This paper is concerned with numerical solutions of time-fractional nonlinear parabolic problems by a class of L1-Galerkin finite element methods.The analysis of L1 methods for time-fractional nonlinear problems is li...This paper is concerned with numerical solutions of time-fractional nonlinear parabolic problems by a class of L1-Galerkin finite element methods.The analysis of L1 methods for time-fractional nonlinear problems is limited mainly due to the lack of a fundamental Gronwall type inequality.In this paper,we establish such a fundamental inequality for the L1 approximation to the Caputo fractional derivative.In terms of the Gronwall type inequality,we provide optimal error estimates of several fully discrete linearized Galerkin finite element methods for nonlinear problems.The theoretical results are illustrated by applying our proposed methods to the time fractional nonlinear Huxley equation and time fractional Fisher equation.展开更多
In recent years,the epidemic model with anomalous diffusion has gained popularity in the literature.However,when introducing anomalous diffusion into epidemic models,they frequently lack physical explanation,in contra...In recent years,the epidemic model with anomalous diffusion has gained popularity in the literature.However,when introducing anomalous diffusion into epidemic models,they frequently lack physical explanation,in contrast to the traditional reaction-diffusion epidemic models.The point of this paper is to guarantee that anomalous diffusion systems on infectious disease spreading remain physically reasonable.Specifically,based on the continuous-time random walk(CTRW),starting from two stochastic processes of the waiting time and the step length,time-fractional space-fractional diffusion,timefractional reaction-diffusion and fractional-order diffusion can all be naturally introduced into the SIR(S:susceptible,I:infectious and R:recovered)epidemic models,respectively.The three models mentioned above can also be applied to create SIR epidemic models with generalized distributed time delays.Distributed time delay systems can also be reduced to existing models,such as the standard SIR model,the fractional infectivity model and others,within the proper bounds.Meanwhile,as an application of the above stochastic modeling method,the physical meaning of anomalous diffusion is also considered by taking the SEIR(E:exposed)epidemic model as an example.Similar methods can be used to build other types of epidemic models,including SIVRS(V:vaccine),SIQRS(Q:quarantined)and others.Finally,this paper describes the transmission of infectious disease in space using the real data of COVID-19.展开更多
A two-grid finite element method with L1 scheme is presented for solving two-dimen-sional time-fractional nonlinear Schrodinger equation.The finite element solution in the L-norm are proved bounded without any time-st...A two-grid finite element method with L1 scheme is presented for solving two-dimen-sional time-fractional nonlinear Schrodinger equation.The finite element solution in the L-norm are proved bounded without any time-step size conditions(dependent on spatial-step size).The classical L1 scheme is considered in the time direction,and the two-grid finite element method is applied in spatial direction.The optimal order error estimations of the two-grid solution in the LP-norm is proved without any time-step size conditions.It is shown,both theoretically and numerically,that the coarse space can be extremely coarse,with no loss in the order of accuracy.展开更多
Fractional initial-value problems(IVPs) and time-fractional initial-boundary value problems(IBVPs), each with a Caputo temporal derivative of order α ∈(0, 1), are considered. An averaged variant of the well-known L1...Fractional initial-value problems(IVPs) and time-fractional initial-boundary value problems(IBVPs), each with a Caputo temporal derivative of order α ∈(0, 1), are considered. An averaged variant of the well-known L1 scheme is proved to be O(N^(-2)) convergent for IVPs on suitably graded meshes with N points, thereby improving the O(N^(-(2-α))) convergence rate of the standard L1 scheme. The analysis relies on a delicate decomposition of the temporal truncation error that yields a sharp dependence of the order of convergence on the degree of mesh grading used. This averaged L1 scheme can be combined with a finite difference or piecewise linear finite element discretization in space for IBVPs, and under a restriction on the temporal mesh width, one gets again O(N^(-2)) convergence in time, together with O(h^(2)) convergence in space,where h is the spatial mesh width. Numerical experiments support our results.展开更多
