An initial value problem was considered for a coupled differential system with multi-term Caputo type fractional derivatives. By means of nonlinear alternative of Leray-Schauder and Banach contraction principle,the ex...An initial value problem was considered for a coupled differential system with multi-term Caputo type fractional derivatives. By means of nonlinear alternative of Leray-Schauder and Banach contraction principle,the existence and uniqueness of solutions for the system were derived. Using a fractional predictorcorrector method, a numerical method was presented for the specified system. An example was given to illustrate the obtained results.展开更多
In this paper,the three-variable shifted Jacobi operational matrix of fractional derivatives is used together with the collocation method for numerical solution of threedimensional multi-term fractional-order PDEs wit...In this paper,the three-variable shifted Jacobi operational matrix of fractional derivatives is used together with the collocation method for numerical solution of threedimensional multi-term fractional-order PDEs with variable coefficients.The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifying the problem.The approximate solutions of nonlinear fractional PDEs with variable coefficients thus obtained by threevariable shifted Jacobi polynomials are compared with the exact solutions.Furthermore some theorems and lemmas are introduced to verify the convergence results of our algorithm.Lastly,several numerical examples are presented to test the superiority and efficiency of the proposed method.展开更多
In this paper, we introduce high-order finite volume methods for the multi-term time fractional sub-diffusion equation. The time fractional derivatives are described in Caputo’s sense. By using some operators, we obt...In this paper, we introduce high-order finite volume methods for the multi-term time fractional sub-diffusion equation. The time fractional derivatives are described in Caputo’s sense. By using some operators, we obtain the compact finite volume scheme have high order accuracy. We use a compact operator to deal with spatial direction;then we can get the compact finite volume scheme. It is proved that the finite volume scheme is unconditionally stable and convergent in L<sub>∞</sub>-norm. The convergence order is O(τ<sup>2-α</sup> + h<sup>4</sup>). Finally, two numerical examples are given to confirm the theoretical results. Some tables listed also can explain the stability and convergence of the scheme.展开更多
In this paper,two numerical schemes for the multi-term fractional mixed diffusion and diffusion-wave equation(of order a,with 0<a<2)are developed to solve the initial value problem.Firstly,we study a direct nume...In this paper,two numerical schemes for the multi-term fractional mixed diffusion and diffusion-wave equation(of order a,with 0<a<2)are developed to solve the initial value problem.Firstly,we study a direct numerical scheme that uses quadratic Charles Hermite and Newton(H2N2)interpolation polynomials approximations in the temporal direction and finite element discretization in the spatial direction.We prove the stability of the direct numerical scheme by the energy method and obtain a priori error estimate of the scheme with an accuracy of order 3a.In order to improve computational efficiency,a new fast numerical scheme based on H2N2 interpolation and an efficient sum-of-exponentials approximation for the kernels is proposed.Numerical examples confirm the error estimation results and the validity of the fast scheme.展开更多
In this manuscript,a class of multi-term delay fractional differential equations(FDEs)under the Hilfer derivative is considered.Some newly updated results are established under boundary conditions.For the required res...In this manuscript,a class of multi-term delay fractional differential equations(FDEs)under the Hilfer derivative is considered.Some newly updated results are established under boundary conditions.For the required results,we utilize the fixed point theory and tools of the nonlinear functional analysis.Further keeping in mind the importance of stability results,we develop some adequate results about the said aspect.The Hyers-Ulam(H-U)-type concept is used to derive the required stability for the solution of the considered problem.Finally,by appropriate test problems,we justify our findings.展开更多
In this paper, high-order numerical analysis of finite element method（FEM） is presented for twodimensional multi-term time-fractional diffusion-wave equation（TFDWE）. First of all, a fully-discrete approximate sche...In this paper, high-order numerical analysis of finite element method（FEM） is presented for twodimensional multi-term time-fractional diffusion-wave equation（TFDWE）. First of all, a fully-discrete approximate scheme for multi-term TFDWE is established, which is based on bilinear FEM in spatial direction and Crank-Nicolson approximation in temporal direction, respectively. Then the proposed scheme is proved to be unconditionally stable and convergent. And then, rigorous proofs are given here for superclose properties in H-1-norm and temporal convergence in L-2-norm with order O（h-2＋ τ-（3-α））, where h and τ are the spatial size and time step, respectively. At the same time, theoretical analysis of global superconvergence in H-1-norm is derived by interpolation postprocessing technique. At last, numerical example is provided to demonstrate the theoretical analysis.展开更多
