Compact finite difference scheme for the solution of a time fractional partial integro‐differential equation with a weakly singular kernel

In this paper, we develop a high‐order finite difference scheme for the solution of a time fractional partial integro‐differential equation with a weakly singular kernel. The fractional derivative is used in the Riemann‐Liouville sense. We prove the unconditional stability and convergence of scheme using energy method and show that the convergence order is . We provide some numerical experiments to confirm the efficiency of suggested scheme. The results of numerical experiments are compared with analytical solutions to show the efficiency of proposed scheme. It is illustrated that the numerical results are in good agreement with theoretical ones.     

B‐spline solution of fractional integro partial differential equation with a weakly singular kernel

The main objective of the paper is to find the approximate solution of fractional integro partial differential equation with a weakly singular kernel. Integro partial differential equation (IPDE) appears in the study of viscoelastic phenomena. Cubic B‐spline collocation method is employed for fractional IPDE. The developed scheme for finding the solution of the considered problem is based on finite difference method and collocation method. Caputo fractional derivative is used for time fractional derivative of order α, . The given problem is discretized in both time and space directions. Backward Euler formula is used for temporal discretization. Collocation method is used for spatial discretization. The developed scheme is proved to be stable and convergent with respect to time. Approximate solutions are examined to check the precision and effectiveness of the presented method.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1565–1581, 2017     

A fast numerical algorithm based on the Taylor wavelets for solving the fractional integro‐differential equations with weakly singular kernels

In this paper, a fast numerical algorithm based on the Taylor wavelets is proposed for finding the numerical solutions of the fractional integro‐differential equations with weakly singular kernels. The properties of Taylor wavelets are given, and the operational matrix of fractional integration is constructed. These wavelets are utilized to reduce the solution of the given fractional integro‐differential equation to the solution of a linear system of algebraic equations. Also, convergence of the proposed method is studied. Illustrative examples are included to demonstrate the validity and applicability of the technique.     

A numerical method to solve a class of linear integro‐differential equations with weakly singular kernel

In this paper, a collocation method based on the Bessel polynomials is introduced for the approximate solution of a class of linear integro‐differential equations with weakly singular kernel under the mixed conditions. The exact solution can be obtained if the exact solution is polynomial. In other cases, increasing number of nodes, a good approximation can be obtained with applicable errors. In addition, the method is presented with error and stability analysis. Copyright © 2012 John Wiley & Sons, Ltd.     

RKM for solving Bratu‐type differential equations of fractional order

The purpose of the present paper is to propose an efficient numerical method for solving the differential equations of Bratu‐type with fractional order in reproducing kernel Hilbert space. The exact solution is calculated in the form of a convergent series with easily computable components. Finally, some examples are given to illustrate the efficiency and applicability of the method. 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.     

Numerical solution of the multiterm time‐fractional diffusion equation based on reproducing kernel theory

In this study, we present an efficient computational method for finding approximate solution of the multi term time‐fractional diffusion equation. The approximate solution is presented in the form of a finite series in a reproducing kernel Hilbert space. The convergence of proposed method is studied under some hypothesis which provides the theoretical basis of proposed method for solving the considered equation. Finally, some numerical experiments are considered to examine the efficiency of proposed method in the sense of accuracy and CPU time.     

Solving nonlocal initial‐boundary value problems for parabolic and hyperbolic integro‐differential equations in reproducing kernel hilbert space

This article is concerned with a method for solving nonlocal initial‐boundary value problems for parabolic and hyperbolic integro‐differential equations in reproducing kernel Hilbert space. Convergence of the proposed method is studied under some hypotheses which provide the theoretical basis of the proposed method and some error estimates for the numerical approximation in reproducing kernel Hilbert space are presented. Finally, two numerical examples are considered to illustrate the computation efficiency and accuracy of the proposed method. © 2016 The Authors Numerical Methods for Partial Differential Equations Published by Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 174–198, 2017     

Spectral collocation methods for nonlinear weakly singular Volterra integro‐differential equations

In this paper, a spectral collocation approximation is proposed for neutral and nonlinear weakly singular Volterra integro‐differential equations (VIDEs) with non‐smooth solutions. We use some suitable variable transformations to change the original equation into a new equation, so that the solution of the resulting equation possesses better regularity, and the the Jacobi orthogonal polynomial theory can be applied conveniently. Under reasonable assumptions on the nonlinearity, we carry out a rigorous error analysis in L^∞ norm and weighted L² norm. To perform the numerical simulations, some test examples (linear and nonlinear) are considered with nonsmooth solutions, and numerical results are presented. Further more, the comparative study of the proposed methods with some existing numerical methods is provided.     

