In this article,we consider the diffusion equation with multi-term time-fractional derivatives.We first derive,by a subordination principle for the solution,that the solution is positive when the initial value is non-...In this article,we consider the diffusion equation with multi-term time-fractional derivatives.We first derive,by a subordination principle for the solution,that the solution is positive when the initial value is non-negative.As an application,we prove the uniqueness of solution to an inverse problem of determination of the temporally varying source term by integral type information in a subdomain.Finally,several numerical experiments are presented to show the accuracy and efficiency of the algorithm.展开更多
In this paper,a local discontinuous Galerkin(LDG)scheme for the time-fractional diffusion equation is proposed and analyzed.The Caputo time-fractional derivative(of orderα,with 0<α<1)is approximated by a finit...In this paper,a local discontinuous Galerkin(LDG)scheme for the time-fractional diffusion equation is proposed and analyzed.The Caputo time-fractional derivative(of orderα,with 0<α<1)is approximated by a finite difference method with an accuracy of order3-α,and the space discretization is based on the LDG method.For the finite difference method,we summarize and supplement some previous work by others,and apply it to the analysis of the convergence and stability of the proposed scheme.The optimal error estimate is obtained in the L2norm,indicating that the scheme has temporal(3-α)th-order accuracy and spatial(k+1)th-order accuracy,where k denotes the highest degree of a piecewise polynomial in discontinuous finite element space.The numerical results are also provided to verify the accuracy and efficiency of the considered scheme.展开更多
In this paper,we develop novel local discontinuous Galerkin(LDG)methods for fractional diffusion equations with non-smooth solutions.We consider such problems,for which the solutions are not smooth at boundary,and the...In this paper,we develop novel local discontinuous Galerkin(LDG)methods for fractional diffusion equations with non-smooth solutions.We consider such problems,for which the solutions are not smooth at boundary,and therefore the traditional LDG methods with piecewise polynomial solutions suffer accuracy degeneracy.The novel LDG methods utilize a solution information enriched basis,simulate the problem on a paired special mesh,and achieve optimal order of accuracy.We analyze the L2 stability and optimal error estimate in L2-norm.Finally,numerical examples are presented for validating the theoretical conclusions.展开更多
After discretization by the finite volume method,the numerical solution of fractional diffusion equations leads to a linear system with the Toeplitz-like structure.The theoretical analysis gives sufficient conditions ...After discretization by the finite volume method,the numerical solution of fractional diffusion equations leads to a linear system with the Toeplitz-like structure.The theoretical analysis gives sufficient conditions to guarantee the positive-definite property of the discretized matrix.Moreover,we develop a class of positive-definite operator splitting iteration methods for the numerical solution of fractional diffusion equations,which is unconditionally convergent for any positive constant.Meanwhile,the iteration methods introduce a new preconditioner for Krylov subspace methods.Numerical experiments verify the convergence of the positive-definite operator splitting iteration methods and show the efficiency of the proposed preconditioner,compared with the existing approaches.展开更多
In this paper,finite difference schemes for solving time-space fractional diffusion equations in one dimension and two dimensions are proposed.The temporal derivative is in the Caputo-Hadamard sense for both cases.The...In this paper,finite difference schemes for solving time-space fractional diffusion equations in one dimension and two dimensions are proposed.The temporal derivative is in the Caputo-Hadamard sense for both cases.The spatial derivative for the one-dimensional equation is of Riesz definition and the two-dimensional spatial derivative is given by the fractional Laplacian.The schemes are proved to be unconditionally stable and convergent.The numerical results are in line with the theoretical analysis.展开更多
We study an indirect finite element approximation for two-sided space-fractional diffusion equations in one space dimension.By the representation formula of the solutions u(x)to the proposed variable coefficient model...We study an indirect finite element approximation for two-sided space-fractional diffusion equations in one space dimension.By the representation formula of the solutions u(x)to the proposed variable coefficient models in terms of v(x),the solutions to the constant coefficient analogues,we apply finite element methods for the constant coefficient fractional diffusion equations to solve for the approximations vh(x)to v(x)and then obtain the approximations uh(x)of u(x)by plugging vh(x)into the representation of u(x).Optimal-order convergence estimates of u(x)−uh(x)are proved in both L2 and Hα∕2 norms.Several numerical experiments are presented to demonstrate the sharpness of the derived error estimates.展开更多