The main contents of this paper are to establish a finite element fully-discrete approximate scheme for multi-term time-fractional mixed sub-diffusion and diffusionwave equation with spatial variable coefficient,which...The main contents of this paper are to establish a finite element fully-discrete approximate scheme for multi-term time-fractional mixed sub-diffusion and diffusionwave equation with spatial variable coefficient,which contains a time-space coupled derivative.The nonconforming EQ^(rot)_(1)element and Raviart-Thomas element are employed for spatial discretization,and L1 time-stepping method combined with the Crank-Nicolson scheme are applied for temporal discretization.Firstly,based on some significant lemmas,the unconditional stability analysis of the fully-discrete scheme is acquired.With the assistance of the interpolation operator I_(h)and projection operator Rh,superclose and convergence results of the variable u in H^(1)-norm and the flux~p=k_(5)(x)ru(x,t)in L^(2)-norm are obtained,respectively.Furthermore,the global superconvergence results are derived by applying the interpolation postprocessing technique.Finally,the availability and accuracy of the theoretical analysis are corroborated by experimental results of numerical examples on anisotropic meshes.展开更多
In this paper,we present a superlinear numerical method for multi-term fractional nonlinear ordinary differential equations(MTFNODEs).First,the presented problem is equivalently transformed into its integral form with...In this paper,we present a superlinear numerical method for multi-term fractional nonlinear ordinary differential equations(MTFNODEs).First,the presented problem is equivalently transformed into its integral form with multi-term Riemann-Liouville integrals.Second,the compound product trapezoidal rule is used to approximate the fractional integrals.Then,the unconditional stability and convergence with the order 1+αN−1−αN−2 of the proposed scheme are strictly established,whereαN−1 andαN−2 are the maximum and the second maximum fractional indexes in the considered MTFNODEs,respectively.Finally,two numerical examples are provided to support the theoretical results.展开更多
In this paper,a class of multi-term time fractional advection diffusion equations(MTFADEs)is considered.By finite difference method in temporal direction and finite element method in spatial direction,two fully discre...In this paper,a class of multi-term time fractional advection diffusion equations(MTFADEs)is considered.By finite difference method in temporal direction and finite element method in spatial direction,two fully discrete schemes of MTFADEs with different definitions on multi-term time fractional derivative are obtained.The stability and convergence of these numerical schemes are discussed.Next,a V-cycle multigrid method is proposed to solve the resulting linear systems.The convergence of the multigrid method is investigated.Finally,some numerical examples are given for verification of our theoretical analysis.展开更多
In this paper,an abstract multi-term time-fractional differential system is considered and the exact controllability results are investigated.In this theory,we tend to implement the basic tools of fractional calculus ...In this paper,an abstract multi-term time-fractional differential system is considered and the exact controllability results are investigated.In this theory,we tend to implement the basic tools of fractional calculus and measure of noncompactness to come up with a new set of sufficient conditions for the exact controllability by utilisation of Mönch fixed point theorem.Finally,an application is given to illustrate the obtained results.展开更多
In this paper,we study the well-posedness and solution regularity of a multi-term variable-order time-fractional diffusion equation,and then develop an optimal Galerkin finite element scheme without any regularity ass...In this paper,we study the well-posedness and solution regularity of a multi-term variable-order time-fractional diffusion equation,and then develop an optimal Galerkin finite element scheme without any regularity assumption on its true solution.We show that the solution regularity of the considered problem can be affected by the maximum value of variable-order at initial time t=0.More precisely,we prove that the solution to the multi-term variable-order time-fractional diffusion equation belongs to C 2([0,T])in time provided that the maximum value has an integer limit near the initial time and the data has sufficient smoothness,otherwise the solution exhibits the same singular behavior like its constant-order counterpart.Based on these regularity results,we prove optimalorder convergence rate of the Galerkin finite element scheme.Furthermore,we develop an efficient parallel-in-time algorithm to reduce the computational costs of the evaluation of multi-term variable-order fractional derivatives.Numerical experiments are put forward to verify the theoretical findings and to demonstrate the efficiency of the proposed scheme.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