Fractional diffusion equations have been the focus of modeling problems in hydrology, biology, viscoelasticity, physics, engineering, and other areas of applications. In this paper, a meshfree method based on the movi...Fractional diffusion equations have been the focus of modeling problems in hydrology, biology, viscoelasticity, physics, engineering, and other areas of applications. In this paper, a meshfree method based on the moving Kriging inter- polation is developed for a two-dimensional time-fractional diffusion equation. The shape function and its derivatives are obtained by the moving Kriging interpolation technique. For possessing the Kronecker delta property, this technique is very efficient in imposing the essential boundary conditions. The governing time-fractional diffusion equations are transformed into a standard weak formulation by the Galerkin method. It is then discretized into a meshfree system of time-dependent equations, which are solved by the standard central difference method. Numerical examples illustrating the applicability and effectiveness of the proposed method are presented and discussed in detail.展开更多
Motivated by the widely used ans¨atz method and starting from the modified Riemann–Liouville derivative together with a fractional complex transformation that can be utilized to transform nonlinear fractional pa...Motivated by the widely used ans¨atz method and starting from the modified Riemann–Liouville derivative together with a fractional complex transformation that can be utilized to transform nonlinear fractional partial differential equations to nonlinear ordinary differential equations, new types of exact traveling wave solutions to three important nonlinear space- and time-fractional partial differential equations are obtained simultaneously in terms of solutions of a Riccati equation. The results are new and first reported in this paper.展开更多
This paper is aimed at solving the nonlinear time-fractional partial differential equation with two small parameters arising from option pricing model in financial economics.The traditional reproducing kernel(RK)metho...This paper is aimed at solving the nonlinear time-fractional partial differential equation with two small parameters arising from option pricing model in financial economics.The traditional reproducing kernel(RK)method which deals with this problem is very troublesome.This paper proposes a new method by adaptive multi-step piecewise interpolation reproducing kernel(AMPIRK)method for the first time.This method has three obvious advantages which are as follows.Firstly,the piecewise number is reduced.Secondly,the calculation accuracy is improved.Finally,the waste time caused by too many fragments is avoided.Then four numerical examples show that this new method has a higher precision and it is a more timesaving numerical method than the others.The research in this paper provides a powerful mathematical tool for solving time-fractional option pricing model which will play an important role in financial economics.展开更多
This paper aims to investigate a new efficient method for solving time fractional partial differential equations.In this orientation,a reliable formable transform decomposition method has been designed and developed,w...This paper aims to investigate a new efficient method for solving time fractional partial differential equations.In this orientation,a reliable formable transform decomposition method has been designed and developed,which is a novel combination of the formable integral transform and the decomposition method.Basically,certain accurate solutions for time-fractional partial differential equations have been presented.Themethod under concern demandsmore simple calculations and fewer efforts compared to the existingmethods.Besides,the posed formable transformdecompositionmethod has been utilized to yield a series solution for given fractional partial differential equations.Moreover,several interesting formulas relevant to the formable integral transform are applied to fractional operators which are performed as an excellent application to the existing theory.Furthermore,the formable transform decomposition method has been employed for finding a series solution to a time-fractional Klein-Gordon equation.Over and above,some numerical simulations are also provided to ensure reliability and accuracy of the new approach.展开更多
This paper proposes a temporal-fractional porous medium model(T-FPMM)for describing the co-current and counter-current imbibition,which arises in a water-wet fractured porous media.The correlation be-tween the co-curr...This paper proposes a temporal-fractional porous medium model(T-FPMM)for describing the co-current and counter-current imbibition,which arises in a water-wet fractured porous media.The correlation be-tween the co-current and counter-current imbibition for the fractures and porous matrix are examined to determine the saturation and recovery rate of the reservoir.For different fractional orders in both porous matrix and fractured porous media,the homotopy analysis technique and its stability analysis are used to explore the parametric behavior of the saturation and recovery rates.Finally,the effects of wettability and inclination on the recovery rate and saturation are studied for distinct fractional values.展开更多
The present paper deals with two reliable efficient methods viz.tanh-sech method and modified Kudryashov method,which are used to solve time-fractional nonlinear evolution equation.For delineating the legitimacy of pr...The present paper deals with two reliable efficient methods viz.tanh-sech method and modified Kudryashov method,which are used to solve time-fractional nonlinear evolution equation.For delineating the legitimacy of proposed methods,we employ it to the time-fractional fifth-order modified Sawada-Kotera equations.As a consequence,we effectively obtained more new exact solutions for time-fractional fifth-order modified Sawada-Kotera equation.We have also presented the numerical simulations for time-fractional fifth-order modified Sawada-Kotera equation by means of three dimensional plots.展开更多