This paper deals with the numerical approximation for the time fractional diffusion problem with fractional dynamic boundary conditions.The well-posedness for the weak solutions is studied.A direct discontinuous Galer...This paper deals with the numerical approximation for the time fractional diffusion problem with fractional dynamic boundary conditions.The well-posedness for the weak solutions is studied.A direct discontinuous Galerkin approach is used in spatial direction under the uniform meshes,together with a second-order Alikhanov scheme is utilized in temporal direction on the graded mesh,and then the fully discrete scheme is constructed.Furthermore,the stability and the error estimate for the full scheme are analyzed in detail.Numerical experiments are also given to illustrate the effectiveness of the proposed method.展开更多
In this paper,an initial boundary value problem of the space-time fractional diffusion equation is studied.Both temporal and spatial directions for this equation are discreted by the Galerkin spectral methods.And then...In this paper,an initial boundary value problem of the space-time fractional diffusion equation is studied.Both temporal and spatial directions for this equation are discreted by the Galerkin spectral methods.And then based on the discretization scheme,reliable a posteriori error estimates for the spectral approximation are derived.Some numerical examples are presented to verify the validity and applicability of the derived a posteriori error estimator.展开更多
In this paper,we have considered the multi-dimensional space fractional diffusion equations with variable coefficients.The fractional operators(derivative/integral)are used based on the Caputo definition.This study pr...In this paper,we have considered the multi-dimensional space fractional diffusion equations with variable coefficients.The fractional operators(derivative/integral)are used based on the Caputo definition.This study provides an analytical approach to determine the analytical solution of the considered problems with the help of the two-step Adomian decomposition method(TSADM).Moreover,new results have been obtained for the existence and uniqueness of a solution by using the Banach contraction principle and a fixed point theorem.We have extended the dimension of the space fractional diffusion equations with variable coefficients into multi-dimensions.Finally,the generalized problems with two different types of the forcing term have been included demonstrating the applicability and high efficiency of the TSADM in comparison to other existing numerical methods.The diffusion coefficients do not require to satisfy any certain conditions/restrictions for using the TSADM.There are no restrictions imposed on the problems for diffusion coefficients,and a similar procedures of the TSADM has followed to the obtained analytical solution for the multi-dimensional space fractional diffusion equations with variable coefficients.展开更多
In this paper,we consider the numerical solutions of the semilinear Riesz space-fractional diffusion equations(RSFDEs)with time delay,which constitute an important class of differential equations of practical signific...In this paper,we consider the numerical solutions of the semilinear Riesz space-fractional diffusion equations(RSFDEs)with time delay,which constitute an important class of differential equations of practical significance.We develop a novel implicit alternating direction method that can effectively and efficiently tackle the RSFDEs in both two and three dimensions.The numerical method is proved to be uniquely solvable,stable and convergent with second order accuracy in both space and time.Numerical results are presented to verify the accuracy and efficiency of the proposed numerical scheme.展开更多
A high order finite difference-spectral method is derived for solving space fractional diffusion equations,by combining the second order finite difference method in time and the spectral Galerkin method in space.The s...A high order finite difference-spectral method is derived for solving space fractional diffusion equations,by combining the second order finite difference method in time and the spectral Galerkin method in space.The stability and error estimates of the temporal semidiscrete scheme are rigorously discussed,and the convergence order of the proposed method is proved to be O(τ2+Nα-m)in L2-norm,whereτ,N,αand m are the time step size,polynomial degree,fractional derivative index and regularity of the exact solution,respectively.Numerical experiments are carried out to demonstrate the theoretical analysis.展开更多
