2012 Finite difference methods for fractional differential equations(分数阶扩散方程)

合集下载

温度曲线

Modeling of simultaneous heat and mass transfer during convective oven ring cake bakingMelike Sakin-Yilmazer a ,⇑,Figen Kaymak-Ertekin a ,Coskan Ilicali ba Ege University,Faculty of Engineering,Department of Food Engineering,35100Bornova,Izmir,TurkeybKyrgyzstan-Turkey Manas University,Faculty of Engineering,Department of Food Engineering,720044Bishkek,Kyrgyzstana r t i c l e i n f o Article history:Received 11November 2011Received in revised form 2February 2012Accepted 11February 2012Available online 20February 2012Keywords:Ring cakeBaking modeling SimulationFinite difference method Heat and mass transfera b s t r a c tThe convective oven ring cake baking process was investigated experimentally and numerically as a simultaneous heat and mass transfer process.The mathematical model described previously by the authors for cup cake baking was modiﬁed to simulate ring cake baking.The heat and mass transfer mech-anisms were deﬁned by Fourier’s and Fick’s second laws,respectively.The implicit alternating direction ﬁnite difference technique was used for the numerical solution of the representative model.Prior to the utilization of the developed model in predicting the temperature and moisture proﬁles for ring cake bak-ing,the results of the numerical model were compared with analytical results involving only heat or mass transfer with constant thermo-physical properties.Excellent agreement was observed.The numerical temperature and moisture contents predicted by the model were compared with the experimental pro-ﬁles.They agreed generally reasonably well with the experimental temperature and moisture proﬁles.Ó2012Elsevier Ltd.All rights reserved.1.IntroductionBaking is a complex process since the physical,chemical and biochemical changes taking place are of complex nature and these changes are simultaneous and coupled.Quantitative understand-ing of processes like volume expansion,evaporation of water,for-mation of a porous structure,denaturation of protein,gelatinization of starch,crust formation and browning reaction are still limited (Sablani et al.,1998;Zhang and Datta,2006).Sim-pliﬁed models for the baking process can be developed by treating baking as simultaneous heat and moisture transfer.Even for such simpliﬁed models,the estimation of the thermo-physical and transport properties,and the transfer coefﬁcients is a challenging task.Nevertheless,mathematical models supply valuable informa-tion on baking process which may be utilized to optimize the bak-ing conditions to reduce energy consumption.As a traditional process,optimization and process control to ob-tain the desired ﬁnal product quality is still largely based on expe-rience and good craftsmanship rather than on predictive calculations (Feyissa et al.,2011).Experimental and theoretical studies for baking in general and cake baking in particular were gi-ven in details in Sakin (2005)and Sakin et al.(2007b)and will not be repeated here.Mathematical models representing the baking process reduce the need for the tedious trial and error baking experiments to achieve a high quality product and they serve as a quick and prac-tical tool for pre-design,optimization and process control (Sakin et al.,2007b ).Ring cake is a popular cake geometry all over the world.Selecting this geometry as the test geometry and developing accurate mathematical models will have practical implications to-wards optimizing the baking conditions for better quality and en-ergy saving.Since there is no published work in literature about ring cake baking and its modeling,a two dimensional simpliﬁed mathemat-ical model for simultaneous heat and moisture transfer will be developed.Volume rise during baking will be considered.The pre-dictions of the numerical model will be compared with experimen-tal temperature and moisture content proﬁles.2.Material and experimentalThe cake batter was prepared by thoroughly mixing of 49.4%(of total weight)ready to bake dry cake mix (Dr.Oetker,containing wheat ﬂour,sugar,corn starch,and baking powder),24.7%pasteur-ized whole liquid egg,16.2%vegetable margarine and 9.7%water.Its preparation was given in details in Sakin et al.(2007b).The ini-tial moisture content of the batter was %0.50kg water/kg dry solid.Teﬂon coated Aluminum was chosen as the material for ring cake mould due to Aluminum’s high thermal conductivity value (206W/mK,Geankoplis,2003).The ring baking mould was an annular ring having an annular radius of 0.04m,radius of 0.11and a height of 0.05m.Baking experiments were carried out in an electrical baking oven (Teba High-01Inox)which was0260-8774/$-see front matter Ó2012Elsevier Ltd.All rights reserved.doi:10.1016/j.jfoodeng.2012.02.020⇑Corresponding author.Tel.:+902323113039;fax:+902323427592.E-mail addresses:melike.sakin@.tr (M.Sakin-Yilmazer),ﬁgen.ertekin@.tr (F.Kaymak-Ertekin),coskan.ilicali@ (C.Ilicali).0.39Â0.44Â0.35m(LÂWÂH)wide.The batter was baked at three different oven temperatures(165,185and205°C)under forced convection conditions.The oven was preheated before the baking experiment to obtain uniform oven baking conditions.The air was circulated by a fan in-stalled on the back side of the oven at a constant speed(0.56m/s, measured by Airﬂow anemometer,LCA6000,UK)and fresh air en-tered steadily into the oven cavity,through a hole(/:28mm)up in the oven to prevent moisture accumulation.Moisture and temperature proﬁles were recorded during baking experiments.The procedures for taking temperature and moisture content data were similar to that given in Sakin et al.(2007b). Moisture proﬁles were measured at the top,bottom,annular and side surface crusts as two parallel measurements.An infrared moisture analyzer(Ohaus,MB200,USA),with a precision of ±0.007g,was used for the moisture content determination of the samples.The results obtained by the instrument were previously validated by the standard air oven method for total solids and moisture in baked products,as1h at130°C(AOAC,1995).Temperature proﬁles were recorded at the top(r=0.063m), bottom(r=0.08m)and annular(z=0.015m)surfaces by using j-type thermocouples(wire size,/:1mm;accuracy:1–1.5°C)that were calibrated in water and ice baths against glass mercury ther-mometer,and by a multi-channel data-logger(Hanna Inst.,HI 98804,Portugal)having an accuracy of±0.5°C,excluding probe er-ror.All the temperature measurements were conducted as two parallels.Techniques used in thermocouple positioning were sim-ilar to Sakin et al.(2007b).The change in the height of the cake batter was measured by a digital caliper(Mitutoyo,Digimatic,Japan).3.Mathematical model descriptionSimultaneous heat and moisture transfer mechanisms in a ring cake during baking can be described by the two dimensional(in r and z directions)unsteady state heat and moisture transfer equa-tions with appropriate boundary conditions.The Fourier’s equation for unsteady state heat conduction for solids with constant thermo-physical properties in aﬁnite cylinder geometry can be modiﬁed by the addition of a phase change term to account for the evaporation of the product moisture(Sakin, 2005;Sakin et al.,2003,2004,2005,2007b;Tong and Lund, 1993;Thorvaldsson and Janestad,1999;Shilton et al.,2002).@T¼a1Á@Tþ@2Tþ@2T!þkPÁ@Xð1Þwhere,T is the temperature(°C),t is the time(s),a is the thermal diffusivity(a=k/q c P,m2/s),r is the distance in the radial direction (m),z is the distance in the upward direction(m),k is latent heat of vaporization of water(J/kg),c p is the speciﬁc heat capacity(J/kgK),X is the moisture content(kg water/kg dry solid),k is the thermal con-ductivity(W/mK)and q is the density(kg/m3).Uniform initial temperature distribution throughout the cake batter was assumed:t¼0;T¼T ini at all r and zð2Þwhere,T ini is the initial batter temperature(°C).The top,bottom,side and annular surfaces boundary conditions for heat transfer were written as Eqs.(3)–(6),respectively as shown below.kÁ@T@zr¼r;z¼H¼hÁT aÀT r;HðÞð3ÞÀkÁ@T@zr¼r;z¼0¼hÁT aÀT r;0ðÞð4ÞkÁ@T@rr¼R;z¼z¼hÁT aÀT R;zðÞð5ÞNomenclatureA(i),B(i),C(i),AA(i),BB(i),CC(i)elements of coefﬁcient matrixBi Biot number for heat transfer,Bi r¼h cÂR=k;Bi z¼h cÂðH=2Þ=kBi m Biot number for moisture transfer,Bi mr¼k cÂRk cÂR=D eff;Bi mz k¼k cÂðH=2ÞD effc p speciﬁc heat(J/kgK)D eff effective moisture diffusivity(m2/s)D(i),DD(i)elements of the right-hand side vectorH thickness of cake batter(m)h surface heat transfer coefﬁcient(W/m2K)k thermal conductivity(W/mK)k c convective mass transfer coefﬁcient(m/s)M number of intervals in z directionMP1M+1,the top surface layerN number of intervals in r directionNH number of intervals in the cake in r directionNH1NH+1,annular surfaceNP1N+1,the radial axisr distance in radial direction(m)R the radius(m)T temperature(°C)T r oven surface temperature(°C)t time(s)U b lower surface overall heat transfer coefﬁcient(W/m2K) U l lateral surface overall heat transfer coefﬁcient(W/m2K) X moisture content(kg water/kg dry solid)z distance in upward direction(m) Subscriptsa ambientannulus annular surfaceb batterbottom bottom surfacecup baking cupdc donecakeeff effectivefﬁnali the i th nodeini initialj the j th noder,z the r and z directionssides side surfacetop top surfaceGreek symbolsa thermal diffusivity(m2/s)k latent heat of vaporization of water(J/kg) D r,D z the spatial increments in r and z directions.D x thickness(m)D t the time step(s)q the product density(kg/m3)290M.Sakin-Yilmazer et al./Journal of Food Engineering111(2012)289–298ÀkÁ@Tr¼R a;z¼z¼hÁT aÀT Ra;zðÞwhere,h is a convective and radiative surface cient(W/m2K),T a is the baking oven thickness of product(m),R is the radius of the R a is the annular radius(m).Fick’s2nd law describes the unsteady statein solids.Inﬁnite cylinder geometry,the change tent with time and position during baking (Crank,1975).@X¼D eff 1Á@Xþ@2Xþ@2X!Initially a uniform moisture distribution waspositionst¼0;X¼X ini at all r and zwhere,X ini is the initial moisture content of cakedry solid).The experimental data obtained in thisthe moisture contents in the top,bottom,sidedropped sharply at the start of the baking process.Although the top surface moisture content was the lowest,the moisture con-tents for the bottom,side and annular crusts were of comparable values.As the cake batter expands in the upward direction,sharp decrease in the moisture contents and sharp increase in tempera-tures were observed in the layers of the batter in contact with the cake mould.When crust formation begins on the surfaces in contact with the mould surfaces,it is believed that the crust sepa-rates slightly from the hot Aluminum surfaces enabling the loss of moisture also from these surfaces.Therefore,to account for the moisture loss in surface layers effective mass transfer coefﬁcients were assigned to these surfaces.The top surface moisture transfer boundary condition was deﬁned as below,by Eq.(9).ÀD effÁ@X@zr¼r;z¼H¼k c;topÁX r;HÀX1ðÞð9Þwhere,k c,top is an effective convective mass transfer coefﬁcient(m/ s),X1is the equilibrium moisture content in the oven medium.The bottom,side and annulus boundary conditions were de-ﬁned similarly as follows by Eqs.(10)–(12),respectively.D effÁ@X@zr¼r;z¼0¼k c;bottomÁX r;0ÀX1ðÞð10ÞÀD effÁ@X@rr¼R;z¼z¼k c;sidesÁX R;zÀX1ðÞð11ÞD effÁ@Xr¼R a;z¼z¼k c;annulusÁX Ra;zÀX1ðÞð12Þ4.Numerical solution4.1.Grid systemThe cake batter in a ring shaped baking mould was a ring of ra-dius R,annular radius R a and height H.A schematic diagram of the coordinate system used is given in Fig.1.Any rectangular cross-section from the annular surface represents the whole product.A grid system where terms i and j were used to index the nodes in r and z directions was formed(Sakin(2005)).D r and D z were spa-tial increments,and N and M were numbers of intervals in the r and z directions.M was taken as20and N was taken as44.The number of intervals within the cake in the radial direction NH was taken as 28.The volume increase and the subsequent decrease during bak-ing were taken into account as linear changes in height.In the numerical solution,the height of the baking product was refreshed at each time step by using the regression equations obtained from experimental height versus time data.4.2.Solution methodThe governing partial differential equations,deﬁning heat and moisture transfer have been approximated byﬁnite differences. The implicit alternating direction method was used for the solu-tion.The tri-diagonal matrix systems were formed and solved by the Gaussian Elimination method for the temperature proﬁle,T i,j and moisture proﬁle,X i,j at all points from i=1to NH+1,j=1to M+1,with a time step(D t)of2s(Carnahan et al.,1969;Chandra and Singh,1995).The elements of the coefﬁcients matrix and the right-hand side vector in tri-diagonal matrix system were deﬁned as A(i,j),B(i,j), C(i,j)and D(i,j)for heat transfer;AA(i,j),BB(i,j),CC(i,j)and DD(i,j) for moisture transfer,respectively by Sakin(2005).These coefﬁ-cients were modiﬁed for ring cake baking.The simulation of the model has been done by a computer pro-gram written on Fortran95.Theﬂow schema of the program is similar to the one given in(Sakin et al.,2007b).4.3.Veriﬁcation of numerical method against analytical solutions for simpliﬁed geometries and materialsThe mathematical model developed for ring cake baking and its numerical results for temperature and concentration proﬁles must be veriﬁed against well known analytical solutions before ring cake baking simulation.The solution of the given coupled partial differ-ential equations,deﬁning the simultaneous heat and mass transfer case of baking,by analytical methods is essentially impossible (Franks,1972).Therefore,the developed model was compared with possible analytical solutions and independently for heat and mass transfer.For heat transfer veriﬁcation Eq.