A new approach by two‐dimensional wavelets operational matrix method for solving variable‐order fractional partial integro‐differential equations

In this paper, a new computational scheme based on operational matrices (OMs) of two‐dimensional wavelets is proposed for the solution of variable‐order (VO) fractional partial integro‐differential equations (PIDEs). To accomplish this method, first OMs of integration and VO fractional derivative (FD) have been derived using two‐dimensional Legendre wavelets. By implementing two‐dimensional wavelets approximations and the OMs of integration and variable‐order fractional derivative (VO‐FD) along with collocation points, the VO fractional partial PIDEs are reduced into the system of algebraic equations. In addition to this, some useful theorems are discussed to establish the convergence analysis and error estimate of the proposed numerical technique. Furthermore, computational efficiency and applicability are examined through some illustrative examples.     

A fractional‐order Jacobi Tau method for a class of time‐fractional PDEs with variable coefficients

This paper presents a shifted fractional‐order Jacobi orthogonal function (SFJF) based on the definition of the classical Jacobi polynomial. A new fractional integral operational matrix of the SFJF is presented and derived. We propose the spectral Tau method, in conjunction with the operational matrices of the Riemann–Liouville fractional integral for SFJF and derivative for Jacobi polynomial, to solve a class of time‐fractional partial differential equations with variable coefficients. In this algorithm, the approximate solution is expanded by means of both SFJFs for temporal discretization and Jacobi polynomials for spatial discretization. The proposed tau scheme, both in temporal and spatial discretizations, successfully reduced such problem into a system of algebraic equations, which is far easier to be solved. Numerical results are provided to demonstrate the high accuracy and superiority of the proposed algorithm over existing ones. Copyright © 2015 John Wiley & Sons, Ltd.     

An efficient numerical approach to solve a class of variable‐order fractional integro‐partial differential equations

The main purpose of this work is to investigate an initial boundary value problem related to a suitable class of variable order fractional integro‐partial differential equations with a weakly singular kernel. To discretize the problem in the time direction, a finite difference method will be used. Then, the Sinc‐collocation approach combined with the double exponential transformation is employed to solve the problem in each time level. The proposed numerical algorithm is completely described and the convergence analysis of the numerical solution is presented. Finally, some illustrative examples are given to demonstrate the pertinent features of the proposed algorithm.     

Quintic spline technique for time fractional fourth‐order partial differential equation

Higher order non‐Fickian diffusion theories involve fourth‐order linear partial differential equations and their solutions. A quintic polynomial spline technique is used for the numerical solutions of fourth‐order partial differential equations with Caputo time fractional derivative on a finite domain. These equations occur in many applications in real life problems such as modeling of plates and thin beams, strain gradient elasticity, and phase separation in binary mixtures, which are basic elements in engineering structures and are of great practical significance to civil, mechanical, and aerospace engineering. The quintic polynomial spline technique is used for space discretization and the time‐stepping is done using a backward Euler method based on the L1 approximation to the Caputo derivative. The stability and convergence analysis are also discussed. The numerical results are given, which demonstrate the effectiveness and accuracy of the numerical method. The numerical results obtained in this article are also compared favorably well with the results of (S. S. Siddiqi and S. Arshed, Int. J. Comput. Math. 92 (2015), 1496–1518). © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 445–466, 2017     

On the fractional derivatives of radial basis functions: Theories and applications

The paper provides the fractional integrals and derivatives of the Riemann‐Liouville and Caputo type for the five kinds of radial basis functions, including the Powers, Gaussian, Multiquadric, Matérn, and Thin‐plate splines, in one dimension. It allows to use high‐order numerical methods for solving fractional differential equations. The results are tested by solving two test problems. The first test case focuses on the discretization of the fractional differential operator while the second considers the solution of a fractional order differential equation.     

Darboux problem for perturbed partial differential equations of fractional order with finite delay

In this paper we investigate the existence of solutions for functional partial perturbed hyperbolic differential equations with fractional order. These results are based upon a fixed point theorem for the sum of contraction and compact operators.     

Fractional‐order orthogonal Bernstein polynomials for numerical solution of nonlinear fractional partial Volterra integro‐differential equations

In this paper, a new two‐dimensional fractional polynomials based on the orthonormal Bernstein polynomials has been introduced to provide an approximate solution of nonlinear fractional partial Volterra integro‐differential equations. For this aim, the fractional‐order orthogonal Bernstein polynomials (FOBPs) are constructed, and its operational matrices of integration, fractional‐order integration, and derivative in the Caputo sense and product operational matrix are derived. These operational matrices are utilized to reduce the under study problem to a nonlinear system of algebraic equations. Using the approximation of FOBPs, the convergence analysis and error estimate associated to the proposed problem have been investigated. Finally, several examples are included to clarify the validity, efficiency, and applicability of the proposed technique via FOBPs approximation.     