We present Alikhanov linearized Galerkin methods for solving the nonlinear time fractional Schrödinger equations.Unconditionally optimal estimates of the fully-discrete scheme are obtained by using the fractional...We present Alikhanov linearized Galerkin methods for solving the nonlinear time fractional Schrödinger equations.Unconditionally optimal estimates of the fully-discrete scheme are obtained by using the fractional time-spatial splitting argument.The convergence results indicate that the error estimates hold without any spatial-temporal stepsize restrictions.Numerical experiments are done to verify the theoretical results.展开更多
This paper aims to extend a space-time spectral method to address the multi-term time-fractional subdiffusion equations with Caputo derivative. In this method, the Jacobi polynomials are adopted as the basis functions...This paper aims to extend a space-time spectral method to address the multi-term time-fractional subdiffusion equations with Caputo derivative. In this method, the Jacobi polynomials are adopted as the basis functions for temporal discretization and the Lagrangian polynomials are used for spatial discretization. An efficient spectral approximation of the weak solution is established. The main work is the demonstration of the well-posedness for the weak problem and the derivation of a posteriori error estimates for the spectral Galerkin approximation. Extensive numerical experiments are presented to perform the validity of a posteriori error estimators, which support our theoretical results.展开更多
In this paper,we propose and analyze high order energy dissipative time-stepping schemes for time-fractional molecular beam epitaxial(MBE)growth model on the nonuniform mesh.More precisely,(2−α)-order,secondorder and...In this paper,we propose and analyze high order energy dissipative time-stepping schemes for time-fractional molecular beam epitaxial(MBE)growth model on the nonuniform mesh.More precisely,(2−α)-order,secondorder and(3−α)-order time-stepping schemes are developed for the timefractional MBE model based on the well known L1,L2-1σ,and L2 formulations in discretization of the time-fractional derivative,which are all proved to be unconditional energy dissipation in the sense of a modified discrete nonlocalenergy on the nonuniform mesh.In order to reduce the computational storage,we apply the sum of exponential technique to approximate the history part of the time-fractional derivative.Moreover,the scalar auxiliary variable(SAV)approach is introduced to deal with the nonlinear potential function and the history part of the fractional derivative.Furthermore,only first order method is used to discretize the introduced SAV equation,which will not affect high order accuracy of the unknown thin film height function by using some proper auxiliary variable functions V(ξ).To our knowledge,it is the first time to unconditionally establish the discrete nonlocal-energy dissipation law for the modified L1-,L2-1σ-,and L2-based high-order schemes on the nonuniform mesh,which is essentially important for such time-fractional MBE models with low regular solutions at initial time.Finally,a series of numerical experiments are carried out to verify the accuracy and efficiency of the proposed schemes.展开更多
In this paper, the new mapping approach and the new extended auxiliary equation approach were used to investigate the exact traveling wave solutions of (2 + 1)-dimensional time-fractional Zoomeron equation with the co...In this paper, the new mapping approach and the new extended auxiliary equation approach were used to investigate the exact traveling wave solutions of (2 + 1)-dimensional time-fractional Zoomeron equation with the conformable fractional derivative. As a result, the singular soliton solutions, kink and anti-kink soliton solutions, periodic function soliton solutions, Jacobi elliptic function solutions and hyperbolic function solutions of (2 + 1)-dimensional time-fractional Zoomeron equation were obtained. Finally, the 3D and 2D graphs of some solutions were drawn by setting the suitable values of parameters with Maple, and analyze the dynamic behaviors of the solutions.展开更多
In the paper under review,we consider the generation of fractional resolvent families by abstract differential operators.Our results can be simply incorporated in the study of corresponding abstract time-fractional eq...In the paper under review,we consider the generation of fractional resolvent families by abstract differential operators.Our results can be simply incorporated in the study of corresponding abstract time-fractional equations with Caputo fractional derivatives.展开更多
In this paper, we approximate the solution to time-fractional telegraph equation by two kinds of difference methods: the Grünwald formula and Caputo fractional difference.