By employing EQ^(ROT)_(1) nonconforming finite element,the numerical approximation is presented for multi-term time-fractional mixed sub-diffusion and diffusion-wave equation on anisotropic meshes.Comparing with the m...By employing EQ^(ROT)_(1) nonconforming finite element,the numerical approximation is presented for multi-term time-fractional mixed sub-diffusion and diffusion-wave equation on anisotropic meshes.Comparing with the multi-term time-fractional sub-diffusion equation or diffusion-wave equation,the mixed case contains a special time-space coupled derivative,which leads to many difficulties in numerical analysis.Firstly,a fully discrete scheme is established by using nonconforming finite element method(FEM)in spatial direction and L1 approximation coupled with Crank-Nicolson(L1-CN)scheme in temporal direction.Furthermore,the fully discrete scheme is proved to be unconditional stable.Besides,convergence and superclose results are derived by using the properties of EQ^(ROT)_(1) nonconforming finite element.What's more,the global superconvergence is obtained via the interpolation postprocessing technique.Finally,several numerical results are provided to demonstrate the theoretical analysis on anisotropic meshes.展开更多
This article deals with numerical inversion for the initial distribution in the multi-term time-fractional diffusion equation using final observations.The inversion problem is of instability,but it is uniquely solvabl...This article deals with numerical inversion for the initial distribution in the multi-term time-fractional diffusion equation using final observations.The inversion problem is of instability,but it is uniquely solvable based on the solution’s expression for the forward problem and estimation to the multivariate Mittag-Leffler function.From view point of optimality,solving the inversion problem is transformed to minimizing a cost functional,and existence of a minimum is proved by the weakly lower semi-continuity of the functional.Furthermore,the homotopy regularization algorithm is introduced based on the minimization problem to perform numerical inversions,and the inversion solutions with noisy data give good approximations to the exact initial distribution demonstrating the efficiency of the inversion algorithm.展开更多
基金National Natural Science Foundation of China(No.11371087)
文摘An initial value problem was considered for a coupled differential system with multi-term Caputo type fractional derivatives. By means of nonlinear alternative of Leray-Schauder and Banach contraction principle,the existence and uniqueness of solutions for the system were derived. Using a fractional predictorcorrector method, a numerical method was presented for the specified system. An example was given to illustrate the obtained results.
基金This work was supported by the Collaborative Innovation Center of Taiyuan Heavy Machinery Equipment,Postdoctoral Startup Fund of Taiyuan University of Science and Technology(20152034)the Natural Science Foundation of Shanxi Province(201701D221135)National College Students Innovation and Entrepreneurship Project(201710109003)and(201610109007).
文摘In this paper,the three-variable shifted Jacobi operational matrix of fractional derivatives is used together with the collocation method for numerical solution of threedimensional multi-term fractional-order PDEs with variable coefficients.The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifying the problem.The approximate solutions of nonlinear fractional PDEs with variable coefficients thus obtained by threevariable shifted Jacobi polynomials are compared with the exact solutions.Furthermore some theorems and lemmas are introduced to verify the convergence results of our algorithm.Lastly,several numerical examples are presented to test the superiority and efficiency of the proposed method.
文摘In this paper, we introduce high-order finite volume methods for the multi-term time fractional sub-diffusion equation. The time fractional derivatives are described in Caputo’s sense. By using some operators, we obtain the compact finite volume scheme have high order accuracy. We use a compact operator to deal with spatial direction;then we can get the compact finite volume scheme. It is proved that the finite volume scheme is unconditionally stable and convergent in L<sub>∞</sub>-norm. The convergence order is O(τ<sup>2-α</sup> + h<sup>4</sup>). Finally, two numerical examples are given to confirm the theoretical results. Some tables listed also can explain the stability and convergence of the scheme.
基金supported by the State Key Program of the National Natural Science Foundation of China(No.11931003)by the Grants(Nos.41974133 and 12271233).