The fractional diffusion equation is one of the most important partial differential equations（PDEs） to model problems in mathematical physics. These PDEs are more practical when those are combined with uncertainties...The fractional diffusion equation is one of the most important partial differential equations（PDEs） to model problems in mathematical physics. These PDEs are more practical when those are combined with uncertainties. Accordingly, this paper investigates the numerical solution of a non-probabilistic viz. fuzzy fractional-order diffusion equation subjected to various external forces. A fuzzy diffusion equation having fractional order 0 〈 α≤ 1 with fuzzy initial condition is taken into consideration. Fuzziness appearing in the initial conditions is modelled through convex normalized triangular and Gaussian fuzzy numbers. A new computational technique is proposed based on double parametric form of fuzzy numbers to handle the fuzzy fractional diffusion equation. Using the single parametric form of fuzzy numbers, the original fuzzy diffusion equation is converted first into an interval-based fuzzy differential equation. Next, this equation is transformed into crisp form by using the proposed double parametric form of fuzzy numbers. Finally, the same is solved by Adomian decomposition method（ADM） symbolically to obtain the uncertain bounds of the solution. Computed results are depicted in terms of plots. Results obtained by the proposed method are compared with the existing results in special cases.展开更多
Due to the successful applications in engineering,physics,biology,finance,etc.,there has been substantial interest in fractional diffusion equations over the past few decades,and literatures on developing and analyzin...Due to the successful applications in engineering,physics,biology,finance,etc.,there has been substantial interest in fractional diffusion equations over the past few decades,and literatures on developing and analyzing efficient and accurate numerical methods for reliably simulating such equations are vast and fast growing.This paper gives a concise overview on finite element methods for these equations,which are divided into time fractional,space fractional and time-space fractional diffusion equations.Besides,we also involve some relevant topics on the regularity theory,the well-posedness,and the fast algorithm.展开更多
In this article,we consider to solve the inverse initial value problem for an inhomogeneous space-time fractional diffusion equation.This problem is ill-posed and the quasi-boundary value method is proposed to deal wi...In this article,we consider to solve the inverse initial value problem for an inhomogeneous space-time fractional diffusion equation.This problem is ill-posed and the quasi-boundary value method is proposed to deal with this inverse problem and obtain the series expression of the regularized solution for the inverse initial value problem.We prove the error estimates between the regularization solution and the exact solution by using an a priori regularization parameter and an a posteriori regularization parameter choice rule.Some numerical results in one-dimensional case and two-dimensional case show that our method is efficient and stable.展开更多
Time fractional diffusion equation is usually used to describe the problems involving non-Markovian random walks. This kind of equation is obtained from the standard diffusion equation by replacing the first-order tim...Time fractional diffusion equation is usually used to describe the problems involving non-Markovian random walks. This kind of equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α∈(0, 1). In this paper, an implicit finite difference scheme for solving the time fractional diffusion equation with source term is presented and analyzed, where the fractional derivative is described in the Caputo sense. Stability and convergence of this scheme are rigorously established by a Fourier analysis. And using numerical experiments illustrates the accuracy and effectiveness of the scheme mentioned in this paper.展开更多
In this paper,we develop a novel fi nite-diff erence scheme for the time-Caputo and space-Riesz fractional diff usion equation with convergence order O(τ^2−α+h^2).The stability and convergence of the scheme are anal...In this paper,we develop a novel fi nite-diff erence scheme for the time-Caputo and space-Riesz fractional diff usion equation with convergence order O(τ^2−α+h^2).The stability and convergence of the scheme are analyzed by mathematical induction.Moreover,some numerical results are provided to verify the eff ectiveness of the developed diff erence scheme.展开更多
Fractional calculus and special functions have contributed a lot to mathematical physics and its various branches. The great use of mathematical physics in distinguished astrophysical problems has attracted astronomer...Fractional calculus and special functions have contributed a lot to mathematical physics and its various branches. The great use of mathematical physics in distinguished astrophysical problems has attracted astronomers and physicists to pay more attention to available mathematical tools that can be widely used in solving several problems of astrophysics/physics. In view of the great importance and usefulness of kinetic equations in certain astrophysical problems, the authors derive a generalized fractional kinetic equation involving the Lorenzo-Hartley function, a generalized function for fractional calculus. The fractional kinetic equation discussed here can be used to investigate a wide class of known （and possibly also new） fractional kinetic equations, hitherto scattered in the literature. A compact and easily computable solution is established in terms of the Lorenzo-Hartley function. Special cases, involving the generalized Mittag-Leffler function and the R-function, are considered. The obtained results imply the known results more precisely.展开更多
