Caputo Time-fractional Derivative Research Articles

Finite Difference Method on Non-Uniform Meshes for Time Fractional Diffusion Problem

Abstract In this paper, we consider the time fractional diffusion equation with Caputo fractional derivative. Due to the singularity of the solution at the initial moment, it is difficult to achieve an ideal convergence rate when the time discretization is performed on uniform meshes. Therefore, in order to improve the convergence order, the Caputo time fractional derivative term is discretized by the L ⁢ 2 - 1 σ {L2-1_{\sigma}} format on non-uniform meshes, with σ = 1 - α 2 {\sigma=1-\frac{\alpha}{2}} , while the spatial derivative term is approximated by the classical central difference scheme on uniform meshes. According to the summation formula of positive integer k power, and considering k = 3 , 4 , 5 {k=3,4,5} , we propose three non-uniform meshes for time discretization. Through theoretical analysis, different time convergence orders O ⁢ ( N - min ⁡ { k ⁢ α , 2 } ) {O(N^{-\min\{k\alpha,2\}})} can be obtained, where N denotes the number of time splits. Finally, the theoretical analysis is verified by several numerical examples.

Numerical Solution of a Quasilinear Parabolic Equation with a Fractional Time Derivative

A homogeneous Dirichlet initial-boundary value problem for a quasilinear parabolic equation with Caputo fractional time derivative is considered. The coefficients of the elliptic part of the equation depend on the derivatives of the solution and satisfy the conditions providing strong monotonicity and Lipschitz-continuity of the corresponding operator. The equation is approximated by two finite-difference schemes: implicit and fractional step scheme. The stability of these finite difference schemes is proved and accuracy estimates are obtained under the condition of sufficient smoothness of the input data and the solution of the differential problem. A number of iterative methods for implementing the constructed nonlinear mesh problems are analyzed. The convergence and convergence rate of the iterative methods are substantiated. The results of numerical experiments confirming the theoretical conclusions are presented.

Determination of Time-Dependent Sources and Parameters of Nonlocal Diffusion and Wave Equations from Final Data

Abstract Two inverse problems with final overdetermination for diffusion and wave equations containing the Caputo fractional time derivative and a fractional Laplacian of distributed order are considered. They are: 1) the problem to reconstruct a time-dependent source term; 2) the problem to recover simultaneously the source term, the order of the time derivative and the fractional Laplacian. Uniqueness of solutions of these problems is proved. Sufficient conditions for the uniqueness are stricter for the 2nd problem than for the 1st problem.

On the existence, uniqueness and stability results for time-fractional parabolic integrodifferential equations

We study the time-fractional parabolic integrodifferential equations with the Caputo fractional time derivative of order α∈(0,1) in a bounded convex polygonal domain in ℝd. We prove the existence and uniqueness of the solution using the eigenfunction expansion and establish a priori bounds for the solution under various regularity assumptions on the initial data and the source function. Our study includes the initial data in the spaces Ḣ2(Ω), H01(Ω) and L2(Ω) while the source function belongs to the class of Hölder continuous and bounded functions. It is shown that the solution of the corresponding homogeneous problem is infinitely differentiable with respect to time t when the initial function is an element of L2(Ω). Finally, we derive a general stability results for the solution of the homogeneous problem.

Finite difference/finite element method for two-dimensional time–space fractional Bloch–Torrey equations with variable coefficients on irregular convex domains

In magnetic resonance imaging of the human brain, the diffusion process of tissue water is considered in the complex tissue environment of cells, membranes and connective tissue. Models based on fractional order Bloch–Torrey equations are known to provide insights into tissue structures and the microenvironment.In this paper, we consider new two-dimensional multi-term time and space fractional Bloch–Torrey equations with variable coefficients on irregular convex domains, which involve the Caputo time fractional derivative and the Riemann–Liouville space fractional derivative. An unstructured-mesh Galerkin finite element method is used to discretize the spatial fractional derivative, while for each time fractional derivative we use the L1 scheme on a temporal graded mesh. The stability and convergence of the fully discrete scheme are proved. Numerical examples are given to verify the efficiency of our method.