Numerical simulations for a variable order fractional cable equation

In this article, Crank-Nicolson method is used to study the variable order fractional cable equation. The variable order fractional derivatives are described in the RiemannLiouville and the Gr¨unwald-Letnikov sense. The stability analysis of the proposed technique is discussed. Numerical results are provided and compared with exact solutions to show the accuracy of the proposed technique.     

Optimality conditions for fractional variational problems with dependence on a combined Caputo derivative of variable order

We establish necessary optimality conditions for variational problems with a Lagrangian depending on a combined Caputo derivative of variable fractional order. The endpoint of the integral is free, and thus transversality conditions are proved. Several particular cases are considered illustrating the new results.     

New predictor-corrector approach for nonlinear fractional differential equations: error analysis and stability

In this paper, the predictor-corrector approach is used to propose two algorithms for the numerical solution of linear and non-linear fractional differential equations (FDE). The fractional order derivative is taken to be in the sense of Caputo and its properties are used to transform FDE into a Volterra-type integral equation. Simpson''s 3/8 rule is used to develop new numerical schemes to obtain the approximate solution of the integral equation associated with the given FDE. The error and stability analysis for the two methods are presented. The proposed methods are compared with the ones available in the literature. Numerical simulation is performed to demonstrate the validity and applicability of both the proposed techniques. As an application, the problem of dynamics of the new fractional order non-linear chaotic system introduced by Bhalekar and Daftardar-Gejji is investigated by means of the obtained numerical algorithms.     

Analysis of differential equations of fractional order

This paper provides a robust convergence checking method for nonlinear differential equations of fractional order with consideration of homotopy perturbation technique. The differential operators are taken in the Caputo sense. Some theorems to prove the existence and uniqueness of the series solutions are presented. Results show that the proposed theoretical analysis is accurate.     

一类分数阶微分方程组积分边值问题的正解

利用锥拉伸和压缩不动点定理,研究了一类具有Riemann-Liouvile分数阶积分条件的分数阶微分方程组边值问题.结合该问题相应Green函数的性质,获得了其正解的存在性条件,并给出了一些应用实例.     

Solution of fractional order heat equation via triple Laplace transform in 2 dimensions

In this article, we introduce the triple Laplace transform for the solution of a class of fractional order partial differential equations. As a consequence, fractional order homogeneous heat equation in 2 dimensions is investigated in detail. The corresponding solution is obtained by using the aforementioned triple Laplace transform, which is the generalization of double Laplace transform. Numerical plots to the concerned solutions are provided to demonstrate our results.     

On a nonlinear distributed order fractional differential equation

We study the existence and the uniqueness of mild and classical solutions for a class of equations of the form . Such equations arise in distributed derivatives models of viscoelasticity and system identification theory. We also formulate a variational principle for a more general equation based on a method of doubling of variables for such equations.     

Inverse problem for a space‐time fractional diffusion equation: Application of fractional Sturm‐Liouville operator

An inverse problem of determining a time‐dependent source term from the total energy measurement of the system (the over‐specified condition) for a space‐time fractional diffusion equation is considered. The space‐time fractional diffusion equation is obtained from classical diffusion equation by replacing time derivative with fractional‐order time derivative and Sturm‐Liouville operator by fractional‐order Sturm‐Liouville operator. The existence and uniqueness results are proved by using eigenfunction expansion method. Several special cases are discussed, and particular examples are provided.     

Fast high order difference schemes for the time fractional telegraph equation

In this paper, a fast high order difference scheme is first proposed to solve the time fractional telegraph equation based on the ℱℒ 2-1_σ formula for the Caputo fractional derivative, which reduces the storage and computational cost for calculation. A compact scheme is then presented to improve the convergence order in space. The unconditional stability and convergence in maximum norm are proved for both schemes, with the accuracy order and , respectively. Difficulty arising from the two Caputo fractional derivatives is overcome by some detailed analysis. Finally, we carry out numerical experiments to show the efficiency and accuracy, by comparing with the ℒ 2-1_σ method.     

Banach空间中一类分数阶微分方程边值问题

研究Banach空间中一类非线性分数阶微分方程边值问题.构建此类方程的Green函数,利用非紧测度和相关的不动点定理,得到了此类方程的mild解存在的几个充分条件,所得结果改进和推广了一些已有的结论.     

Polynomial spectral collocation method for space fractional advection–diffusion equation

This article discusses the spectral collocation method for numerically solving nonlocal problems: one‐dimensional space fractional advection–diffusion equation; and two‐dimensional linear/nonlinear space fractional advection–diffusion equation. The differentiation matrixes of the left and right Riemann–Liouville and Caputo fractional derivatives are derived for any collocation points within any given bounded interval. Several numerical examples with different boundary conditions are computed to verify the efficiency of the numerical schemes and confirm the exponential convergence; the physical simulations for Lévy–Feller advection–diffusion equation and space fractional Fokker–Planck equation with initial δ‐peak and reflecting boundary conditions are performed; and the eigenvalue distributions of the iterative matrix for a variety of systems are displayed to illustrate the stabilities of the numerical schemes in more general cases. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 514–535, 2014