Anomalous transport in magnetically confined plasmas is investigated by radial fractional transport equations.It is shown that for fractional transport models,hollow density profiles are formed and uphill transports c...Anomalous transport in magnetically confined plasmas is investigated by radial fractional transport equations.It is shown that for fractional transport models,hollow density profiles are formed and uphill transports can be observed regardless of whether the fractional diffusion coefficients(FDCs)are radially dependent or not.When a radially dependent FDC<D_(α)(r)1 is imposed,compared with the case under=D_(α)(r)1.0,it is observed that the position of the peak of the density profile is closer to the core.Further,it is found that when FDCs at the positions of source injections increase,the peak values of density profiles decrease.The non-local effect becomes significant as the order of fractional derivative a 1 and causes the uphill transport.However,as a 2,the fractional diffusion model returns to the standard model governed by Fick’s law.展开更多
Recently,Zhang and Ding developed a novel finite difference scheme for the time-Caputo and space-Riesz fractional diffusion equation with the convergence order 0(ι^(2-a)+h^(2))in Zhang and Ding(Commun.Appl.Math.Compu...Recently,Zhang and Ding developed a novel finite difference scheme for the time-Caputo and space-Riesz fractional diffusion equation with the convergence order 0(ι^(2-a)+h^(2))in Zhang and Ding(Commun.Appl.Math.Comput.2(1):57-72,2020).Unfortunately,they only gave the stability and convergence results for a∈(0,1)andβ∈[7/8+^(3)√621+48√87+19/8^(3)√621+48√87,2]In this paper,using a new analysis method,we find that the original difference scheme is unconditionally stable and convergent with orderΟ(ι^(2-a)+h^(2))for all a∈(0,1)andβ∈(1,2].Finally,some numerical examples are given to verify the correctness of the results.展开更多
In this paper, we use the Mittag-Leffler addition formula to solve the Green function of generalized time fractional diffusion equation in the whole plane and prove the convergence of the Green function.
基金supported by National Natural Science Foundation of China(12271277)the Open Research Fund of Key Laboratory of Nonlinear Analysis&Applications(Central China Normal University),Ministry of Education,China.
文摘In this article,we consider the diffusion equation with multi-term time-fractional derivatives.We first derive,by a subordination principle for the solution,that the solution is positive when the initial value is non-negative.As an application,we prove the uniqueness of solution to an inverse problem of determination of the temporally varying source term by integral type information in a subdomain.Finally,several numerical experiments are presented to show the accuracy and efficiency of the algorithm.
基金supported by the State Key Program of National Natural Science Foundation of China(11931003)the National Natural Science Foundation of China(41974133)。
文摘In this paper,a local discontinuous Galerkin(LDG)scheme for the time-fractional diffusion equation is proposed and analyzed.The Caputo time-fractional derivative(of orderα,with 0<α<1)is approximated by a finite difference method with an accuracy of order3-α,and the space discretization is based on the LDG method.For the finite difference method,we summarize and supplement some previous work by others,and apply it to the analysis of the convergence and stability of the proposed scheme.The optimal error estimate is obtained in the L2norm,indicating that the scheme has temporal(3-α)th-order accuracy and spatial(k+1)th-order accuracy,where k denotes the highest degree of a piecewise polynomial in discontinuous finite element space.The numerical results are also provided to verify the accuracy and efficiency of the considered scheme.
文摘In this paper,we develop novel local discontinuous Galerkin(LDG)methods for fractional diffusion equations with non-smooth solutions.We consider such problems,for which the solutions are not smooth at boundary,and therefore the traditional LDG methods with piecewise polynomial solutions suffer accuracy degeneracy.The novel LDG methods utilize a solution information enriched basis,simulate the problem on a paired special mesh,and achieve optimal order of accuracy.We analyze the L2 stability and optimal error estimate in L2-norm.Finally,numerical examples are presented for validating the theoretical conclusions.