(1)without the phase change term was used.To eliminate the effect of mass trans-fer,all the mass transfer coefﬁcients were taken as zero.To trans-form the ring cake to an inﬁnite slab,the height of the cake was taken to be much smaller than the radius.When this is the case,M.the ring cake becomes an inﬁnite slab with thickness being equal to the height of the ring cake.Predictions of the numerical model were compared with analytical solutions (Cengel,2003)for a wide range of Biot numbers.Excellent agreement was observed.To the knowledge of the authors,no analytical solution exists for unstea-dy state heat conduction in an annulus,with convective boundary conditions.Therefore,for radial testing of the model,the height of the ring cake was taken to be much larger than the radius and the number of radial intervals in the ring cake,NH,was taken to be equal to the total number of intervals À1,N À1.Furthermore,the heat transfer coefﬁcient for the annular surface was taken to be zero (adiabatic axis).Under these conditions,the resulting geome-try will closely simulate an inﬁnite cylinder geometry.The predic-tions of the ring cake numerical model were compared with the analytical solutions for inﬁnite cylinder.A very high level of agree-ment between analytical and numerical models was obtained.A ﬁ-nal check for the heat transfer section was performed by comparing the predictions of the numerical model with analytical solutions for a ﬁnite cylinder.The method of superposition was used for the analytical solution of the two dimensional partial dif-ferential equations.Input data used in the veriﬁcations for a ﬁnite cylinder are given in Table parison of the predictions of the numerical model for the center temperature with analytical solu-tions for a ﬁnite cylinder at Bi r =Bi z =1and Bi r =Bi z =10are shown in Fig.2.Excellent agreements were observed.For mass transfer veriﬁcations of the model,different cases were considered:In all these cases,the ambient temperature was as-sumed to be equal to the initial temperature and the latent heat was taken to be zero to eliminate heat transfer effects.Initially,the radius of the cake was taken to be much larger than the height and only convective mass transfer on the top surface was consid-ered.Under these conditions,the ring cake becomes an inﬁnite slabwith half thickness being equal to the height of cake.For the second case used in mass transfer veriﬁcations,the radius of the cake was taken to be much larger than the height and convective mass trans-fer was considered for the top and bottom surfaces.The same mass transfer coefﬁcients were used for the both surfaces.For this case,the ring cake becomes an inﬁnite slab with thickness equal to height of the cake.Numerical predictions for the surface,symmetry axis and average moisture content were compared with the analyt-ical solutions for a wide range of mass transfer Biot numbers (Bi m ).Very good agreements were observed for all cases.For radial testing of the model,the height of the ring cake was taken to be much lar-ger than the radius and the number of radial intervals in the ring cake,NH,was taken to be equal to the total number of intervals À1,N À1,similar to the heat transfer case.Furthermore,the mass transfer coefﬁcient for the annular surface was taken to be zero.Un-der these conditions,the resulting geometry will closely simulate an inﬁnite cylinder geometry.Again,very good results were ob-tained for these cases for center,surface and average moisture con-tents.Final veriﬁcation of the model for mass transfer was carried out for an impermeable hollow short cylinder where the annular ra-dius was 1/40times the cylindrical radius.This geometry is practi-cally a short cylinder.The input parameters used for mass transfer veriﬁcations are also given in Table 1.Constant transport properties and constant product thickness were parison of the predictions of the numerical model with analytical solutions for Bi mr =Bi mz =1and Bi mr =Bi mz =10for center and average moisture contents are shown in Figs.3and 4,respectively.The average moisture content was calculated by the following equation;X ave ¼2R 2ÀR 2aZRR aXrdr ð13ÞAs can be observed from Figs.2–4,the numerical model devel-oped predicts accurately the analytical temperature and moistureproﬁles.The slight differences in the average moisture contents in Fig.4were attributed to numerical integration error in Eq.(13).Therefore,it was concluded that the numerical model devel-oped for ring cake baking can be used to simulate experimental temperature and moisture content proﬁles.4.4.Input parameters to ring cake baking simulation and method of validation of simulation resultsThe ring cake baking process parameters used during the numerical solution were the product dimensions,thermo-physicalTable 1Input parameters used in the veriﬁcation of the numerical model.Cake dimensions (m)H :0.0462,R =0.0231Initial temperature,(°C)T ini =25.7Ambient temperature,(°C)T a =185Heat transfer coefﬁcients (W/m 2K)h :10.39,103.9Thermal conductivity (W/mK)k :0.24Density (kg/m 3)q =945Speciﬁc heat capacity (J/kg K)Cp =2650Initial moisture content (kg water/kg dry solids)X ini =0.56Ambient moisture content (kg water/kg dry solids)X 1:0Mass transfer coefﬁcient (m/s)k c =1.3Â10À5Effective diffusivity (m 2/s)D ef =3Â10À7,3Â10À8292M.Sakin-Yilmazer et al./Journal of Food Engineering 111(2012)289–298and mass transport properties,baking oven temperatures,the sur-face heat and mass transfer coefﬁcients,the initial temperatureand moisture content of the cake batter which are illustrated by Table 2.They were either experimentally determined,estimated or taken from the literature to simulate the experimental ring cake baking process.The mass transfer coefﬁcients for oven tempera-ture of 205°C were taken to be 20%higher than the corresponding values at 165and 185°C since the baking time to approximately the same ﬁnal moisture content was 50min at 205°C,compared to 60min at 165and 185°C.The product density and thermal con-ductivity was assumed to change linearly with cake height accord-ing to the following equations:q ¼q b þðq dc Àq b ÞðH ÀH ini ÞðH f ÀH ini Þð14Þk ¼k b þðk dc Àk b ÞðH ÀH ini ÞðH f ÀH ini Þð15ÞThe speciﬁc heat capacity was also evaluated separately and then the thermal diffusivities were calculated.The cake height was measured experimentally.The change in cake height with time was represented by regression straight lines.The %absolute deviation (p ,%)between the experimental and numerical results,for temperature and moisture proﬁles,were cal-culated by Eq.(16)to validate the simulation results (Boquet,Chi-rife,Iglesias,1978).Table 2Baking process parameters and input variables.Ring cake dimensions (m)H a :0.025,R a a :0.04,R a :0.11Baking temperatures (o C)T a a :165,185,205Initial moisture content (kg moisture/kg solids)X ini a :0.543–0.567Initial temperature (o C)T ini a :23.7–25.7Batter density (kg/m 3)q b a :947.8Donecake density (kg/m 3)q dc a :320Batter thermal conductivity(W/mK)k b b :0.24k b b :0.24Donecake thermalconductivity(W/mK)k dc b :0.121Speciﬁc heat capacity (J/kg K)C p b :2650Heat transfer coefﬁcients (W/m 2K)h a :16.3Mass transfer coefﬁcients (m/s)k c ,top a ,c :2.40Â10À5,k c ,bottom a :1.68Â10À5,k c ,sides a :1.56Â10À5,k c ,annulus a :1.56Â10À5k c ,205a =1.2k c,165Effective moisture diffusivity (m 2/s)D eff a ,d :1.7Â10À8a Experimentally measured or estimated values.b Baik et al.(1999).c Demirkol et al.(2006).dSakin et al.(2007a).M.Sakin-Yilmazer et al./Journal of Food Engineering 111(2012)289–298293p ð%Þ¼100N ÁX Ni ¼1X exp ÀXnum j jX expð16Þ5.Results and DiscussionTemperature and moisture proﬁles at certain positions of thering cake during baking were determined.The difference between the experimental and numerical moisture proﬁles were compared by p%values;where,a value below 10%indicates the good ﬁt be-tween both results (Boquet,Chirife,Iglesias,1978).The experimentally observed and numerically predicted tem-perature proﬁles at the positions of the product top surfaceaccuracy of the predictions.For this reason,the experimental data for the center temperature and model predictions were not re-ported in this research.Althoughmeasurements were performed at three oven temperatures,165,185and 205°C,only the results at 165and 205°C were shown to make the ﬁgures less crowded.The top surface numerical temperature proﬁle predictions were in good agreement with the experimental results,especially to-wards the completion of baking period (Fig.5).The p %values were calculated as 6.5%and 3.8%,for 165and 205°C,respectively.The speciﬁcation of the heat transfer coefﬁcient is crucial in heat trans-fer calculations especially at small Biot numbers.When we have a ring cake,the characteristic dimension to be used in the Biot num-ber is ambigous,since we have an annular geometry and the height of the cake also changes.However,if we take (R –R a )is the charac-294M.Sakin-Yilmazer et al./Journal of Food Engineering 111(2012)289–298h¼_mWkAðT aÀTÞtð17Þwhere_m W is the total moisture evaporated(kg)during a com-plete baking run,A is the total heat transfer area(m2),T is the sur-face temperature(o C)and t is the baking time(s);3600s for165°C and3000s for205°C.The surface temperature was assumed to be the mean of ambient temperature and100°C.A single heat trans-fer coefﬁcient was used in all runs.Radiative contribution to heat transfer was not considered.Sakin et al.(2009)have shown that the oven wall temperature and the oven medium temperature dif-fer appreciably depending on oven medium temperature.For an oven medium temperature of160°C,the oven wall average tem-perature was181.6°C in the forced convection oven.At higher oven temperatures,the wall and medium temperature approaches to each other.In the present work,lack of experimental data on heat transfer coefﬁcients will lead to inaccuracies in temperature predictions.The predicted bottom temperatures(Fig.6)were in good agree-ment with the experimental temperatures especially at an oven medium temperature of205°C.The p%values were calculated as 6.1%and3.9%,for165and205°C,respectively.The same heat transfer coefﬁcient was used for all surfaces.However,the convec-tive and radiative heat transfer coefﬁcients will be different for dif-ferent surfaces.Accurate determination of the heat transfer coefﬁcients will be essential for more accurate predictions.Annular surface temperature predictions by the present model were com-pared with experimental values in Fig.7.Satisfactory agreement was observed for oven medium temperature of165°C,where the p%value was calculated as5.5%.However,experimental tempera-tures were greater than the predicted temperatures for205°C.The p%value was calculated as10.9%.This larger deviation was again attributed to the use of the same heat transfer coefﬁcient for all surfaces.The estimation of thermo-physical properties of the ring cake and the transfer coefﬁcients were very important in the numerical solution.They were mostly taken from the related liter-ature(Table2).The differences between the experimental temper-ature proﬁles and the numerical predictions can also be attributed to these inaccuracies in the thermo-physical properties.More com-prehensive studies on the thermo-physical property estimation of a baking cake batter and the estimation of heat transfer coefﬁcients at different surfaces on a ring cake will result in more accurate numerical models.Similarly,the effective moisture diffusivity term and the mass transfer coefﬁcients have a strong effect on the moisture proﬁle prediction.The relative importance of the estimation of these two parameters will depend on the mass transfer Biot number.It may be shown from the data in Table2that mass transfer Biot numbers vary between11and66.For mass transfer,the internal resistance controls the diffusion of moisture.Therefore,accurate estimation of the effective diffusivity is essential for accurate mass transfer modeling.The available literature for this property for cake baking was limited mostly to the model equations generated for a thin layer of cake batter during drying/baking(Baik and Marcotte,2002;Sakin et al.,2005,2007a)and cup cake baking (Sakin et al.,2007b).As pointed out by Sakin et al.(2007b)the use of the proposed effective moisture diffusivity expressions by Baik and Marcotte(2002)and Sakin(2005)resulted in higher moisture proﬁles than experimental results.Although the effective diffusivity does not depend on the system geometry,for the same material,the geometry chosen in the estimation of the diffusivity affects the diffusion behaviour.Thus,the effective diffusivities cal-culated from experimental moisture data obtained from the drying or baking of a model system,will be indirectly affected by the geometry,in this case the ring cake.Rather than modifying these expressions,a constant effective diffusivity value was used for the sake of simplicity.The effective moisture diffusivity was taken as1.7Â10À8m2/s in line with the observations of Sakin(2005).When the moisture contents of different crusts of the ring cake were examined,it was observed that generally the top crust had the lowest moisture content and the bottom crust had the second lowest moisture.Moisture proﬁles for the side and annular crusts were higher than the top and bottom moisture contents and were very similar.Considering the moisture proﬁles,the top surface con-vective mass transfer coefﬁcient was taken as 1.3Â10À5m/s (Demirkol et al.,2006).The effective convective mass transfer coef-ﬁcients for the other surfaces were taken as shown in Table2.The experimentally observed and numerically predicted moisture pro-ﬁles at the top,bottom,side and annular surface positions for oven temperatures of165and205°C were shown in Figs.8–11.In almost all cases,theﬁnal numerical moisture contents were in close agreement with the experimental values.For the top crust moisture content(Fig.8)the numerical and experimental moisture proﬁles agreed quite well at165°C(p%value is9.1%).For oven temperature of205°C,the experimental proﬁles were lower than the numerical proﬁles;with a p%value larger than10%,indicating that the temperature dependence of the mass transfer coefﬁcient and the effective diffusivity should be considered for this surface. For the bottom surface(Fig.9),the experimental and numerical moisture proﬁles agreed very well at205°C with a p%8.2.How-ever,at165°C,experimental proﬁles were higher than the numer-ical ones(the p%value equals to10.0).This can again beattributed M.Sakin-Yilmazer et al./Journal of Food Engineering111(2012)289–298295。