文摘In this paper,two numerical schemes for the multi-term fractional mixed diffusion and diffusion-wave equation(of order a,with 0<a<2)are developed to solve the initial value problem.Firstly,we study a direct numerical scheme that uses quadratic Charles Hermite and Newton(H2N2)interpolation polynomials approximations in the temporal direction and finite element discretization in the spatial direction.We prove the stability of the direct numerical scheme by the energy method and obtain a priori error estimate of the scheme with an accuracy of order 3a.In order to improve computational efficiency,a new fast numerical scheme based on H2N2 interpolation and an efficient sum-of-exponentials approximation for the kernels is proposed.Numerical examples confirm the error estimation results and the validity of the fast scheme.
文摘In this manuscript,a class of multi-term delay fractional differential equations(FDEs)under the Hilfer derivative is considered.Some newly updated results are established under boundary conditions.For the required results,we utilize the fixed point theory and tools of the nonlinear functional analysis.Further keeping in mind the importance of stability results,we develop some adequate results about the said aspect.The Hyers-Ulam(H-U)-type concept is used to derive the required stability for the solution of the considered problem.Finally,by appropriate test problems,we justify our findings.
基金Supported by the National Natural Science Foundation of China(Nos.11771438,11471296)the Key Scientific Research Projects in Universities of Henan Province(No.19B110013)the Program for Scientific and Technological Innovation Talents in Universities of Henan Province(No.19HASTIT025)
文摘In this paper, high-order numerical analysis of finite element method（FEM） is presented for twodimensional multi-term time-fractional diffusion-wave equation（TFDWE）. First of all, a fully-discrete approximate scheme for multi-term TFDWE is established, which is based on bilinear FEM in spatial direction and Crank-Nicolson approximation in temporal direction, respectively. Then the proposed scheme is proved to be unconditionally stable and convergent. And then, rigorous proofs are given here for superclose properties in H-1-norm and temporal convergence in L-2-norm with order O（h-2＋ τ-（3-α））, where h and τ are the spatial size and time step, respectively. At the same time, theoretical analysis of global superconvergence in H-1-norm is derived by interpolation postprocessing technique. At last, numerical example is provided to demonstrate the theoretical analysis.
基金The work is supported by the National Natural Science Foundation of China(Nos.11971416 and 11871441)the Scientific Research Innovation Team of Xuchang University(No.2022CXTD002)the Foundation for University Key Young Teacher of Henan Province(No.2019GGJS214).
文摘The main contents of this paper are to establish a finite element fully-discrete approximate scheme for multi-term time-fractional mixed sub-diffusion and diffusionwave equation with spatial variable coefficient,which contains a time-space coupled derivative.The nonconforming EQ^(rot)_(1)element and Raviart-Thomas element are employed for spatial discretization,and L1 time-stepping method combined with the Crank-Nicolson scheme are applied for temporal discretization.Firstly,based on some significant lemmas,the unconditional stability analysis of the fully-discrete scheme is acquired.With the assistance of the interpolation operator I_(h)and projection operator Rh,superclose and convergence results of the variable u in H^(1)-norm and the flux~p=k_(5)(x)ru(x,t)in L^(2)-norm are obtained,respectively.Furthermore,the global superconvergence results are derived by applying the interpolation postprocessing technique.Finally,the availability and accuracy of the theoretical analysis are corroborated by experimental results of numerical examples on anisotropic meshes.
基金supported by the National Natural Science Foundation of China(Grant Nos.11701502 and 11871065).
文摘In this paper,we present a superlinear numerical method for multi-term fractional nonlinear ordinary differential equations(MTFNODEs).First,the presented problem is equivalently transformed into its integral form with multi-term Riemann-Liouville integrals.Second,the compound product trapezoidal rule is used to approximate the fractional integrals.Then,the unconditional stability and convergence with the order 1+αN−1−αN−2 of the proposed scheme are strictly established,whereαN−1 andαN−2 are the maximum and the second maximum fractional indexes in the considered MTFNODEs,respectively.Finally,two numerical examples are provided to support the theoretical results.
基金This research is supported by the National Center for Mathematics and Interdisciplinary Sciences,CAS,and by the National Natural Science Foundation of China(Grant No.11371357).