基金This work was supported by the National Natural Science Foundation of China(No.11971354)The author Yi-Shu Du acknowledges the financial support from the China Scholarship Council(File No.201906260146).
文摘After discretization by the finite volume method,the numerical solution of fractional diffusion equations leads to a linear system with the Toeplitz-like structure.The theoretical analysis gives sufficient conditions to guarantee the positive-definite property of the discretized matrix.Moreover,we develop a class of positive-definite operator splitting iteration methods for the numerical solution of fractional diffusion equations,which is unconditionally convergent for any positive constant.Meanwhile,the iteration methods introduce a new preconditioner for Krylov subspace methods.Numerical experiments verify the convergence of the positive-definite operator splitting iteration methods and show the efficiency of the proposed preconditioner,compared with the existing approaches.
基金the National Natural Science Foundation of China under Grant Nos.12271339 and 12201391.
文摘In this paper,finite difference schemes for solving time-space fractional diffusion equations in one dimension and two dimensions are proposed.The temporal derivative is in the Caputo-Hadamard sense for both cases.The spatial derivative for the one-dimensional equation is of Riesz definition and the two-dimensional spatial derivative is given by the fractional Laplacian.The schemes are proved to be unconditionally stable and convergent.The numerical results are in line with the theoretical analysis.
基金the OSD/ARO MURI Grant W911NF-15-1-0562the National Science Foundation under Grant DMS-1620194.
文摘We study an indirect finite element approximation for two-sided space-fractional diffusion equations in one space dimension.By the representation formula of the solutions u(x)to the proposed variable coefficient models in terms of v(x),the solutions to the constant coefficient analogues,we apply finite element methods for the constant coefficient fractional diffusion equations to solve for the approximations vh(x)to v(x)and then obtain the approximations uh(x)of u(x)by plugging vh(x)into the representation of u(x).Optimal-order convergence estimates of u(x)−uh(x)are proved in both L2 and Hα∕2 norms.Several numerical experiments are presented to demonstrate the sharpness of the derived error estimates.
基金supported by the National Natural Science Foundation of China(Grant Nos.11771112,12071100)by the Fundamental Research Funds for the Central Universities(Grant No.2022FRFK060019).
文摘This paper deals with the numerical approximation for the time fractional diffusion problem with fractional dynamic boundary conditions.The well-posedness for the weak solutions is studied.A direct discontinuous Galerkin approach is used in spatial direction under the uniform meshes,together with a second-order Alikhanov scheme is utilized in temporal direction on the graded mesh,and then the fully discrete scheme is constructed.Furthermore,the stability and the error estimate for the full scheme are analyzed in detail.Numerical experiments are also given to illustrate the effectiveness of the proposed method.
基金supported by the State Key Program of National Natural Science Foundation of China(No.11931003)National Natural Science Foundation of China(Nos.41974133,11671157 and 11971410)supported by the Innovation Project of Graduate School of South China Normal University(No.2018LKXM008).
文摘In this paper,an initial boundary value problem of the space-time fractional diffusion equation is studied.Both temporal and spatial directions for this equation are discreted by the Galerkin spectral methods.And then based on the discretization scheme,reliable a posteriori error estimates for the spectral approximation are derived.Some numerical examples are presented to verify the validity and applicability of the derived a posteriori error estimator.
文摘In this paper,we have considered the multi-dimensional space fractional diffusion equations with variable coefficients.The fractional operators(derivative/integral)are used based on the Caputo definition.This study provides an analytical approach to determine the analytical solution of the considered problems with the help of the two-step Adomian decomposition method(TSADM).Moreover,new results have been obtained for the existence and uniqueness of a solution by using the Banach contraction principle and a fixed point theorem.We have extended the dimension of the space fractional diffusion equations with variable coefficients into multi-dimensions.Finally,the generalized problems with two different types of the forcing term have been included demonstrating the applicability and high efficiency of the TSADM in comparison to other existing numerical methods.The diffusion coefficients do not require to satisfy any certain conditions/restrictions for using the TSADM.There are no restrictions imposed on the problems for diffusion coefficients,and a similar procedures of the TSADM has followed to the obtained analytical solution for the multi-dimensional space fractional diffusion equations with variable coefficients.