IMF 金融风险,financial risk,英文版

WP/12/210
Effects of Culture on Firm Risk-Taking: A Cross-Country and Cross-Industry Analysis
Roxana Mihet

© 2012 International Monetary Fund
WP/12/
IMF Working Paper Research Department Effects of Culture on Firm Risk-Taking: A Cross-Country and Cross-Industry Analysis Prepared by Roxana Mihet* Authorized for distribution by Stijn Claessens August 2012
* I have benefited especially from detailed discussions with Stijn Claessens and Luc Laeven. I am grateful to Mohsan Bilal for his extensive help with the data. I would like to thank George Akerlof, Christopher Baum, John Beshears, Nathan Nunn, Lev Ratnovski, Fabian Valencia, and Francis Vitek in particular, and participants to 17th International Conference on Cultural Economics for their interesting comments on the topic, and useful and valuable suggestions on this paper. This paper is the winner of the President Prize for the best graduate student paper presented at the ACEI's 17th International Conference on Cultural Economics in Kyoto, Japan.

lecture21[1]

NUMERICAL SOLUTION OF PDES 7314.A finite element method for the transport equationWe consider the approximation of the initial value problem:β·∇u +σu =f,in Ω,u =g on Γin (Ω),where βis a unit vector,σ≥c >0andΓin (Ω)={x ∈∂Ω:β·n <0},where n is the unit outward normal to Ω.We shall assume that Ωis a polygon and T h is a triangulation of Ω.Then it is possible to order the triangles {T 1,T 2,···}such that for each k ,the domain of dependence of T k consists of some subset of Γin (Ω)and {T 1,···,T k −1}.With such an ordering,one can develop an ﬁnite element method in an explicit fashion,triangle by triangle.Letting P n (T )denote the polynomials of degree ≤n on T ,the Discontinuous Galerkin method for this problem has the form:For T ∈T h ,ﬁnd u h ∈P n (T )such that u −h =g h on Γin (Ω)and satisﬁes (β·∇u h +σu h ,v )T − Γin (T )(u +h −u −h )v β·n ds =(f,v )T ,v ∈P n (T ),where for x ∈Γin (T ),u ±h (x )=lim ǫ→0u h (x ±ǫβ).If we solve these equations using theordering discussed above,then u −h is known at the time it is needed to compute the solutionon triangle T .On each triangle,we need to solve a simple square linear system of equations.Note that the solution produced is a piecewise polynomial,but one that is discontinuous across triangle edges.The key to the analysis of this method is the following identity.Lemma 9.Assume that βis a constant vector.Then(β·∇u,u )T − Γin (T )(u +−u −)u +β·n ds=12 Γout (T )(u −)2|β·n |ds +12 Γin (T )(u +−u −)2|β·n |ds −12 Γin (T )(u −)2|β·n |ds.One important implication of this identity is that it easily follows that the linear system on each triangle has a unique solution for σ>0.To see this,we need only show that if f =0and u −h =0,then u h =0.Choosing v =u +h and using the above identity,we get 12 Γout (T )(u −)2|β·n |ds +12 Γin (T )(u +−u −)2|β·n |ds +σ u h 2T =0,and so u h =0.In analyzing this problem,it is helpful to think of u h as evolving in layers S i ,deﬁned byS 0=∅,S i ={T ∈T h :Γin (T )⊂Γin (Ω−∪j<i S j },j =1,2,···.Within a layer,u h can be developed in parallel.We can also deﬁne a sequence of fronts F i ,to which u h has advanced after it has been computed in Ωi =∪j ≤i S j .74NUMERICAL SOLUTION OF PDESIn the case when f =0,we also have a very simple stability analysis that can be expressed in the above terms.Theorem 16.12|u −h |2F i +σ u h 2Ωi ≤12|u −h |2Γin (Ω).Proof.Applying our identity with f =0,we obtain 12 Γout (T )(u −)2|β·n |ds +12 Γin (T )(u +−u −)2|β·n |ds +σ u h 2T =12 Γin (T )(u −)2|β·n |ds.Summing over all the triangles in the layer S i and omitting the positive jump terms,we get 12|u −h |2F i +σ T ∈S iu h 2T ≤12|u −h |2F i −1.The theorem follows by iterating this inequality.Also using this key identity,we are able to show thatu −u h L 2(Ω)≤Ch n +1/2 u n +1.Note that this is not an optimal order error estimate,since the best approximation by polynomials of degree ≤n ,would be O (h n +1).15.The finite volume method for elliptic problemsWe consider the approximation of the problem−div(a ∇u )=f in Ω,u =0on ∂Ω.Finite volume methods are based on an approximation of the balance equation − ∂b a ∇u ·n ds = bf dx,valid for any subdomain b ⊂Ω,where n denotes the unit outward normal to the boundary of b .Note this equation can be obtained by integrating the partial diﬀerential equation over the subdoiman b and applying the divergence theorem.One approach to the ﬁnite volume method,and the one we will discuss,uses a ﬁnite element partition of Ω,where the approximate solution space consists of piecewise linear functions,a collection of vertex-centered control volumes,and a test space consisting of piecewise constant functions over the control volumes.More precisely,we begin with a family of triangulations {T h }of the domain Ωand letX h ={v ∈H 10(Ω):v |T ∈P 1,∀T ∈T h }.To construct the control volumes,we let z T denote the barycenter of T and connect z T to the midpoints of the edges of T with line segments.This partitions each triangle T into three quadrilaterals.We use the notation K z,T to denote the quadrilateral in T which shares the vertex z of the triangle T .To each vertex z of the triangulation,we associate a controlNUMERICAL SOLUTION OF PDES75 volume b z consisting of the union of the quadrilaterals K z,T,where the union is taken over all triangles T containing the vertex z.Theﬁnite volume method is then toﬁnd u h∈X h such that− ∂b z a∇u h·n ds= b z f dx,for all z∈Z0h,where Z0h denotes the set of interior vertices of the mesh T h.There is a large literature on the use ofﬁnite volume methods to approximate partial diﬀerential equations.An error analysis for the method presented here can be found in the paper“Error estimates for aﬁnite volume element method for elliptic pdes in nonconvex polygonal domains,”by P.Chatzipantelidis and zarov,SIAM J.Numer.Anal.,(42), no5.,2005,pp.1932-1958.。