Error analysis of the unstructured mesh finite element method for the two-dimensional time-space fractional Schrödinger equation with a time-independent potential

In this paper, error analysis of the unstructured mesh Galerkin finite element method for the two-dimensional time-space fractional Schrödinger equation with a time-independent potential defined on a finite domain is studied. The finite difference method is used to discretize the Caputo time fractional derivative, while the finite element method using unstructured mesh is used to deal with the Riesz fractional operators in space. Both the stability and convergence analysis of the numerical scheme are constructed. Numerical example is conducted to testify the validity of the proposed method. The conservation of the space fractional Schrödinger equation and the non-conservation of the time fractional Schrödinger equation in quantum mechanical system are achieved. This paper proposes an efficient numerical method as well as its theoretical analysis for the two-dimensional time-space fractional Schrödinger equation with time-independent potentials.

Thermal investigation for electrified convection flow of Newtonian fluid subjected to damped thermal flux on a permeable medium

The main purpose of this manuscript is to study an unsteady, MHD and free convection flow with the thermal transport for an electrical conducted, incompressible, and linearly viscous fluid flow close to a moving vertical plate in the existence of an exponential heating process. The flow area is considered permeable half-space as well as the utilization of an electromagnetic field perpendicular to fluid motion. The Caputo time-fractional derivative is utilized to present a heat flux constitutive-equation through a weak memory. The Laplace transformation technique is utilized to achieve the exact solutions of the dimensionless governing equations for the temperature field, Nusselt number, fluid velocity, and skin friction coefficient and stated in connection with Wright functions as well as a generalized G-Lorenzo-Hartley function. As a limiting case, a comparison with (Awan et al 2019 Chin. J. Phys. 60 98–106) equation (38) is done to verify the applied technique as well as for validity of our results. The influences of a variety of fractional as well as material factors are pictured with MathCad15. For low estimations of time, Fluid temperature and velocity are increased by decreasing fractional factor γ and both illustrate a contrary behavior for larger estimations of the time The fluid velocity is decreased through the large estimations of the magnetic factor

A compact quadratic spline collocation method for the time-fractional Black–Scholes model

A compact quadratic spline collocation (QSC) method for the time-fractional Black–Scholes model governing European option pricing is presented. Firstly, after eliminating the convection term by an exponential transformation, the time-fractional Black–Scholes equation is transformed to a time-fractional sub-diffusion equation. Then applying $$L1 - 2$$ formula for the Caputo time-fractional derivative and using a collocation method based on quadratic B-spline basic functions for the space discretization, we establish a higher accuracy numerical scheme which yields $$3-\alpha $$ order convergence in time and fourth-order convergence in space. Furthermore, the uniqueness of the numerical solution and the convergence of the algorithm are investigated. Finally, numerical experiments are carried out to verify the theoretical order of accuracy and demonstrate the effectiveness of the new technique. Moreover, we also study the effect of different parameters on option price in time-fractional Black–Scholes model.

Analytical solutions for Navier–Stokes equations with Caputo fractional derivative

This work aims to use the homotopy analysis method to obtain analytical solutions of the linear time-fractional Navier–Stokes equation with cylindrical coordinates and also of a system of nonlinear time-fractional Navier–Stokes equations with Cartesian coordinates. These equations are described by means of $$\psi $$ -Caputo fractional derivative in time and contain results of particular cases given in the literature. The solutions obtained for time-fractional Navier–Stokes equations are graphically presented.

Convergence analysis of an L1-continuous Galerkin method for nonlinear time-space fractional Schrödinger equations

This paper develops and analyses a finite difference/spectral-Galerkin scheme for the nonlinear fractional Schrödinger equations with Riesz space- and Caputo time-fractional derivatives. The finite difference approximation is used for the discretization of the Caputo fractional derivative and the Legendre-Galerkin spectral method is used for the spatial approximation. Additionally, by using a proper form of discrete Grönwall inequality, the scheme is proved to be unconditionally stable and convergent with accuracy in time and spectral accuracy in space in case of smooth solutions. Finally, some numerical tests are preformed to distinguish the validity of our theoretical results.