基金supported by National Natural Science Foundation(NSF)of China(Grant No.11501238)NSF of Guangdong Province(Grant No.2016A030313119)and NSF of Huizhou University(Grant No.hzu201806)+2 种基金supported by the startup fund from City University of Hong Kong and the Hong Kong RGC General Research Fund(projects Nos.12301420,12302919 and 12301218)supported by the NSF of China No.11971221the Shenzhen Sci-Tech Fund No.JCYJ20190809150413261,JCYJ20180307151603959,and JCYJ20170818153840322,and Guangdong Provincial Key Laboratory of Computational Science and Material Design(No.2019B030301001).
文摘In this paper,we consider the numerical solutions of the semilinear Riesz space-fractional diffusion equations(RSFDEs)with time delay,which constitute an important class of differential equations of practical significance.We develop a novel implicit alternating direction method that can effectively and efficiently tackle the RSFDEs in both two and three dimensions.The numerical method is proved to be uniquely solvable,stable and convergent with second order accuracy in both space and time.Numerical results are presented to verify the accuracy and efficiency of the proposed numerical scheme.
基金supported by National Center for Mathematics and Interdisciplinary Sciences,CASNational Natural Science Foundation of China (Grant Nos. 60931002 and 91130019)
文摘A high order finite difference-spectral method is derived for solving space fractional diffusion equations,by combining the second order finite difference method in time and the spectral Galerkin method in space.The stability and error estimates of the temporal semidiscrete scheme are rigorously discussed,and the convergence order of the proposed method is proved to be O(τ2+Nα-m)in L2-norm,whereτ,N,αand m are the time step size,polynomial degree,fractional derivative index and regularity of the exact solution,respectively.Numerical experiments are carried out to demonstrate the theoretical analysis.
基金the UGC,Government of India,for financial support under Rajiv Gandhi National Fellowship(RGNF)
文摘The fractional diffusion equation is one of the most important partial differential equations（PDEs） to model problems in mathematical physics. These PDEs are more practical when those are combined with uncertainties. Accordingly, this paper investigates the numerical solution of a non-probabilistic viz. fuzzy fractional-order diffusion equation subjected to various external forces. A fuzzy diffusion equation having fractional order 0 〈 α≤ 1 with fuzzy initial condition is taken into consideration. Fuzziness appearing in the initial conditions is modelled through convex normalized triangular and Gaussian fuzzy numbers. A new computational technique is proposed based on double parametric form of fuzzy numbers to handle the fuzzy fractional diffusion equation. Using the single parametric form of fuzzy numbers, the original fuzzy diffusion equation is converted first into an interval-based fuzzy differential equation. Next, this equation is transformed into crisp form by using the proposed double parametric form of fuzzy numbers. Finally, the same is solved by Adomian decomposition method（ADM） symbolically to obtain the uncertain bounds of the solution. Computed results are depicted in terms of plots. Results obtained by the proposed method are compared with the existing results in special cases.
基金supported by the Major Project on New Generation of Artificial Intelligence from MOST of China(Grant No.2018AAA0101002)the National Natural Science Foundation of China(Grant Nos.11771438 and 11601460)+1 种基金the Natural Science Foundation of Hunan Province of China(Grant No.2018JJ3491)the Research Foundation of Education Commission of Hunan Province of China(Grant No.19B565).
文摘Due to the successful applications in engineering,physics,biology,finance,etc.,there has been substantial interest in fractional diffusion equations over the past few decades,and literatures on developing and analyzing efficient and accurate numerical methods for reliably simulating such equations are vast and fast growing.This paper gives a concise overview on finite element methods for these equations,which are divided into time fractional,space fractional and time-space fractional diffusion equations.Besides,we also involve some relevant topics on the regularity theory,the well-posedness,and the fast algorithm.
基金The project is supported by the National Natural Science Foundation of China(11561045,11961044)the Doctor Fund of Lan Zhou University of Technology.