金融工程常用术语(中英对照)

金融工程常用术语中英对照AABS Asset-Backed Security 资产支持证券ABS CDO 由ABS所派生出的份额产品Accrual Swap 计息互换Accrued Interest 应计利息Actuaries 保险精算师Adaptive Mesh Model 自适应网格模型Adjusted Present Value 调整现值法Adverse Selection 逆向选择After-tax Interest Rate 税后利润Agency Costs 代理费用American Option 美式期权Amortization 分期偿付Amortization Schedule 分期偿付时间表Amortizing Swap 分期偿还互换Analytic Result 解析结果APR Annual Percentage Rate 年度百分率Annualized Capital Cost 按年折算的资本成本Arbitrage 套利Arbitrageur 套利者Asian Option 亚式期权Ask Price 卖盘价Asset 资产Asset Allocation 资产分配Asset-or-Nothing Call Option 资产或空手看涨期权Asset-or-Nothing Put Option 资产或空手看跌期权Asset Swap 资产互换As-You-Like-It Option 任选期权At-the-Money Option 平值期权Average Price Call Option 平均价格看涨期权Average Price Put Option 平均价格看跌期权Average Strike Option 平均执行价格期权BBackdating 倒填日期Back Testing 回顾测试Backwards Induction 倒推归纳Barrier Option 障碍期权Base Correlation 基础相关系数Basel Committee 巴塞尔委员会Basis 基差Basis Point 基点Basis Risk 基差风险Basis Swap 基差互换Basket Credit Default Swap 篮筐式信用违约互换Basket Option 篮筐式期权Bear Spread 熊市差价Bermudan Option 百慕大式期权Before-tax Interest Rate 税前利率Beta 贝塔Bid-Ask Spread 买入卖出差价Bid Price 买入价Bilateral Clearing 双边结算Binary Credit Default Swap 两点式信用违约互换Binary Option 两值期权Binomial Model 二项式模型Binomial Option Pricing Model 二项期权定价模型Binomial Tree 二叉树Bivariate Normal Distribution 二元正态分布Black’s Approximation 布莱克近似Black’s Model 布莱克模型Black-Scholes-Merton Model 布莱克-斯科尔斯-莫顿模型Bond Option 债券期权Bond Yield 债券收益率Book Value 账面价值Bootstrap Method 票息剥离方法Boston Option 波士顿期权BOT Build-Operate-Transfer 建设-经营-转让Box Spread 合式差价Break-even point 盈亏平衡点Break Forward 断点远期Brownian Motion 布朗运动Bull Spread 牛市差价Butterfly Spread 蝶式差价CCalendar Spread 日历差价Calibration 校正Callable Bond 可赎回债券Call Option 看涨期权Cancelable Swap 可取消互换Cap 上限Cap Rate 上限利率CAPM Capital Asset Pricing Model 资本资产定价模型Caplet 上限单元Capital gain 资本收藏Capital less 资本损失Capital Market 资本市场Capital Market Line 资本市场线Caps 赔付限额Case-Shiller Index 凯斯-席勒指数Cash budget 现金预算Cash cycle time 现金周转时间Cash Dividend 现金股利Cash Flow Mapping 现金流映射Cash-or-Nothing Call Option 现金或空手看涨期权Cash-or-Nothing Put Option 现金或空手看跌期权Cash Settlement 现金交割或现金清算CCP Central Clearing Party 中央结算对手CDD Cooling Degree Days 降温天数CDO Collateralized Debt Obligation 债务抵押债券CDS Credit Default Swap 信用违约互换CEBO Credit Event Binary Option 信用事件两点式期权Central Clearing 中心结算Central Clearing Party 中央结算对手Central Counterparty 中央交易对手CEV Model Constant Elasticity of Variance Model 常方差弹性模型Cheapest-to-Deliver Bond 最便宜可交割债券Cholesky Decomposition 乔里斯基分解Chooser Option 选择人期权Class of Options 期权分类Clean Price of Bond 债券除息价格Clearing House 结算中心Clearing Margin 结算保证金Cliquet Option 棘轮期权CMO Collateralized Mortgage Obligation 见房产抵押债券CMS Constant Maturity Swap 固定期限国债互换Collar 双限Collateral 抵押品Collateralization 抵押品策略Collateralized Debt Obligation 债务抵押债券Collateralized Mortgage Obligation 房产抵押债券Combination 组合Commercial Banks 商业银行Commercial Loan Rate 商业贷款利率Commodity Futures Trading Commission 商品期货交易管理委员会Commodity Swap 商品互换Compound Interest 复利Compounding 复利计息Compounding Frequency 复利利率Compound Correlation 复合相关系数Compound Option 复合期权Confidence interval 置信区间Continuous Probability Distribution 连续概率分布Confirmation 交易确认书Consumption Asset 消费资产Contango 期货溢价Continuous Compounding 连续复利Control Variate Technique 控制变量技术Convenience Yield 便利收益率Conversion Factor 转换因子Convertible Bond 可转换债券Convexity 曲率Convexity Adjustment 曲率调整Cornish-Fisher Expansion 科尼什-费雪展开Copayments 赔付比例Corporation 公司Correlation 相关性Cost of Capital 资本成本Cost of Carry 持有成本Controller 审计官Counterparty 交易对手Coupon 券息Coupon bond 付息债券Covariance 协方差Covarance Matrix 协方差矩阵Covered Call 备保看涨期权Crash phobia 暴跌恐惧症Credit Contagion 信用蔓延Credit Default Swap 信用违约互换Credit Derivative 信用衍生产品Credit Event 信用事件Credit Event Binary Option 信用事件两点式期权Credit Index 信用指数Credit Rating 信用等级Credit Ratings Transition Matrix 信用评级转移矩阵Credit Risk 信用风险Credit Spread Option 信用差价期权CSA Credit Support Annex 信用支持附约CVA Credit Value Adjustment 信用价值调节量Credit Value at Risk 信用风险价值度Cross Hedging 交叉对冲Currency Swap 货币互换Current yield 本期收益率DDay Count 计天方式Day Trade 即日交易DCF Discounted Cash flow Model 现金流折现模型DDM Dividend Discount Model 股利贴现模型DVA Debt Value Adjustment 债务价值调节量Decision Tree 决策树Deductible 免赔额Default Risk 违约风险Defined-benefit Pension Plan 规定受益型养老金计划Defined-contribution Pension Plan 规定缴费型养老金计划Delivery Price 交割价格Delta Hedging Delta对冲Delta-Neutral Portfolio Delta 中性交易组合Derivative 衍生产品Deterministic Variable 确定性变量Diagonal Spread 对角差价Differential Swap 交叉货币度量互换Diffusion Process 扩散过程Dirty Price of Bond 带息价格Discount Bond 折扣债券Discount Instrument 折扣产品Discounted Cash Flow Analysis 贴现现金流分析Discounted Dividend Model 股利贴现模型Diversifiable Risk 可分散风险Diversification 分散化Diversification Principle 分散化原则Diversifying 分散投资Discount Rate 贴现率Dividend 股息Dividend Yield 股息收益率Dodd-Frank Act 多德-弗兰克法案Dollar Duration 绝对额久期DOOM Option DOOM期权Down-and-In Option 下降-敲入期权Down-and-Out Option 下降-敲出期权Downgrade Trigger 降级触发Drift Rate 漂移变化率Duration 久期Duration Matching 久期匹配Dynamic Hedging 动态对冲EEAR/ EFF Effective Annual Rate 实际年利率Early Exercise 提前行使EBIT Earnings Before Interest and Tax 息税前利润Effective Federal Funds Rate 有效联邦基金利率Efficient Portfolio 有效投资组合Efficient Portfolio Frontier 有效投资组合边界Electronic Trading 电子交易Embedded Option 内含期权EMH Efficient Markets Hypothesis 有效市场假说Empirical Research 实证研究Employee Stock Option 雇员股票期权Equilibrium Model 均衡模型Equity Swap 股权互换Equity Tranche 股权份额Equivalent Annual Interest Rate 等价年利率Eurocurrency 欧洲货币Eurodollar 欧洲美元Eurodollar Futures Contract 欧洲美元期货合约Eurodollar Interest Rate 欧洲美元利率Euro LIBOR 欧元同业拆借利率European Option 欧式期权Exchange Option 互换期权Exchange Rate 汇率Exclusions 免赔条款Ex-dividend Date 除息日Exercise Limit 行使限额Exercise Multiple 行使倍数Exercise Price 执行价格Exotic Option 特种期权Expectations Theory 预期理论Expected Shortfall 预期亏损Expected Rate of Return 预期回报率Expected Value of a Variable 变量的期望值Expiration Date 到期日Explicit Finite Difference Method 显式有限差分方法Exponentially Weighted Moving Average Model 指数加权移动平均模型Exponential Weighting 指数加权Exposure 风险敞口Extendable Bond 可展期债券Extendable Swap 可延期互换External Financing 外部投资FFace Value 面值Factor 因子Factor Analysis 因子分析Federal Funds Rate 联邦基金利率FEI Financial Executives Institute 财务执行官组织Finance 金融学Financial Futures 金融期货Financial Guarantees 财务担保Financial Intermediary 金融媒介Financial System 金融系统Finite Difference Method 有限差分法Fixed-Income Instrument 固定收益证券Flat Volatility 