Numerical Treatment of Time-Fractional Klein–Gordon Equation Using Redefined Extended Cubic B-Spline Functions

In this work, we have developed a numerical algorithm based on redefined extended cubic B-spline functions to explore the approximate solution for time fractional Klein--Gorden equation. The presented technique engages finite difference formulation for discretizing the Caputo time fractional derivative of order $\alpha \in (1,2]$ and redefined extended cubic B-spline functions to interpolate the solution curve along spatial grid. A stability analysis of the scheme is set up to affirm that the errors do not amplify during the execution of numerical procedure. The derivation of uniform convergence reveals that the scheme is $O(h^{2}+\Delta t^{2-\alpha})$ accurate. Some computational experiments are carried out to verify the theoretical considerations. Numerical results are compared with the existing techniques on the topic and it is concluded that the present scheme returns superior outcomes.

On high-order compact schemes for the multidimensional time-fractional Schr\xf6dinger equation

In this article, some high-order compact finite difference schemes are presented and analyzed to numerically solve one- and two-dimensional time fractional Schrödinger equations. The time Caputo fractional derivative is evaluated by the L1 and L1-2 approximation. The space discretization is based on the fourth-order compact finite difference method. For the one-dimensional problem, the rates of the presented schemes are of order O(tau ^{2-alpha }+h^{4}) and O(tau ^{3-alpha }+h^{4}), respectively, with the temporal step size τ and the spatial step size h, and alpha in (0,1). For the two-dimensional problem, the high-order compact alternating direction implicit method is used. Moreover, unconditional stability of the proposed schemes is discussed by using the Fourier analysis method. Numerical tests are performed to support the theoretical results, and these show the accuracy and efficiency of the proposed schemes.

A Crank–Nicolson Finite Volume Element Method for Time Fractional Sobolev Equations on Triangular Grids

In this paper, a finite volume element (FVE) method is proposed for the time fractional Sobolev equations with the Caputo time fractional derivative. Based on the L1-formula and the Crank–Nicolson scheme, a fully discrete Crank–Nicolson FVE scheme is established by using an interpolation operator Ih*. The unconditional stability result and the optimal a priori error estimate in the L2(Ω)-norm for the Crank–Nicolson FVE scheme are obtained by using the direct recursive method. Finally, some numerical results are given to verify the time and space convergence accuracy, and to examine the feasibility and effectiveness for the proposed scheme.

Time Fractional Fisher-KPP and Fitzhugh-Nagumo Equations.

A standard reaction–diffusion equation consists of two additive terms, a diffusion term and a reaction rate term. The latter term is obtained directly from a reaction rate equation which is itself derived from known reaction kinetics, together with modelling assumptions such as the law of mass action for well-mixed systems. In formulating a reaction–subdiffusion equation, it is not sufficient to know the reaction rate equation. It is also necessary to know details of the reaction kinetics, even in well-mixed systems where reactions are not diffusion limited. This is because, at a fundamental level, birth and death processes need to be dealt with differently in subdiffusive environments. While there has been some discussion of this in the published literature, few examples have been provided, and there are still very many papers being published with Caputo fractional time derivatives simply replacing first order time derivatives in reaction–diffusion equations. In this paper, we formulate clear examples of reaction–subdiffusion systems, based on; equal birth and death rate dynamics, Fisher–Kolmogorov, Petrovsky and Piskunov (Fisher–KPP) equation dynamics, and Fitzhugh–Nagumo equation dynamics. These examples illustrate how to incorporate considerations of reaction kinetics into fractional reaction–diffusion equations. We also show how the dynamics of a system with birth rates and death rates cancelling, in an otherwise subdiffusive environment, are governed by a mass-conserving tempered time fractional diffusion equation that is subdiffusive for short times but standard diffusion for long times.

An efficient cubic trigonometric B-spline collocation scheme for the time-fractional telegraph equation

In this paper, a proficient numerical technique for the time-fractional telegraph equation (TFTE) is proposed. The chief aim of this paper is to utilize a relatively new type of B-spline called the cubic trigonometric B-spline for the proposed scheme. This technique is based on finite difference formulation for the Caputo time-fractional derivative and cubic trigonometric B-splines based technique for the derivatives in space. A stability analysis of the scheme is presented to confirm that the errors do not amplify. A convergence analysis is also presented. Computational experiments are carried out in addition to verify the theoretical analysis. Numerical results are contrasted with a few present techniques and it is concluded that the presented scheme is progressively right and more compelling.