文摘In this article,we consider to solve the inverse initial value problem for an inhomogeneous space-time fractional diffusion equation.This problem is ill-posed and the quasi-boundary value method is proposed to deal with this inverse problem and obtain the series expression of the regularized solution for the inverse initial value problem.We prove the error estimates between the regularization solution and the exact solution by using an a priori regularization parameter and an a posteriori regularization parameter choice rule.Some numerical results in one-dimensional case and two-dimensional case show that our method is efficient and stable.
基金Supported by the Discipline Construction and Teaching Research Fund of LUTcte(20140089)
文摘Time fractional diffusion equation is usually used to describe the problems involving non-Markovian random walks. This kind of equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α∈(0, 1). In this paper, an implicit finite difference scheme for solving the time fractional diffusion equation with source term is presented and analyzed, where the fractional derivative is described in the Caputo sense. Stability and convergence of this scheme are rigorously established by a Fourier analysis. And using numerical experiments illustrates the accuracy and effectiveness of the scheme mentioned in this paper.
基金the National Natural Science Foundation of China(no.11561060).
文摘In this paper,we develop a novel fi nite-diff erence scheme for the time-Caputo and space-Riesz fractional diff usion equation with convergence order O(τ^2−α+h^2).The stability and convergence of the scheme are analyzed by mathematical induction.Moreover,some numerical results are provided to verify the eff ectiveness of the developed diff erence scheme.
文摘Fractional calculus and special functions have contributed a lot to mathematical physics and its various branches. The great use of mathematical physics in distinguished astrophysical problems has attracted astronomers and physicists to pay more attention to available mathematical tools that can be widely used in solving several problems of astrophysics/physics. In view of the great importance and usefulness of kinetic equations in certain astrophysical problems, the authors derive a generalized fractional kinetic equation involving the Lorenzo-Hartley function, a generalized function for fractional calculus. The fractional kinetic equation discussed here can be used to investigate a wide class of known （and possibly also new） fractional kinetic equations, hitherto scattered in the literature. A compact and easily computable solution is established in terms of the Lorenzo-Hartley function. Special cases, involving the generalized Mittag-Leffler function and the R-function, are considered. The obtained results imply the known results more precisely.
基金supported by the National MCF Energy R&D Program of China(No.2019YFE03090300)National Natural Science Foundation of China(No.11925501)Fundamental Research Funds for the Central Universities(No.DUT21GJ204)。
文摘Anomalous transport in magnetically confined plasmas is investigated by radial fractional transport equations.It is shown that for fractional transport models,hollow density profiles are formed and uphill transports can be observed regardless of whether the fractional diffusion coefficients(FDCs)are radially dependent or not.When a radially dependent FDC<D_(α)(r)1 is imposed,compared with the case under=D_(α)(r)1.0,it is observed that the position of the peak of the density profile is closer to the core.Further,it is found that when FDCs at the positions of source injections increase,the peak values of density profiles decrease.The non-local effect becomes significant as the order of fractional derivative a 1 and causes the uphill transport.However,as a 2,the fractional diffusion model returns to the standard model governed by Fick’s law.
基金supported by the National Natural Science Foundation of China(Nos.11901057 and 11561060).
文摘Recently,Zhang and Ding developed a novel finite difference scheme for the time-Caputo and space-Riesz fractional diffusion equation with the convergence order 0(ι^(2-a)+h^(2))in Zhang and Ding(Commun.Appl.Math.Comput.2(1):57-72,2020).Unfortunately,they only gave the stability and convergence results for a∈(0,1)andβ∈[7/8+^(3)√621+48√87+19/8^(3)√621+48√87,2]In this paper,using a new analysis method,we find that the original difference scheme is unconditionally stable and convergent with orderΟ(ι^(2-a)+h^(2))for all a∈(0,1)andβ∈(1,2].Finally,some numerical examples are given to verify the correctness of the results.
文摘In this paper, we use the Mittag-Leffler addition formula to solve the Green function of generalized time fractional diffusion equation in the whole plane and prove the convergence of the Green function.