In the current paper, based on fractional complex transformation, the GG2-expansion method which is used to solve differential equations of integer order is developed for finding exact solutions of nonlinear fractiona...In the current paper, based on fractional complex transformation, the GG2-expansion method which is used to solve differential equations of integer order is developed for finding exact solutions of nonlinear fractional differential equations with Jumarie's modified Riemann-Liouville derivative. And then, time-fractional Burgers equation and space-fractional coupled Konopelchenko-Dubrovsky equations are provided to show that this method is effective in solving nonlinear fractional differential equations.展开更多
The analytical solution of the multi-dimensional,time-fractional model of Navier-Stokes equation using the triple and quadruple Elzaki transformdecompositionmethod is presented in this article.The aforesaidmodel is an...The analytical solution of the multi-dimensional,time-fractional model of Navier-Stokes equation using the triple and quadruple Elzaki transformdecompositionmethod is presented in this article.The aforesaidmodel is analyzed by employing Caputo fractional derivative.We deliberated three stimulating examples that correspond to the triple and quadruple Elzaki transform decomposition methods,respectively.The findings illustrate that the established approaches are extremely helpful in obtaining exact and approximate solutions to the problems.The exact and estimated solutions are delineated via numerical simulation.The proposed analysis indicates that the projected configuration is extremely meticulous,highly efficient,and precise in understanding the behavior of complex evolutionary problems of both fractional and integer order that classify affiliated scientific fields and technology.展开更多
In this article,we study the energy dissipation property of time-fractional Allen–Cahn equation.On the continuous level,we propose an upper bound of energy that decreases with respect to time and coincides with the o...In this article,we study the energy dissipation property of time-fractional Allen–Cahn equation.On the continuous level,we propose an upper bound of energy that decreases with respect to time and coincides with the original energy at t=0 and as t tends to∞.This upper bound can also be viewed as a nonlocal-in-time modified energy which is the summation of the original energy and an accumulation term due to the memory effect of time-fractional derivative.In particular,the decrease of the modified energy indicates that the original energy indeed decays w.r.t.time in a small neighborhood at t=0.We illustrate the theory mainly with the time-fractional Allen-Cahn equation but it could also be applied to other time-fractional phase-field models such as the Cahn-Hilliard equation.On the discrete level,the decreasing upper bound of energy is useful for proving energy dissipation of numerical schemes.First-order L1 and second-order L2 schemes for the time-fractional Allen-Cahn equation have similar decreasing modified energies,so that stability can be established.Some numerical results are provided to illustrate the behavior of this modified energy and to verify our theoretical results.展开更多
基金This work is supported by NSFC(Grant Nos.11771035,11771162,11571128,61473126,91430216,91530204,11372354 and U1530401),a grant from the RGC of HK 11300517,China(Project No.CityU 11302915),China Postdoctoral Science Foundation under grant No.2016M602273,a grant DRA2015518 from 333 High-level Personal Training Project of Jiangsu Province,and the USA National Science Foundation grant DMS-1315259the USA Air Force Office of Scientific Research grant FA9550-15-1-0001.Jiwei Zhang also thanks the hospitality of Hong Kong City University during the period of his visiting.
文摘This paper is concerned with numerical solutions of time-fractional nonlinear parabolic problems by a class of L1-Galerkin finite element methods.The analysis of L1 methods for time-fractional nonlinear problems is limited mainly due to the lack of a fundamental Gronwall type inequality.In this paper,we establish such a fundamental inequality for the L1 approximation to the Caputo fractional derivative.In terms of the Gronwall type inequality,we provide optimal error estimates of several fully discrete linearized Galerkin finite element methods for nonlinear problems.The theoretical results are illustrated by applying our proposed methods to the time fractional nonlinear Huxley equation and time fractional Fisher equation.
