有关非一致格子超几何差分方程与离散分数阶差和分的联系，更深入的内容参见笔者著作[16]及文献[19-21].(42)(43)参考文献：[1]Kilbas A 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theory[M].New York:Springer-Verlag,1992:1-35.[18]Suslov S K.On the theory of difference analogues of special functions of hypergeo-metric type[J].Russian Math Surveys,1989,44:227-278.[19]Cheng J F,Jia L K.Generalizations of rodrigues type formulas for hypergeometric difference equations on nonuniform[J].Journal of Difference Equations and Applications,2020,26(4):435-457.[20]Cheng J F,Dai W Z.Adjoint difference equation for a Nikiforov-Uvarov-Suslov difference equation of hypergeometric typeon non-uniform Lattices[J].Ramanujan Journal,2020,53:285-318.[21]Cheng J F.On the complex difference equation of hypergeometric type on non-uniform lattices[J].Acta Mathematical Sinica,English Series,2020,36(5):487–511.[责任编辑：钟国翔]程金发：非一致格子上离散分数阶差分与分数阶和分第1期13。

a r X i v :g r-qc/125v12Feb21Einstein’s Field Equations for the Interior of a Uniformly Rotating Stationary Axisymmetric Perfect Fluid E.Kyriakopoulos Department of Physics National Technical University 15780Zografou,Athens,GREECE Abstract We reduce Einstein’s field equations for the interior of a uniformly rotating,axisymmetric perfect fluid to a system of six second order partial differential equations for the pressure p the energy density µand four dependent variables.Four of these equations do not depend on p and µand the other two determine p and µ.PACS number(s):04.20.Jb,04.20.-q 1Introduction The number of solutions of Einstein’s field equations for the interior of a uniformly rotating stationary axisymmetric perfect fluid is very limited con-trary to what happens in the stationary axisymmetric vacuum case [1].This is mainly due to the fact that the equations we have to solve in the first case are complicated [2],while in the second case they have been reduced to the equivalent and much simpler Ernst’s equation [3].Reduction of the”interior”problem is known for Einstein-Mawxell field [4]and dust [5].Also Einstein’s field equations for the interior of a uniformly rotating stationary axisymmetric perfect fluid have been reduced to a system of two second order partial differential equations for two unknown functions,which however are very complicated [6].In this work the six Einstein’s field equations for the interior of a uniformly rotating stationary axisymmetric perfect fluid,which depend on the pressure p the energy density µof the fluid and on four dependent variables,will be divided into a set of four equations which depend on the four dependent variables but not on p and µand a set of two equations which give m p and µas functions of the four dependent variables.Therefore solving the1first system of equations we can determine p and µalgebraically using the second set.Introducing a potential and redefining variables we write the first system in a relatively simple form,which is very convenient for finding the B¨a cklund transformations the equations of the system may posses.Also we eliminate one of the dependent variables of the system increasing however the complexity of the problem.2The Field EquationsThe line element of stationaryaxisymmetric fields admitting 2-spaces orthog-onal to the Killing vectors ξ=∂t and η=∂φcan be written in the form [7],[1]ds 2=e −2U {e 2K (dρ2+dz 2)+F 2dφ2}−e 2U (dt +Adφ)2(1)where U =U (ρ,z ),K =K (ρ,z ),F =F (ρ,z )and A =A (ρ,z ).The decomposition of the metric into orthogonal 2-spaces whose points are labeled by ρand z is possible for perfect fluid solutions provided that the 4-velocity of the fluid satisfies the circularity condition u [a ξb ηc ]=0[1].For a perfect fluid source we have the energy-momentum tensorT ab =(µ+p )u a u b +pg ab (2)where u a are the components of the 4-velocity and µand p are the mass density and the pressure of the fluid respectively.We shall introduce the notation∂ρ=∂∂z,∂=1∂=1∂F =pF e 2K −2U (4)2∂F (∂U ∂U∂F )+1∂A =1∂A −1∂F +∂U +∂F ∂∂U 2F e 4U ∂A =0(7)∂∂U +1∂A =1Also from the conservation relation T ab;b=0we get equations∂p+(µ+p)∂U=0(9) which are obtained from Eq.(4)-(8)and will be omitted in the following.Eqs(4),(5)and(8)are equivalent to the systemp=2∂F e2U−2K(10)µ=e2U−2K{8∂F (∂U∂U∂F)+4∂A−12∂∂U−3∂A}(11)4∂∂U+1∂A−2∂F=0(12)We shall replace Eqs(4),(5)and(8)by Eqs(10)-(12).If we putA=−ω,e2U=T(13) Eq(6)takes the form∂(T2∂ω)+F∂ω)=0or∂ρ(T2Fωz)=0(14)where a letter as an index in a function means differentiation with respect to the corresponding variable e.g.