单一波动率Flex Option 灵活期权Flexi Cap Flexi上限Floor 下限Floor-Ceiling Agreement 下限上限协议Floor let 下限单元Floor Rate 下限利率Flow of funds 资金流Foreign Currency Option 外汇期权Forward Contract 远期合约Forward Exchange Rate 远期汇率Forward Interest Rate 远期利率Forward Price 远期价格Forward Rate 远期率FRA Forward Rate Agreement 远期利率合约Forward Risk-Neutral World 远期风险中性世界Forward Start Option 远期开始期权Forward Swap 远期互换Fundamental Value 基本价值Futures Commission Merchants 期货佣金经纪人Futures Contract 期货合约Futures Option 期货期权Futures-Style Option 期货式期权FV Final Value 终值GGrowth annuity 增长年金GAP Management 制品管理Gap Option 缺口期权Gaussian Copula Model 高斯Copula 模型Gaussian Quadrature 高斯求积公式Generalize Wiener Process 广义维纳过程Geometric Average 几何平均Geometric Brownian Motion 几何布朗运动Girsanov’s Theorem 哥萨诺夫定理Guaranty Fund 担保基金HHaircut 折扣Hazard Rate 风险率Hedge 对冲Hedge Funds 对冲基金Hedger 对冲者Hedge Ratio 对冲比率Hedgers 套期保值者Historical Simulation 历史模拟Historical Volatility 历史波动率Holiday Calendar 假期日历Human Capital 人力资本IImmediate Annuity 即时年金Implicit Finite Difference Method 隐式有限差分Implied Correlation 隐含相关系数Implied Distribution 隐含分布Implied Dividend 隐含股利Implied Tree 隐含树形Implied Volatility 隐含波动率Inception Profit 起始盈利Index Amortizing Swap 指数递减互换Index Arbitrage 指数套利Index Futures 指数期货Index-linked Bonds 指数化债券Index Option 指数期权Index Principal Swap 指数本金互换Initial Margin 初始保证金Instantaneous Forward Rate 瞬时远期利率Insuring 保险Intangible Assets 无形资产Interest-rate Arbitrage 利率套利Interest Rate Cap 利率上限Interest Rate Collar 利率双限Interest Rate Derivative 利率衍生产品Interest Rate Floor 利率下限Interest Rate Swap 利率互换Internal Financing 内部融资International Swap and Derivatives Association 国际互换和衍生产品协会In-the-Money Option 实值期权Intrinsic Value 内涵价值Inverted Market 反向市场Investment Asset 投资资产Investment Banks 投资银行ISDA International Swap and Derivatives Association 国际互换和衍生产品协会IRR Internal Rate of Return 内部收益率JJump-Diffusion Model 跳跃扩散模型Jump Process 跳跃过程LLaw of One Price 一价原则Liability 负债LIBID London Inter Bank Bid Rate 伦敦同业借款利率LIBOR London Inter Bank Offered Rate 伦敦同业拆出利率LIBOR Curve LIBOR曲线LIBOR-in-Arrears Swap LIBOR后置互换Life Annuity 人寿年金Limited Liability 有限责任Limit Move 涨跌停版变动Limit Order 限价指令Liquidity 流动性Liquidity Preference Theory 流动性偏好理论Liquidity Premium 流动性溢价Liquidity Risk 流动性风险Locals 自营经纪人Lognormal Distribution 对数正态分布Long Hedge 多头对冲Long Position 多头Look back option 回望期权Low Discrepancy Sequence 低偏差序列MMaintenance Margin 维持保证金Margin 保证金Margin Call 保证金催付Market Capitalization Rate 市场资本化利率MSU Market-Leveraged Stock Unit 市场股票凭据Market Maker 做市商Market Model 市场模型Market Portfolio 市场投资组合Market Price of Risk 风险市场价格Market Segmentation Theory 市场分隔理论Market-weighted Stock Indexes 市场加权股票指数Marking to Market 按市场定价Markov Process 马尔科夫过程Martingale 鞅Maturity 期限Maturity Date 到期日Maximum Likelihood Method 极大似然方法Mean Reversion 均值回归Measure 测度Merger 合并Mezzanine Tranche 中层份额Minimum Variance 最小方差组合Modified Duration 修正久期Money Market 货币市场Money Market Account 货币市场帐户Monte Carlo Simulation 蒙特卡罗模拟Moral Hazard 道德风险Mortgage-Backed Security 房产抵押贷款证券Mutual Fund 共同基金NNaked Position 裸露期权Netting 净额结算Net Present Value 净现值Net Worth 净资产No-Arbitrage Assumption 无套利假设No-Arbitrage Interest Rate Model 无套利假设Nominal Future Value 名义终值Nominal Interest Rate 名义利率Nominal Prices 名义价格Nondiversifiable Risk 不可分割风险Nonstationary Model 非平稳模型Non Systemic Risk 非系统风险Normal Backwardation 正常现货溢价NPV Net Present Value 净现值Normal Distribution 正态分布Normal Market 正常市场Notional Principal 面值（本金）Numeraire 计价单位Numerical Procedure 数值方法OOCC Option Clearing Corporation 期权结算中心Offer Price 卖出价格Open Interest 未平仓合约Open Outcry 公开喊价Opportunity Cost of Capital 资金的机会成本Optimal Combination of risky assets 风险资产的最优组合Option 期权Option-Adjusted Spread 期权调整差价Option Class 期权种类Ordinary Annuity 普通年金Out-of-the-Money Option 虚值期权Overnight Indexed Swap 隔夜指数互换Over-the-Counter Market 场外交易市场PPackage 组合期权Par bonds 等价债券Par Value 面值Par Yield 面值收益Parallel Shift 平行移动Parisian Option 巴黎期权Partnership 合伙制Path-Dependent Option 路径依赖型期权Payoff 收益Pay off Diagram 收益图Percent-of-sales method 销售收入百分比法Permanent Income 持久收入Perpetuity 永续年金Perpetual Derivatives 永续衍生品Portfolio Immunization 组合免疫Portfolio Insurance 证券组合保险Portfolio selection 投资组合选择Portfolio theory 投资组合理论Position Limit 头寸限额Premium 期权付费Premium Bond 溢价债券Present Value 现值Principal-agent Problem 委托人-代理人问题Prepayment Function 提前偿付函数Principal 本金Principal Components Analysis 主因子分析Principal Protected Notes 保本型证券Probability Distributions 概念分布Program Trading 程序交易Protective Put 保护看跌期权Pull-to-Par 收敛于面值现象Purchasing-power Parity 购买力评价Pure Discount Bonds 纯贴现债券Put-Call Parity 看跌-看涨期权平价关系式Put Option 看跌期权Puttable Bond 可提前退还债券Puttable Swap 可赎回互换President 总裁PMT Payment（Returns the periodic payment for an annuity）年金PPP Public-Private Partnership 政府和社会资本合作PV Present Value 现值QQuansi-Random Sequences 伪随机序列RRate of Return on capital 资本收益率Rainbow Option 彩虹期权Range-Forward Contract 远期范围合约Ratchet Cap 执行价格调整上限Real Future Value 实际终值Real Interest Rate 实际利率Real Option 实物期权Real Prices 实际价格Rebalancing 再平衡Recovery Rate 回收率Reference Entity 参考实体Reinvestment Rate 再投资利率Residual Claim 剩余索取权Repo 再回购Repo Rate 再回购利率Reset Date 重置日（定息日）RSU Restricted Stock Unit 受限股票单位Reversion Level 回归水平Risk Aversion 风险厌恶Risk-adjusted discount rate 风险调整贴现率Risk Exposure 风险暴露Rights Issue 优先权证Risk-Free Rate 无风险利率Risk Management 风险管理Risk Management Process 风险管理过程Risk-Neutral Valuation 风险中性定价Risk-Neutral World 风险中性世界Roll Back 倒推ROS Ratio of income as percentage of sales 销售利润率ROA Return On Assets 资产收益率ROE Rate of Return on Common Stockholders’ Equity 净资产收益率SScalper 投机者Scenario Analysis 情形分析Securitization 证券化Security Market Line 证券市场线Sensitivity Analysis 敏感性分析Self-financing Investment Strategy 自筹资金投资策略Settlement Price 结算价格Share Repurchase 股票回购Short Hedge 空头头寸对冲Short Position 空头头寸Short Rate 短期利率Short Selling 卖空交易Short-Term Risk-Free Rate 短期无风险利率Shout Option 喊价期权Simple Interest 单利Sole Proprietorship 独资企业Specialist 专家Speculator 投机者Spot futures price parity relation 现货期货价格平价关系。

对分数阶扩散方程的紧致差分方法

Article history: Received 28 April 2009 Received in revised form 17 July 2009 Accepted 23 July 2009 Available online 3 August 2009 MSC: 65M06 65M12 65M15 35Q51 35Q53 78M20 Keywords: Fractional diffusion equation Finite difference Compact scheme Padé approximant Stability Convergence Fourier analysis
Journal of Computational Physics 228 (2009) 7792–7804