基金This work is supported in part by the National Natural Science Foundation of China(Grant Nos.62173027,62003026 and 61973329)the Natural Science Foundation of Beijing Municipality(Grant No.Z180005)Alianza UCMX seed funding(2020-2022)on Binational Collaborative Projects addressing COVID-19.
文摘In recent years,the epidemic model with anomalous diffusion has gained popularity in the literature.However,when introducing anomalous diffusion into epidemic models,they frequently lack physical explanation,in contrast to the traditional reaction-diffusion epidemic models.The point of this paper is to guarantee that anomalous diffusion systems on infectious disease spreading remain physically reasonable.Specifically,based on the continuous-time random walk(CTRW),starting from two stochastic processes of the waiting time and the step length,time-fractional space-fractional diffusion,timefractional reaction-diffusion and fractional-order diffusion can all be naturally introduced into the SIR(S:susceptible,I:infectious and R:recovered)epidemic models,respectively.The three models mentioned above can also be applied to create SIR epidemic models with generalized distributed time delays.Distributed time delay systems can also be reduced to existing models,such as the standard SIR model,the fractional infectivity model and others,within the proper bounds.Meanwhile,as an application of the above stochastic modeling method,the physical meaning of anomalous diffusion is also considered by taking the SEIR(E:exposed)epidemic model as an example.Similar methods can be used to build other types of epidemic models,including SIVRS(V:vaccine),SIQRS(Q:quarantined)and others.Finally,this paper describes the transmission of infectious disease in space using the real data of COVID-19.
基金supported by the State Key Program of National Natural Science Foundation of China(Grant No.11931003)by the National Natural Science Foundation of China(Grant Nos.41974133,12271233)+1 种基金The work of first author was supported by the Guangdong Basic and Applied Basic Research Foundation(Grant Nos.2024A1515012430,2020A1515011032)by the Educational Commission of Guangdong Province,China(Grant No.2019KTSCX174).
文摘A two-grid finite element method with L1 scheme is presented for solving two-dimen-sional time-fractional nonlinear Schrodinger equation.The finite element solution in the L-norm are proved bounded without any time-step size conditions(dependent on spatial-step size).The classical L1 scheme is considered in the time direction,and the two-grid finite element method is applied in spatial direction.The optimal order error estimations of the two-grid solution in the LP-norm is proved without any time-step size conditions.It is shown,both theoretically and numerically,that the coarse space can be extremely coarse,with no loss in the order of accuracy.
基金supported by National Natural Science Foundation of China (Grant Nos. 12101509, 12171283, 12171025 and NSAF-U1930402)the Science Foundation Program for Distinguished Young Scholars of Shandong (Overseas) (Grant No. 2022HWYQ-045)。
文摘Fractional initial-value problems(IVPs) and time-fractional initial-boundary value problems(IBVPs), each with a Caputo temporal derivative of order α ∈(0, 1), are considered. An averaged variant of the well-known L1 scheme is proved to be O(N^(-2)) convergent for IVPs on suitably graded meshes with N points, thereby improving the O(N^(-(2-α))) convergence rate of the standard L1 scheme. The analysis relies on a delicate decomposition of the temporal truncation error that yields a sharp dependence of the order of convergence on the degree of mesh grading used. This averaged L1 scheme can be combined with a finite difference or piecewise linear finite element discretization in space for IBVPs, and under a restriction on the temporal mesh width, one gets again O(N^(-2)) convergence in time, together with O(h^(2)) convergence in space,where h is the spatial mesh width. Numerical experiments support our results.