文摘In this paper,a class of multi-term time fractional advection diffusion equations(MTFADEs)is considered.By finite difference method in temporal direction and finite element method in spatial direction,two fully discrete schemes of MTFADEs with different definitions on multi-term time fractional derivative are obtained.The stability and convergence of these numerical schemes are discussed.Next,a V-cycle multigrid method is proposed to solve the resulting linear systems.The convergence of the multigrid method is investigated.Finally,some numerical examples are given for verification of our theoretical analysis.
基金The work of the first author(Vikram Singh)is supported by the Ministry of Human Resource Development,India under[grant number:MHR-02-23-200-44]’.
文摘In this paper,an abstract multi-term time-fractional differential system is considered and the exact controllability results are investigated.In this theory,we tend to implement the basic tools of fractional calculus and measure of noncompactness to come up with a new set of sufficient conditions for the exact controllability by utilisation of Mönch fixed point theorem.Finally,an application is given to illustrate the obtained results.
基金the National Natural Science Foundation of China(No.11971482)the Natural Science Foundation of Shandong Province(No.ZR2017MA006)+2 种基金the National Science Foundation(No.DMS-1620194)the China Postdoctoral Science Foundation(Nos.2020M681136,2021TQ0017,2021T140129)the International Postdoctoral Exchange Fellowship Program(Talent-Introduction Program)(No.YJ20210019).
文摘In this paper,we study the well-posedness and solution regularity of a multi-term variable-order time-fractional diffusion equation,and then develop an optimal Galerkin finite element scheme without any regularity assumption on its true solution.We show that the solution regularity of the considered problem can be affected by the maximum value of variable-order at initial time t=0.More precisely,we prove that the solution to the multi-term variable-order time-fractional diffusion equation belongs to C 2([0,T])in time provided that the maximum value has an integer limit near the initial time and the data has sufficient smoothness,otherwise the solution exhibits the same singular behavior like its constant-order counterpart.Based on these regularity results,we prove optimalorder convergence rate of the Galerkin finite element scheme.Furthermore,we develop an efficient parallel-in-time algorithm to reduce the computational costs of the evaluation of multi-term variable-order fractional derivatives.Numerical experiments are put forward to verify the theoretical findings and to demonstrate the efficiency of the proposed scheme.
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National Natural Science Foundation of China(No.11971416)Scientific Research Innovation Team of Xuchang University(No.2022CXTD002)+3 种基金Foundation for University Key Young Teacher of Henan Province(No.2019GGJS214)Key Scientific Research Projects in Universities of Henan Province(Nos.21B110007,22A110022)National Natural Science Foundation of China(International cooperation key project:No.12120101001)Australian Research Council via the Discovery Project(DP190101889).
文摘By employing EQ^(ROT)_(1) nonconforming finite element,the numerical approximation is presented for multi-term time-fractional mixed sub-diffusion and diffusion-wave equation on anisotropic meshes.Comparing with the multi-term time-fractional sub-diffusion equation or diffusion-wave equation,the mixed case contains a special time-space coupled derivative,which leads to many difficulties in numerical analysis.Firstly,a fully discrete scheme is established by using nonconforming finite element method(FEM)in spatial direction and L1 approximation coupled with Crank-Nicolson(L1-CN)scheme in temporal direction.Furthermore,the fully discrete scheme is proved to be unconditional stable.Besides,convergence and superclose results are derived by using the properties of EQ^(ROT)_(1) nonconforming finite element.What's more,the global superconvergence is obtained via the interpolation postprocessing technique.Finally,several numerical results are provided to demonstrate the theoretical analysis on anisotropic meshes.
基金This work is supported by the National Natural Science Foundation of China(Nos.11371231 and 11071148).
文摘This article deals with numerical inversion for the initial distribution in the multi-term time-fractional diffusion equation using final observations.The inversion problem is of instability,but it is uniquely solvable based on the solution’s expression for the forward problem and estimation to the multivariate Mittag-Leffler function.From view point of optimality,solving the inversion problem is transformed to minimizing a cost functional,and existence of a minimum is proved by the weakly lower semi-continuity of the functional.Furthermore,the homotopy regularization algorithm is introduced based on the minimization problem to perform numerical inversions,and the inversion solutions with noisy data give good approximations to the exact initial distribution demonstrating the efficiency of the inversion algorithm.