Contents lists available at ScienceDirect
Journal of Computational Physics
journal homepage: /locate/jcp
E-mail address: mrcui@ 0021-9991/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jcp.2009.07.021
M.R. Cui / Journal of Computational Physics 228 (2009) 7792–7804
1. Introduction Fractional differential equations (FDEs) have attracted increasing attention because they have applications in various ﬁelds of science and engineering. For example, they can describe many physical and chemical processes, biological systems, etc. The main physical purpose for investigating diffusion equations of fractional order is to describe phenomena of anomalous diffusion in transport processes through complex and/or disordered systems including fractal media, and fractional kinetic equations have proved particularly useful in the context of anomalous slow diffusion, see, for example, the excellent review paper [1]. An overview of the basic theory of fractional differentiation, fractional-order differential equations, methods of their solution and applications can be found in the book [2]. There have been several numerical methods proposed for solving the space and/or time FDEs up to now. Lynch et al. [3] developed two numerical schemes, one explicit and another one semi-implicit, for solving the transport problem with anomalous diffusion modeled by a partial differential equation of fractional order. Meerschaert and Tadjeran [4] studied the onedimensional radial ﬂow model, they found that the fractional derivative describes more accurately the early arrival that cannot be explained by the classical advection–dispersion equations and they presented an implicit Euler method, based on a

【doc】有限差分法在一维输运方程定解中的运用

有限差分法在一维输运方程定解中的运用2012年2月第12卷第1期廊坊师范学院(自然科学版)JournalofLangfangTeachersCollege(NaturalScienceEdition)Feb.2012V0l_12No.1有限差分法在一维输运方程定解中的运用林喜季(福建江夏学院,福建福州350108)【摘要】有限差分方法就是一种数值解法,在一维输运方程定解中可以巧用它来解题,把表示变量连续变化关系的偏微分方程离散为有限个代数方程,然后利用电子计算机求此线性代数方程组的解.【关键词】一维输运方程;有限差分法;定解问题FiniteDifferenceMethodin0ne—DimensionalTransport EquationintheUseofDefiniteSolutionLINXii【Abstract】Thefinitedifferencemethodisanumericalmethod,one-dimensionaltransportequationinth esolutioncanbeskillfullyusedittosolveproblems,thecontinuousvariationofthatvariablepartialdifferential equationisdiscretizedintoafinitenumberofalgebraicequations,andthenusethecomputertosolveThishnearalgebraice quations.【Keywords】one-dimensionaltransportequation;finitedifferencemethod;definitesolutionproblem[中图分类号]0175.2(文献标识码]A[文章编号]1674—3229(2012)01—0016—03 在数学中,有限差分法的内涵是指用差商代替微商,即用泰勒级数展开式将变量的导数写成变量在不同时间或空间点值的差分形式的方法.它的基本思想是按时间步长和空间步长将时间和空间区域剖分成若干方格网,用泰勒级数展开近似式代替所用偏微分方程中出现的各阶导数,从而把表示变量连续变化关系的偏微分方程离散为有限个代数方程,然后,解此线性代数方程组.l导数用泰勒级数展开近似式导数(微商)y:=m0=±,是无限小的微分m0△),除以无限小的微分是△的商.它可以分别近似为:,,=dxAx=(1)y=Ax=(2)y=dxAx=坐(3)式(1),(2)相当于把泰勒级数y(+~xx)=y()+(Ax)y+1(△)+…(—Ax)=y()一(△)y+1(△)+…截断于(Ax)v项,把(Ax)项以及更高幂次的项全部略去.式(3)相当于把泰勒级数y(+Ax)一Y(一△)=2(Ax)Y+(△)y,-+..截断于2(Ax)项,把(Ax)项以及更高幂次的项全部略去.因此,式(3)的误差小于式(1)和(2).二阶导数类似的可近似为差商的差商,一X[dx…一血dx【一止]:志[(+△)+y(—Ax)一2y()](4)[收稿日期]2011—11—21[作者简介]林喜季(1977一),女,福建江夏学院讲师,研究方向:代数表示论.16?第12卷?第1期林喜季:有限差分法在一维输运方程定解中的运用2012年2月这相当于把泰勒级数Y(+△)一v(一△)=2y()+(△)+(△)Y+..'截断于(△)项,把(△)项以及更高幂次的项全部略去.偏导数也可仿照式(1)一(4)近似为商差.这样一来,偏微分方程就成了差分方程.2一维输运方程的定解问题如,在区间(0,L)上求解一维输运方程"='axx.分析:(1)把整个空间分为.,个"步子",每一步的长度=I/J.于是,自变量以步长跳跃,它的取值是(i=0,1,2,…,.,).把时间步长取为zI,即自变量t取值t=kv(k=0,1,2,…,).(2)仿照式(1)和(4),一维输运方程可近似为(1)=(1—2lM(￡),2+l_!["(+l,t)+u(一1,t)】(5)这样只要知道某个时刻t的u在各个地点的值(,t),代人式(5)就可以得到下个时刻t…的的各个地点的值u(i,t).但这种解法时间t的步长z.不能太大,必须满足条件≤1,否则,由于舍入误差,会在其后各步的计算中产生雪崩影响,以致计算结果完全失去意义. (3)仿照式(2)和(4),一维输运方程可近似为u(,t)一U(i,t一1)r2(+1,t)+u(i一1,t)一2U(,t)即"(¨)=(1+2竿)Ⅱ(),2一旦j三[(+1,t)+"(,t)】(6)这样做可以取消对步长r的限制.但是知道某个时刻t的Ⅱ在各个地点的值(,t),并不能代入式(6)直接得到下个时刻t川的的各个地点的值(,t),且必须把i=1,2,3,…,.,一1的共计J一1个同式(6)的方程联立起来求解u(t,t+1),u(2,t+1),…,u(J一1,t%+1),当然这种联立方程的计算依靠电子计算机还是很方便的.(4)仿照式(3),偏导数近似为u(Xi'tk+1):,从而一维输运方程可近似为M(,t+1)一u(,t):u(Xi+?,tk+1)+u(—t,+.)一2u(,t+吉).上式中(,tk+)可理解为:[配(,t+.)+u(,t)】/2,于是,上例差分方程即为窘u(…)一(+字)"()+au("~i-1+)=一Ⅱ()一(1一窘)"(3Ci~tk)一骞H+1)(7)知道某个时刻t的在各个地点的值M(,t)后,必须把i=1,2,3,…,J一1的共计.,一1个同式(7)的方程联立起来求解"(.,t…),U(2,t+1),…,(J—l,t+1),当然这种联立方程的计算依靠电子计算机还是很方便的.这种解法对时间的步长也有限制,应满足2≤1,但与解法(2)相比限制要宽些.3用近似式求一维波动方程的定解问题如,在区间(0,L)上求解一维波动方程%一a"=0.把整个空间分为.,个"步子",每一步的长度=l/J.把时间步长取为r,仿照式(4),一维波动方程可近似为"(戈,t+1)+(,t一1)一2u(,t)即(.):2(1一譬)u(+旦}[(+1,tk)+(;一1,)一¨(,一1)】.这就是说,只要知道某时刻t及其以前时刻的U在各个地点i的值u(,t),代入上式,就可以得到下个时刻tk+.的的各个地点的值"(,t…).在此f青景下,时间的步长r的限制条件为≤1.17?,J,L2一,J一,L2"r,+,L22012年2月廊坊师范学院(自然科学版)第12卷?第1期从上述例子可以看出,在研究一维输运方程定解问题时,可采用有限差分法,按适当的数学变换把定解问题中的微商换成差商,从而把原问题离散化为差分格式,进而求出数值解.该方法具有简单,灵活,容易在计算机上实现的特点.并且该方法还有很强的通用性,如热传导过程,气体扩散过程这类定解问题,其过程都与时间有关,利用差分法解这类问题,就是从初始值出发,通过差分格式沿时间增加的方向,逐步求出微分方程的近似解.再如弹性力学中的平衡,电磁场及引力场等问题,其特征均为椭圆型方程,利用差分法解这类问题,就是合理选定的差分方格网,建立差分格式,最后求解代数方程组.[参考文献][1]吴顺唐,邓之光.有限差分法方程[M].南京:河海大学出版社.1993.[2]陈祖墀.偏微分方程[M].合肥:中国科学技术大学出版社.2004.[3]王晓东.算法与数据结构[M].北京:电子工业出版, 1998.[4]王震,谢树森.解四阶拟线性波动方程的一类二阶差分格式[J].中国海洋大学,2004,(34).(上接15页)将上述n为奇数与n为偶数两种情况统一起来,可得数列{a)的通项公式为.=1[(口.+.)+(一1)(.一n.)】?r.2)看P≠1,根据文献l3j结论司知,b=+()p,即有an+lan=+【一)p,从而anan_l=+(?一-qp)p.若令一1)=+(一,(n≥2),贝0ana一:f(一1).当p≠±,g≠o时,一)+(g一)p=一p棚=qp(1一p)≠0(n≥2),从而,('一p)≠0(n≥).由%a一.=/(n一1)递推可得:当n为大于1奇数时,(n一1)(一1)厂(//,一3).n':(—.—.::—0n一:je'.——.::—'':e';:—:二—;0n一=?…?f1al;一厂(n一2)厂(一4)厂(3)'()'当ll,为大于2的偶数时,n=:;{—;——{.一:=.一18?=船?…?n一12Ⅱ[,(2Jl})]即当n为大于1奇数时,a=丁_—一a.;Ⅱ厂()Ⅱ()当n为大于2的偶数时,.=T或改写为a=Ⅱ[,(2)]二者可统一为n=—1[(nl+口2)+(一1)(n2一口1)]?{{一吉【3+(一1)】)n一吉[3+(一1)】[Ⅱ[(2)]/Ⅱ厂(+)](-1),nEN+且n≥3,其中f(n)=(1一p),n∈N+.[参考文献][1]劳建祥.递推数列求通项大观[J].数学教学,2005,(3): 41—42.[2]高焕江.也谈二阶线性递推数列的周期?t:ff-[J].廊坊师范学院,2009,9(6):8—10.[3]高焕江.二阶线性递推数列的通项公式[J].保定学院学报,2010,23(3):34—37.。