基金Project supported by the National Natural Science Foundation of China(Grant No.11072117)the Natural Science Foundation of Ningbo City,China(GrantNo.2013A610103)+2 种基金the Natural Science Foundation of Zhejiang Province,China(Grant No.Y6090131)the Disciplinary Project of Ningbo City,China(GrantNo.SZXL1067)the K.C.Wong Magna Fund in Ningbo University,China
文摘Fractional diffusion equations have been the focus of modeling problems in hydrology, biology, viscoelasticity, physics, engineering, and other areas of applications. In this paper, a meshfree method based on the moving Kriging inter- polation is developed for a two-dimensional time-fractional diffusion equation. The shape function and its derivatives are obtained by the moving Kriging interpolation technique. For possessing the Kronecker delta property, this technique is very efficient in imposing the essential boundary conditions. The governing time-fractional diffusion equations are transformed into a standard weak formulation by the Galerkin method. It is then discretized into a meshfree system of time-dependent equations, which are solved by the standard central difference method. Numerical examples illustrating the applicability and effectiveness of the proposed method are presented and discussed in detail.
基金Supported by National Natural Science Foundation of China under Grant Nos.11071278,111471004the Fundamental Research Funds for the Central Universities of GK201302026 and GK201102007
文摘Motivated by the widely used ans¨atz method and starting from the modified Riemann–Liouville derivative together with a fractional complex transformation that can be utilized to transform nonlinear fractional partial differential equations to nonlinear ordinary differential equations, new types of exact traveling wave solutions to three important nonlinear space- and time-fractional partial differential equations are obtained simultaneously in terms of solutions of a Riccati equation. The results are new and first reported in this paper.
基金the National Natural Science Foundation of China(Grant Nos.71961022,11902163,12265020,and 12262024)the Natural Science Foundation of Inner Mongolia Autonomous Region of China(Grant Nos.2019BS01011 and 2022MS01003)+5 种基金2022 Inner Mongolia Autonomous Region Grassland Talents Project-Young Innovative and Entrepreneurial Talents(Mingjing Du)2022 Talent Development Foundation of Inner Mongolia Autonomous Region of China(Ming-Jing Du)the Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region Program(Grant No.NJYT-20-B18)the Key Project of High-quality Economic Development Research Base of Yellow River Basin in 2022(Grant No.21HZD03)2022 Inner Mongolia Autonomous Region International Science and Technology Cooperation High-end Foreign Experts Introduction Project(Ge Kai)MOE(Ministry of Education in China)Humanities and Social Sciences Foundation(Grants No.20YJC860005).
文摘This paper is aimed at solving the nonlinear time-fractional partial differential equation with two small parameters arising from option pricing model in financial economics.The traditional reproducing kernel(RK)method which deals with this problem is very troublesome.This paper proposes a new method by adaptive multi-step piecewise interpolation reproducing kernel(AMPIRK)method for the first time.This method has three obvious advantages which are as follows.Firstly,the piecewise number is reduced.Secondly,the calculation accuracy is improved.Finally,the waste time caused by too many fragments is avoided.Then four numerical examples show that this new method has a higher precision and it is a more timesaving numerical method than the others.The research in this paper provides a powerful mathematical tool for solving time-fractional option pricing model which will play an important role in financial economics.
基金funded by the Deanship of Research in Zarqa University,Jordan。
文摘This paper aims to investigate a new efficient method for solving time fractional partial differential equations.In this orientation,a reliable formable transform decomposition method has been designed and developed,which is a novel combination of the formable integral transform and the decomposition method.Basically,certain accurate solutions for time-fractional partial differential equations have been presented.Themethod under concern demandsmore simple calculations and fewer efforts compared to the existingmethods.Besides,the posed formable transformdecompositionmethod has been utilized to yield a series solution for given fractional partial differential equations.Moreover,several interesting formulas relevant to the formable integral transform are applied to fractional operators which are performed as an excellent application to the existing theory.Furthermore,the formable transform decomposition method has been employed for finding a series solution to a time-fractional Klein-Gordon equation.Over and above,some numerical simulations are also provided to ensure reliability and accuracy of the new approach.