Inverse problem of determining an order of the Caputo time-fractional derivative for a subdiffusion equation

Abstract An inverse problem for determining the order of the Caputo time-fractional derivative in a subdiffusion equation with an arbitrary positive self-adjoint operator A with discrete spectrum is considered. By the Fourier method it is proved that the value of ∥ A ⁢ u ⁢ ( t ) ∥ {\|Au(t)\|} , where u ⁢ ( t ) {u(t)} is the solution of the forward problem, at a fixed time instance recovers uniquely the order of derivative. A list of examples is discussed, including linear systems of fractional differential equations, differential models with involution, fractional Sturm–Liouville operators, and many others.

Graded mesh discretization for coupled system of nonlinear multi-term time-space fractional diffusion equations

In this paper, we develop an efficient finite difference/spectral method to solve a coupled system of nonlinear multi-term time-space fractional diffusion equations. In general, the solutions of such equations typically exhibit a weak singularity at the initial time. Based on the L1 formula on nonuniform meshes for time stepping and the Legendre–Galerkin spectral method for space discretization, a fully discrete numerical scheme is constructed. Taking into account the initial weak singularity of the solution, the convergence of the method is proved. The optimal error estimate is obtained by providing a generalized discrete form of the fractional Gronwall inequality which enables us to overcome the difficulties caused by the sum of Caputo time-fractional derivatives and and the positivity of the reaction term over the nonuniform time mesh. The error estimate reveals how to select an appropriate mesh parameter to obtain the temporal optimal convergence. Furthermore, numerical experiments are presented to confirm the theoretical claims.

The couple of Hermite-based approach and Crank-Nicolson scheme to approximate the solution of two dimensional stochastic diffusion-wave equation of fractional order

The main aim of this study is presenting a semi-discretization scheme to find the numerical solution of two dimensional (2D) stochastic time fractional diffusion-wave equation, which obtains from classical 2D diffusion-wave equation by replacing integer time derivative with Caputo fractional time derivative of order α (1 < α ≤ 2) and inserting some stochastic factors. In this scheme, first Crank-Nicolson and linear spline techniques are used to discrete mentioned problem in the time direction and then Hermite-based approach is applied to obtain the approximate solution in each time step. It is not required any discretization in the spatial directions and therefore this approach is an efficient tool to solve various problems which have been defined on irregular domains. Finally, to confirm this claim that obtained numerical results are accurate and in reliable agreement with the theoretical discussion, some test problems are included in the numerical example section. The values of maximum error, error associated with norm 2, and RMS–error are reported to demonstrate accuracy and reliability of the proposed method. Also, the domain of last example is considered in a long range of the time interval to study the stability of our method on time variable.

An effective pure meshfree method for 1D/2D time fractional convection-diffusion problems on irregular geometry

An easy implemented and effective pure meshfree method is first developed to solve the 1D/2D constant/variable-order time fractional convection-diffusion equation (TF-CDE) on non-regular domain with two boundary conditions in this paper. The proposed method (CSPH-FDM) is derived from that the finite difference scheme (FDM) for Caputo time fractional derivative and a corrected smoothed particle hydrodynamics (CSPH) without kernel derivative for spatial derivatives. In the proposed CSPH-FDM, the high-order spatial derivative is divided into multi first-order derivatives and solved continuously by the CSPH, the Neumann boundary condition can be accurately treated, the two distribution cases of the local refinement and irregular particles or the arbitrary irregular shape domain can be easily and effectively implemented by the CSPH. To demonstrate the validity and numerical convergent order of the proposed method, several 1D/2D analytical examples with local refinement and irregular particles distributions or on complex geometries are first investigated, in which a four-order derivate problem with Neumann boundary is also considered. Subsequently, the CSPH-FDM is extended to predict the solute moving process versus time by two TF-CDEs on different irregular domains and compared with other numerical results. All the numerical results show the flexible application ability and reliability of the present method.

A computational approach for solving time fractional differential equation via spline functions

A computational approach based on finite difference scheme and a redefined extended B-spline functions is presented to study the approximate solution of time fractional advection diffusion equation. The Caputo time-fractional derivative and redefined extended B-spline functions have been used for the time and spatial discretization, respectively. The numerical scheme is shown to be O(h2+Δt2-α) accurate and unconditionally stable. The proposed method is tested through some numerical experiments involving homogeneous/non-homogeneous boundary conditions which concluded that it is more accurate than existing methods. The simulation results show superior agreement with the exact solution as compared to existing methods.