物理专业英语词汇F汇总

物理专业英语词汇(F) f band f 带f center f 心f center laser f 心激光器f center maser f心微波激射器f layer f 层f value振子强度f/d ratio f/d 比face centered crystal 面心晶体face centered cubic lattice 面心立方晶格face centered cubic structure 面心立方结构face centered lattice 面心点阵factor因子factor group 剩余群facula光斑faddeev equation法捷耶夫方程faddeev popov ghost法捷耶夫波波夫鬼态fading衰退;褪色fahrenheit scale 华氏温标fahrenheit temperature scale 华氏温标fahrenheit thermometer 华氏温度计fall降低fall of potential 电压降fall time下降时间falling sphere viscometer 落球粘度计fallout放射性沉降物false image 鬼象family 族family of asteroids 小行星族family of comets 彗星族fano factor 法诺因子far field pattern 远场图样far infrared远红外区far infrared laser远红外激光器far infrared radiation 远红外卜辐射far infrared rays 远红外卜线far point 远点far ultraviolet 远紫外区far ultraviolet laser 远紫外激光器far ultraviolet radiation 紫外线辐射far ultraviolet rays 远紫夕卜线farad 法faraday法拉第faraday cage法拉第笼faraday cell 法拉第盒faraday constant 法拉第常数faraday cup法拉第笼faraday dark space 法拉第暗区faraday effect 磁致旋光faraday rotation 磁致旋光faraday tube法拉第管faraday,s law法拉第定律farvitron线振质谱仪fast迅速的fast breeder reactor 快增殖堆fast fission快中子裂变fast fission effect快中子裂变效应fast fourier transform快速傅里叶变换fast neutron 快中子fast neutron reactor 快中子堆fast nova快新星fast reactor 快中子堆fatigue 疲劳fatigue limit疲劳极限fatigue strength 疲劳强度fcc structure面心立方结构feedback 反馈feedback amplifier 反馈放大器feedback circuit 反馈电路feedback control 反馈控制feedback factor 反馈系数feedback ratio 反馈比feeder馈电线femto 飞femtometer 费密femtosecond 飞秒femtosecond region 飞秒区域fermat,s principle 费马原理fermi费密fermi acceleration 费密加速fermi age费密年龄fermi dirac statistics 费密统计法fermi distribution 费密分布fermi energy 费密能fermi gas费密气体fermi glass费密玻璃fermi hole费密空穴fermi interaction 费密相互酌fermi level费密能级fermi liquid费密液体fermi particle 费密子fermi pasta ulam problem费密巴斯德乌拉姆问题fermi resonance 费密共振fermi selection rule 费密选择定则fermi statistics费密统计法fermi surface 费密面fermi temperature 费密温度fermi transition 费密跃迁fermi's golden rule费密黄金定律fermiology费密面学fermion费密子fermionic dark matter 费密子暗物质fermium 镄ferrielectricity 亚电性ferrimagnetic resonance 亚铁磁共振ferrimagnetism铁氧体磁性ferrite铁氧体ferrite magnetostrictive vibrator 铁氧体磁致伸缩振子ferro resonance 铁磁共振ferroelastic phase transition 铁弹性相变ferroelasticity 铁弹性ferroelectric 铁电的ferroelectric domain 铁电畴ferroelectric mode 铁电模ferroelectric phase transition 铁电相变ferroelectric semiconductor 铁电半导体ferroelectric substance 铁电性材料ferroelectricity 铁电性ferrofluid 铁磁铃ferromagnet 铁磁体ferromagnetic 铁磁的ferromagnetic dielectrics 铁磁电介质ferromagnetic fluid 铁磁铃ferromagnetic material 铁磁性材料ferromagnetic resonance 铁磁共振ferromagnetic substance 铁磁性材料ferromagnetic superconductor 铁磁超导体ferromagnetic thin film 铁磁薄膜ferromagnetism 铁磁性feynman diagram 费因曼图feynman path integral费因曼的路径积分feynman spectrum 费因曼谱ffag synchrotron固定场交变梯度回旋加速器fiber纤维fiber bundle 纤维丛fiber electrometer 悬丝静电计fiber laser纤维激光器fiber optics纤维光学fiber structure 纤维结构fibonacci semiconductor superlattice 菲博纳奇半导体超晶格fibril小纤维fibril structure 纤维结构fick,s law斐克定律fictitious spin quantum number 假想自旋量子数fictitious year 假年field场;体field adsorption 场吸附field current 励磁电流field density 场密度field desorption 场致退吸field distortion 场畸变field distribution 场的分布field effect transistor 场效应晶体管field electron emission 场致电子发射field emission 场致发射field emission microscope场致发射显微镜；自动电子显微镜field emitted electron 场致发射电子field equation 场方程field intensity 场强field ion microscope场致离子显微镜field lens 场镜field magnet 场磁铁field of forces 力场field of gravity 重力场field of view 视场field of vision 视场field operator 场算符field particle 场粒子field quantum 场量子field stop 场栏field strength 场强field structure 场结构field theory 场论fifth force第五种力fifth fundamental force 第五种基本力figure图形figure of merit 优值figure of noise 噪声指数file文件filled band 满带filled level满带能级filled shell 满壳层film软片film badge胶片剂量计film boiling膜态沸腾film dosimeter胶片剂量计film type condensation 膜状凝结filter滤器滤光器滤波器filtration 过滤final product最终产物final state interaction 终态相互酌final vacuum极限真空finder取景器finding telescope 寻星镜fine particle 微粒子fine structure 精细结构fine structure constant 精细结构常数fine structure splitting 精细结构劈裂finesse 锐度finite difference method 差分法finite element method 有限元法finite group 有限群finite universe 有限宇宙fireball 火球fireball model 火球模型first harmonics 基波first integral 初积分first law of thermodynamics 热力学第一定律first order phase transition 一级相转变first point of aries 春分点first quarter 上弦first resonance 一次共振first sound第一次声波fish finder鱼群探测器fissible可裂变的fissile可裂变的fissile nucleus可分裂的核fissility可分变性fission 分裂fission chain reaction 裂变链式反应fission chamber 裂变室fission counter 裂变计数器fission energy 裂变能量fission event裂变事件fission fragment 裂变碎片fission neutron 裂变中子fission yield裂变产额fissionable可裂变的fissionable nucleus 可裂变核 fissioning isomer 裂变同质异能素 fit适合five minute oscillations 五分钟振动 fixation 定影fixed capacitor固定电容器fixed point 定点fixed resistor固定电阻器fixed star 恒星fixing定影fixing solution 定影液fizeau fringe 菲佐条纹fizeau interferometer 菲佐干涉仪flame火焰flame photometer 火焰光度计flame photometry火焰光度法flame spectrum 火焰光谱flare太阳耀斑flare star 耀星flash闪光flash of light 闪光flash photolysis 闪烁光解flash spectrum 闪光光谱flat surface 平面flatness problem平坦性问题flattening 扁率flavor 味flexibility 挠性flexible polymer挠性聚合体flexural rigidity 弯曲刚度flexural strength 抗挠强度flexure 挠曲flexure vibration 弯曲振动flicker 闪烁flicker effect 闪烁效应flicker noise闪变噪声flicker photometer 闪烁光度计flight path飞越距离flight time飞越时间flint glass火石玻璃flip flop触发器flip flop circuit双稳态触发电路floating body 浮体floating point representation 浮点表示floating zone melting method 浮区熔炼法flocculation 凝聚flood and ebb 潮汐flooding 溢流floppy disk 软磁盘flow 流flow birefringence 怜双折射flow counter柳式计数管flow dichroism 怜二色性flow equation 怜方程flow parameter 怜参数flow pattern 镣flow proportional counter 怜正比计数器flow rate 量flow visualization 辽视化flowmeter 量计fluctuating force 涨落力fluctuation 起伏fluctuation dissipation theorem 涨落耗散定理fluctuon起伏量子fluent变数fluid 铃fluid dynamics 铃动力学fluid elasticity 水弹性fluid model 铃模型fluidal 铃的fluidic 铃的fluidity 怜性fluorescence 荧光fluorescence center 荧光中心fluorescence dosimeter 荧光剂量计fluorescence spectrophotometer 荧光分光光度计fluorescence spectrum 荧光光谱fluorescence yield 荧光产额fluorescent 萤光的fluorescent indicator tube 荧光指示管fluorescent lamp 荧光灯fluorescent material 萤光材料fluorescent radiation 特狰射fluorescent scattering 荧光散射fluorescent screen 荧光屏fluorescent x rays 荧光 x 射线fluorimeter 荧光计fluorine 氟fluorite structure 萤石型结构fluorography荧光照相法fluorometer 荧光计fluorometric analysis 荧光分析fluoroscope 荧光镜fluoroscopy荧光检查法fluorspar structure 萤石型结构flux通量flux creep磁通量蠕变flux density通量密度flux flow磁通量流flux jump磁通量跃变flux method 熔剂法flux motion 磁通量运动flux of energy 能通量flux of light 光通量flux of radiation 辐射通量flux pinning 通量锁住flux pump 通量泵flux quantization 磁通量量子化flux quantum磁通量子fluxmeter磁通计fluxoid磁通量子fluxon磁通量子fly苍蝇座flying spot electron microscope 扫描电子显微镜fm receiver档接受器focal distance 焦星巨focal length 焦星巨focal line 焦线focal plane 焦面focal point 焦点focal power 光焦度focal surface 焦曲面fock representation 福克表示fock space福克空间focometer焦距计focon聚焦锥focus焦点focused ion beam聚焦离子束focusing 倒focusing camera 倒照相机focusing coil聚焦线圈focusing cone 聚焦锥focusing lens 聚焦透镜focusing quadrupole magnet 聚焦四极磁铁focuson聚焦子foil 箔fokker planck equation 福克普朗克方程follow up control 随动控制foot pound second system 英尺磅秒单位制forbidden禁戒的forbidden band 禁带forbidden decay 禁戒衰变forbidden line 禁线forbidden lines in astrophysics 天体物理学中的禁线forbidden reflection 禁戒反射forbidden transition 禁戒跃迁forbiddenness 禁戒force 力force function 力函数force of attraction 弓|力force of gravity 重力force of inertia 惯性力force of repulsion 斥力force of rolling friction 滚动摩擦力force of sliding friction 滑动摩擦力force polygon力的多边形force triangle力的三角形forced circulation 强制循环forced convection 强制对流forced emission 强迫发射forced oscillations 受迫振荡forced vibration 受迫振荡forced vortex 强迫涡流fore vacuum 前级真空forecast 预报forecast of solar activity 太阳活动预告foreign atom 杂质原子forepump预备真空泵fork mounting 叉式装置form drag 型阻form factor形状因子formant共振峰formation of order 秩序形成formula 公式formulation 公式化fornax天炉座fortran程序语言forward scattering 前方散射foucault currents 涡电流foucault knife edge test 傅科刀口检验foucault's pendulum 傅科摆fountain effect 喷水效应four current 四维电流four dimensional space 四维空间four dimensional structure of the universe 宇宙四维结构four dimensionality 四维性four factor formula 四因子公式four force四维力four momentum 四维动量four potential 四维势four terminal network 四端网络four vector 四维矢four velocity四维速度four wave mixing 四波混合fourier component 谐波分量fourier integral傅里叶积分fourier number 傅里叶数fourier series傅里叶级数fourier spectroscopy 傅里叶光谱学fourier transform hologram傅里叶变换全息图fourier transform spectrometer 傅里叶光谱仪fourier transformation 傅里叶变换fourth sound第四次声波fractal 分形fractal dimension 分形维数fractional charge 分数电荷fractional crystallization 分级结晶fractional quantum hall effect 分数量子霍尔效应fracton分形子fracture 破裂fracture mechanics 断裂力学fragility 脆性frame antenna环形天线frame of reference 参考系francium 钫franck condon principle 富兰克康登原理franck hertz,s experiment 富兰克赫兹实验 frank read source弗朗克里德源franklin富兰克林fraunhofer diffraction 夫琅和费衍射fraunhofer hologram夫琅和费全息图fraunhofer line夫琅和费谱线fre量子free自由的free charge自由电荷free convection自由对流自然对流free electron自由电子free electron laser自由电子激光器free energy 自由能free energy of activation 激活的自由能free field自由场free free transition 自由自由跃迁free group自由群free gyroscope自由陀螺仪free motion自由运动free neutron自由中子free oscillation 自由振动free path自由程free pendulum 自由摆free rotation自由旋转free space自由空间free state自由态free surface 自由面free system 自由系free vibration 自由振动free volume自由体积free volume theory自由体积理论freedom 自由freezing 凝固freezing mixture 冷冻剂freezing point 凝固点frenkel defect夫伦克尔缺陷frenkel exciton 夫伦克尔激子frequency 频率frequency analysis 频率分析frequency band 频带frequency characteristic 频率特性frequency converter 变频器frequency converter tube 变频管frequency counter 频率计数器frequency divider 分频器frequency domain 频域frequency factor 频率因子frequency meter 频率计frequency modulation 档frequency multiplier倍频器频率倍增器frequency range 频率范围frequency response 频率响应frequency response method 频率特性法frequency shift 移频frequency spectrum 频谱frequency stability 频率稳定度frequency stabilized laser 稳频激光器frequency transfer function 频率传递函数fresnel diffraction 菲涅耳衍射fresnel half period zones 菲涅耳半周期带fresnel hologram菲涅耳全息图fresnel lens菲涅耳透镜fresnel prism 菲涅耳棱镜fresnel rhomb菲涅耳斜方系fresnel zone 菲涅耳带fresnel,s biprism菲涅耳双棱镜fresnel's dragging coefficient 菲涅耳曳引系数fresnel,s zone plate 菲涅耳波带片friction 摩擦friction coefficient 摩擦系数friction cone 摩擦锥friction layer 摩擦层friction loss摩擦损失friction of fluid lubrication 液体润滑摩擦frictional drag 摩擦阻力frictional electricity 摩擦电frictional force 摩擦力frictional oscillation 摩擦振动frictional resistance 摩擦阻力 friedel sum rule弗里德尔的求和定则 friedmann equation 弗里德曼方程friedmann universe弗里德曼宇宙frigorimeter 深冷温度计fringes with white light 白光干涉条纹froude number 弗劳德数frozen in magnetic field 冻结磁场frustrated total internal reflection 衰减全内反射frustration 抑止ft value ft 值fuel assembly燃料组件fuel cell燃料电池fuel cycle燃料循环fuel regeneration 燃料再生fuel reprocessing 燃料再生fuel rod燃料元件棒fugacity挥发性fulcrum 支点full load 满载full moon 望月full wave rectification 全波整流full width at half maximum 半宽度fullerene球壳状碳分子function 函数functional 泛函functional analysis 泛函分析functional ceramics 机能陶瓷functional derivative 泛函微分fundamental absorption 基本吸收fundamental catalog 基本星表fundamental constants 基本常数fundamental doublet 基本双重线fundamental frequency 基频率fundamental interaction 基本相互酌fundamental law 基本定律fundamental magnitude 基本量fundamental mode 关Bfundamental particle 基础粒子fundamental research 基础研究fundamental series 伯格曼系fundamental star 基本星fundamental theorem 基本定理fundamental tone 基音fundamental unit 基本单位furnace 炉furry,s theorem 弗里定理fuse熔断器保险丝fused quartz 熔融石英fusible alloy易熔合金fusing point 熔点fusion熔化fusion fission hybrid reactor核聚变裂变混合反应堆fusion point 熔点fusion reaction 聚变反应fusion reactor 热核堆fusion temperature 聚变温度。