文摘This paper proposes a temporal-fractional porous medium model(T-FPMM)for describing the co-current and counter-current imbibition,which arises in a water-wet fractured porous media.The correlation be-tween the co-current and counter-current imbibition for the fractures and porous matrix are examined to determine the saturation and recovery rate of the reservoir.For different fractional orders in both porous matrix and fractured porous media,the homotopy analysis technique and its stability analysis are used to explore the parametric behavior of the saturation and recovery rates.Finally,the effects of wettability and inclination on the recovery rate and saturation are studied for distinct fractional values.
文摘The present paper deals with two reliable efficient methods viz.tanh-sech method and modified Kudryashov method,which are used to solve time-fractional nonlinear evolution equation.For delineating the legitimacy of proposed methods,we employ it to the time-fractional fifth-order modified Sawada-Kotera equations.As a consequence,we effectively obtained more new exact solutions for time-fractional fifth-order modified Sawada-Kotera equation.We have also presented the numerical simulations for time-fractional fifth-order modified Sawada-Kotera equation by means of three dimensional plots.
基金supported by the NSFC (No.12001067)by the Natural Science Foundation of Chongqing,China (No.cstc2019jcyj-bshX0038)by the China Postdoctoral Science Foundation funded Project No.2019M653333.
文摘We present Alikhanov linearized Galerkin methods for solving the nonlinear time fractional Schrödinger equations.Unconditionally optimal estimates of the fully-discrete scheme are obtained by using the fractional time-spatial splitting argument.The convergence results indicate that the error estimates hold without any spatial-temporal stepsize restrictions.Numerical experiments are done to verify the theoretical results.
基金supported by the State Key Program of National Natural Science Foundation of China(Nos.11931003)National Natural Science Foundation of China(Nos.41974133)。
文摘This paper aims to extend a space-time spectral method to address the multi-term time-fractional subdiffusion equations with Caputo derivative. In this method, the Jacobi polynomials are adopted as the basis functions for temporal discretization and the Lagrangian polynomials are used for spatial discretization. An efficient spectral approximation of the weak solution is established. The main work is the demonstration of the well-posedness for the weak problem and the derivation of a posteriori error estimates for the spectral Galerkin approximation. Extensive numerical experiments are presented to perform the validity of a posteriori error estimators, which support our theoretical results.
基金supported by NSFC grant 12001248,the NSF of Jiangsu Province grant BK20201020the NSF of Universities in Jiangsu Province of China grant 20KJB110013+3 种基金the Hong Kong Polytechnic University grant 1-W00Dsupported by Hong Kong Research Grants Council RFS grant RFS2021-5S03 and GRF grant 15302122,the Hong Kong Polytechnic University grant 1-9BCTCAS AMSS-PolyU Joint Laboratory of Applied Mathematicssupported by the Guangdong Provincial Key Laboratory of Interdisciplinary Research and Application for Data Science under UIC 2022B1212010006.
文摘In this paper,we propose and analyze high order energy dissipative time-stepping schemes for time-fractional molecular beam epitaxial(MBE)growth model on the nonuniform mesh.More precisely,(2−α)-order,secondorder and(3−α)-order time-stepping schemes are developed for the timefractional MBE model based on the well known L1,L2-1σ,and L2 formulations in discretization of the time-fractional derivative,which are all proved to be unconditional energy dissipation in the sense of a modified discrete nonlocalenergy on the nonuniform mesh.In order to reduce the computational storage,we apply the sum of exponential technique to approximate the history part of the time-fractional derivative.Moreover,the scalar auxiliary variable(SAV)approach is introduced to deal with the nonlinear potential function and the history part of the fractional derivative.Furthermore,only first order method is used to discretize the introduced SAV equation,which will not affect high order accuracy of the unknown thin film height function by using some proper auxiliary variable functions V(ξ).To our knowledge,it is the first time to unconditionally establish the discrete nonlocal-energy dissipation law for the modified L1-,L2-1σ-,and L2-based high-order schemes on the nonuniform mesh,which is essentially important for such time-fractional MBE models with low regular solutions at initial time.Finally,a series of numerical experiments are carried out to verify the accuracy and efficiency of the proposed schemes.