ωρ=∂ωF ∂ω=i∂φ(15)orωρ=FT2φρ(16)whereφis an arbitrary function.Also we have T{φρρ+φzz+1F {∂ρ(FT2φz)}(17) 3But if Eqs(16)are satisfied the above relation vanish.Therefore we can substitute Eq(14)by the relationT{φρρ+φzz+12(E+F∇F· ∇E)− ∇E· ∇E}=ImΛ=0(20)2∂K−∂F−2FE)2∂∂K−1∂F+1E)2(∂E E+E)=0(22)p=1E)e−2K∂E+E+2(E+F∇F· ∇E)− ∇E· ∇E(26)N=6(E+∂K+3E(∂E E+E)=3pe2K(27)The expressions ImΛand ReΛare the imaginary part and the real part of Λrespectively,andˆρandˆz are the unit vectors in the direction of theρand z axis respectively.Also in deriving Eq(27)we have used Eqs(22)and(23). Thus we have reduced the solution of the problem to the solution of Eqs(20) -(22),which form a system of four equations for the four unknowns F,K, and E=T+iφ.Having solved this system we can use Eqs(23)and(24)to calculate p andµ.Our system of equations(20)-(24)is simplified considerably in the vac-uum case.In this case we have p=µ=0and we can introduced Weyl’s4canonical coordinates with F=ρ.Then Eq(23)is satisfied and Eq(24) implies the relation ReΛ=0.This combined with Eq(20)givesΛ=1E)(∇2E+14T2(T2ρ+φ2ρ−T2z−φ2z),K z=ρF∂a∂b F=∂a∂b G+∂a G∂b G(31)Then Eqs(21)and(22)become∂G+∂G−2∂K+2E)2∂E=0(32)2∂∂G−∂G(E+∂∂E∂2T2(T2η+φ2η)=0(36) Mηξ−GηGξ+12(Gηφξ+Gξφη)−1Also we can easily write Eqs(23)and(24)in the variables introduced above.One way offinding solutions is to solve the system of four Eqs(36)-(38) with four unknowns and then use Eqs(23)and(24)to get p andµ.This form of the system of equations i.e.four equations without p andµand two giving p andµin term of the four dependent variables is very convenient in the search for B¨a cklund transformations[8].Of course one can approach the problem in a different way.3Another Form of the SystemWe shall write now the system we want to solve in a different way.Eqs(32) and(33)become in the variablesξandηof Eqs(34)Gηη+G2η−2GηKη+2E)2Eη(E+Eξ+EξN since G is real.Then solving Eq(39)for Kηand using the resulting expression to eliminate Kηfrom Eq(40)we getKη=Nη2+1E)2NEη∂ξ{Nη(E+Eη}=N(E+Eη+Eη2(NEξ+E+since G is real.We can show that any complex number N which satisfies the first of Eqs(45)is of the form of Eq(41)with G real.Also Eq(43)implies the relationImKηξ=0(46) which means that Kρand K z coming from Eqs(42)satisfy the integrability condition Kρz=K zρ.Using Eq(44)we can write Eq(43)in the form(NηN+2(E n+N(E+2(NEξ+ E+(E+N2+N)Eη。

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 ofstrain 应变协调方程equation of straincompatibility 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Kolosoff-利法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 plasticMaterial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 bendingMoment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 plasticMaterial 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 admissibleField 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Haigh-Westergaard 间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 velocityEquation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 eliminationMethod 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 fractureMechanics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 fracturemecha-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 singularityField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-integral J阻力曲线J-resistance curve断裂韧度fracture toughness应力强度因子stress intensity factor HRR场Hutchinson-Rice-RosengrenField 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 transitiontempe- rature 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 tensionspecimen, 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, crackPropagation 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-temperaturecompensating gage半导体应变计semiconductor strainGage 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 coatingMethod光弹性夹片法photoelastic sandwichMethod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 birefringentEffect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 birefringentEffect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 variationalPrinciple 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-ReissnerPrinciple 修正变分原理modified variationalPrinciple 约束变分原理constrained variationalPrinciple 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。