大全,MCM备用数学方法中英文对照

MCM备用数学方法中英文对照MCM临近了，专业规范的表达是必不可少的，大家都来热热身吧！各位模友们自己有想到的数学方法或相关方面的专业词汇也可以贴出中英对照的版本！[B]计算数学方法：[/B]内插法：interpolation method有限元方法：method of finite element有限差分法：method of finite difference曲线拟合法：method of curve fitting迭代法：method of iteration误差分析法：error analysis样条函数法：method of spline function最小二乘法：method of least square[B]概率论方法：[/B]可靠性分析法：method of reliability analysis时间序列分析法： method of time-series analysis抽样调查法：method of sampling investigation非参数统计法：method of nonparametric statistics实验设计法：method of experiment design故障分析法：method of fault analysis相关分析法：method of correlation analysis测度论方法：method of measure theory统计假设检验法： method of statistical hypothesis testing随机服务法（排队论法）：stochastic service system数理统计法：method of mathematical statistics 蒙特卡洛法：Monte Carlo method[B]运筹学方法：[/B]共轭函数法：method of conjugate function动态规划法：method of dynamic programming网络法：method of network优选法：method of optimum seeking图论法：method of graph theory爬山法：method of climbing线性规划法：method of linear programming罚函数法： method of penalty function统筹法：method of overall planning乘子法：method of multiplier最速下降法：method of steepest descent整数规划法：method of integer programming。

1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。

C. Li & Fபைடு நூலகம் Zeng
However, the diﬀerences exist in that the fractional derivatives are globally deﬁned compared with the classical one deﬁned in a pointwise manner, which attributes to the diﬃculties in the construction of numerical methods and analysis (including consistency, convergence and stability) for the FDEs, especially for the nonlinear ones. At the same time, the computation cost and storage are much expensive, since the coeﬃcient matrices derived from the fractional systems can be full. At present, there are several numerical methods to solve FDEs, such as the ﬁnite diﬀerence method (rigorously speaking, the fractional ﬁnite diﬀerence method, here we just use it for brevity), the ﬁnite element method and the spectral method which are relatively rare [Roop, 2006, 2008; Zheng et al., 2010a, 2010b; Lin & Xu, 2007; Li & Xu, 2009] in literature, while the work on the ﬁnite diﬀerence method for FDEs are very rich and continues to be developed. This is mainly due to the following reasons. The ﬁnite diﬀerence method is a powerful tool and widely used to solve the DEs as well as the FDEs in science and engineering, which is also easy to be understood. Meanwhile, the implementation of the ﬁnite diﬀerence scheme is simple and easy to be put into practice in computer programs. In this paper, we just review almost all the existing FDMs for the FDEs of diﬀerent types. If some important references have been omitted, we do apologize for those omissions. In recent publications, FDMs for the FDEs almost concentrate on the kinetic equations of diﬀusion (subdiﬀusion and superdiﬀusion [Li & Zhao, 2011a]), diﬀusion– advection, and Fokker–planck type, etc. with partial fractional derivatives that are derived asymptotically from basic random walk models, the generalized master and Langevin equations. We will outline those numerical schemes for some typical FDEs and their main results (including consistency, stability and convergence). Simultaneously, the unresolved problems are also pointed out. To some extent, some of FDMs for FDEs can be seen as the generalizations of the corresponding methods for classical equations [Yuste, 2006; Meerschaert & Tadjeran, 2006]. The tools, say von Neumann stability analysis [Yuste & Acedo, 2005; Ghazizadeh et al., 2010], Fourier analysis method [Chen et al., 2007; Sousa, 2009] and energy method [Zhuang et al., 2008; Gao & Sun, 2011], which are used for the numerical analysis of the classical PDEs, can be extended to the cases for
FINITE DIFFERENCE METHODS FOR FRACTIONAL DIFFERENTIAL EQUATIONS*
CHANGPIN LI† and FANHAI ZENG Department of Mathematics, Shanghai University, Shanghai 200444, P. R. China Received March 15, 2011; Revised May 6, 2011
∗
[Metzler & Klafter, 2000]. Analytical methods, such as the Fourier transform method, the Laplace transform method, the Mellin transform method and the Green function method, etc. have been developed to seek the closed-form analytical solutions of the FDEs [Podlubny, 1999], while there are very few simple cases (such as some linear FDEs) in which the analytical solutions are available [Podlubny, 1999; Shen et al., 2008a]. Therefore, developing eﬃcient and reliable numerical methods for FDEs is of great importance. Numerical methods for the classical diﬀerential equations (DEs) are abundant, while numerical methods for the FDEs are very limited. Loosely but not mathematically speaking, the fractional derivatives can be seen as a generalization of the classical derivative, so some numerical methods for the classical DEs can be extended to the FDEs in some way.
In this review paper, the ﬁnite diﬀerence methods (FDMs) for the fractional diﬀerential equations are displayed. The considered equations mainly include the fractional kinetic equations of diﬀusion or dispersion with time, space and time-space derivatives. In some way, these numerical methods have similar form as the case for classical equations, some of which can be seen as the generalizations of the FDMs for the typical diﬀerential equations. And the classical tools, such as the von Neumann analysis method, the energy method and the Fourier method are extended to numerical methods for fractional diﬀerential equations accordingly. At the same time, the techniques for improving the accuracy and reducing the computation and storage are also introduced. Keywords : Fractional diﬀerential equations; ﬁnite diﬀerence method; extrapolation method; alternating direction implicit method.