文摘In this paper, the new mapping approach and the new extended auxiliary equation approach were used to investigate the exact traveling wave solutions of (2 + 1)-dimensional time-fractional Zoomeron equation with the conformable fractional derivative. As a result, the singular soliton solutions, kink and anti-kink soliton solutions, periodic function soliton solutions, Jacobi elliptic function solutions and hyperbolic function solutions of (2 + 1)-dimensional time-fractional Zoomeron equation were obtained. Finally, the 3D and 2D graphs of some solutions were drawn by setting the suitable values of parameters with Maple, and analyze the dynamic behaviors of the solutions.
基金Supported by Ministry of Science and Technological Development,Republic of Serbia(Grant No.174024)
文摘In the paper under review,we consider the generation of fractional resolvent families by abstract differential operators.Our results can be simply incorporated in the study of corresponding abstract time-fractional equations with Caputo fractional derivatives.
文摘In this paper, we approximate the solution to time-fractional telegraph equation by two kinds of difference methods: the Grünwald formula and Caputo fractional difference.
基金Supported by the National Natural Science Foundation of China（11462019） Supported by the Scientific Research Foundation of Inner Mongolia University for Nationalities（NMDYB1750, NMDGP1713）
文摘In the current paper, based on fractional complex transformation, the GG2-expansion method which is used to solve differential equations of integer order is developed for finding exact solutions of nonlinear fractional differential equations with Jumarie's modified Riemann-Liouville derivative. And then, time-fractional Burgers equation and space-fractional coupled Konopelchenko-Dubrovsky equations are provided to show that this method is effective in solving nonlinear fractional differential equations.
基金supported by the Natural Science Foundation of China(GrantNos.61673169,11301127,11701176,11626101,11601485).
文摘The analytical solution of the multi-dimensional,time-fractional model of Navier-Stokes equation using the triple and quadruple Elzaki transformdecompositionmethod is presented in this article.The aforesaidmodel is analyzed by employing Caputo fractional derivative.We deliberated three stimulating examples that correspond to the triple and quadruple Elzaki transform decomposition methods,respectively.The findings illustrate that the established approaches are extremely helpful in obtaining exact and approximate solutions to the problems.The exact and estimated solutions are delineated via numerical simulation.The proposed analysis indicates that the projected configuration is extremely meticulous,highly efficient,and precise in understanding the behavior of complex evolutionary problems of both fractional and integer order that classify affiliated scientific fields and technology.
基金partially supported by the National Natural Science Foundation of China/Hong Kong RGC Joint Research Scheme(NSFC/RGC 11961160718)the fund of the Guangdong Provincial Key Laboratory of Computational Science And Material Design(No.2019B030301001)+4 种基金supported in part by the Guangdong Provincial Key Laboratory of Interdisciplinary Research and Application for Data Science under UIC 2022B1212010006supported by the National Science Foundation of China(NSFC)Grant No.12271240supported by NSFC Grant 12271241Guangdong Basic and Applied Basic Research Foundation(No.2023B1515020030)Shenzhen Science and Technology Program(Grant No.RCYX20210609104358076).
文摘In this article,we study the energy dissipation property of time-fractional Allen–Cahn equation.On the continuous level,we propose an upper bound of energy that decreases with respect to time and coincides with the original energy at t=0 and as t tends to∞.This upper bound can also be viewed as a nonlocal-in-time modified energy which is the summation of the original energy and an accumulation term due to the memory effect of time-fractional derivative.In particular,the decrease of the modified energy indicates that the original energy indeed decays w.r.t.time in a small neighborhood at t=0.We illustrate the theory mainly with the time-fractional Allen-Cahn equation but it could also be applied to other time-fractional phase-field models such as the Cahn-Hilliard equation.On the discrete level,the decreasing upper bound of energy is useful for proving energy dissipation of numerical schemes.First-order L1 and second-order L2 schemes for the time-fractional Allen-Cahn equation have similar decreasing modified energies,so that stability can be established.Some numerical results are provided to illustrate the behavior of this modified energy and to verify our theoretical results.