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.     

Numerical solution of delay integro‐differential equations by using Taylor collocation method

This paper is concerned with the numerical solution of delay integro‐differential equations. The main purpose of this work is to provide a new numerical approach based on the use of continuous collocation Taylor polynomials for the numerical solution of delay integro‐differential equations. It is shown that this method is convergent. Numerical illustrations confirm our theoretical analysis. Copyright © 2013 John Wiley & Sons, Ltd.     

Analysis of positive solution and Hyers‐Ulam stability for a class of singular fractional differential equations with p‐Laplacian in Banach space

This paper deals with 2 core aspects of fractional calculus including existence of positive solution and Hyers‐Ulam stability for a class of singular fractional differential equations with nonlinear p‐Laplacian operator in Caputo sense. For these aims, the suggested problem is converted into an integral equation via Green function , for ε∈(n−1,n], where n≥4. Then, the Green function is examined whether it is increasing or decreasing and positive or negative function. After these properties, some classical fixed‐point theorems are used for the existence of positive solution. Hyers‐Ulam stability of the proposed problem is also considered. For the application of the results, an expressive example is included.     

Simplified reproducing kernel method for fractional differential equations with delay

This paper is devoted to the numerical scheme for the delay initial value problems of a fractional order. The main idea of this method is to establish a novel reproducing kernel space that satisfies the initial conditions. Based on the properties of the new reproducing kernel space, the simplified reproducing kernel method (SRKM for short) is applied to obtain accurate approximation. The Schmidt orthogoralization process which requires a large number of calculation is less likely to be employed. Numerical experiments are provided to illustrate the performance of the method.     

Relative controllability of nonlinear neutral fractional integro‐differential systems with distributed delays in control

In this paper, we establish sufficient conditions for the global relative controllability of nonlinear neutral fractional Volterra integro‐differential systems with distributed delays in control. The results are obtained by using the Mittag–Leffler functions and the Schauder fixed‐point theorem. Examples are presented to illustrate the results. Copyright © 2015 John Wiley & Sons, Ltd.     

An efficient method for fractional nonlinear differential equations by quasi‐Newton's method and simplified reproducing kernel method

An efficient method for nonlinear fractional differential equations is proposed in this paper. This method consists of 2 steps. First, we linearize the nonlinear operator equation by quasi‐Newton's method, which is based on Fréchet derivative. Then we solve the linear fractional differential equations by the simplified reproducing kernel method. The convergence of the quasi‐Newton's method is discussed for the general nonlinear case as well. Finally, some numerical examples are presented to illustrate accuracy, efficiency, and simplicity of the method.     

Simplified reproducing kernel method for impulsive delay differential equations

In this paper, a new algorithm is presented to solve the impulsive delay initial value problems. This is the first time to propose the simplified reproducing kernel method (SRKM for short) to solve the impulsive delay differential equations. Then the uniform convergence of the numerical solution is proved, and the time consuming Schmidt orthogonalization process is avoided. The proposed method is proved to be stable and is not less than second order convergence. The algorithm is employed successfully in some numerical examples.     

The singular kernel coagulation equation with multifragmentation

In this article, we prove the existence of solutions to singular coagulation equations with multifragmentation. We use weighted L¹ spaces to deal with the singularities and to obtain regular solutions. The Smoluchowski kernel is covered by our proof. The weak L¹ compactness methods are applied to suitably chosen approximating equations as a base of our proof. A more restrictive uniqueness result is also given. Copyright © 2014 John Wiley & Sons, Ltd.     

A numerical solution to nonlinear multi‐point boundary value problems in the reproducing kernel space

In this paper, a new numerical algorithm is provided to solve nonlinear multi‐point boundary value problems in a very favorable reproducing kernel space, which satisfies all complex boundary conditions. Its reproducing kernel function is discussed in detail. The theorem proves that the approximate solution and its first‐ and second‐order derivatives all converge uniformly. The numerical experiments show that the algorithm is quite accurate and efficient for solving nonlinear multi‐point boundary value problems. Copyright © 2010 John Wiley & Sons, Ltd.     

Solving the reaction–diffusion equations with nonlocal boundary conditions based on reproducing kernel space

The reaction–diffusion equations with initial condition and nonlocal boundary conditions are discussed in this article. A reproducing kernel space is constructed, in which an arbitrary function satisfies the initial condition and nonlocal boundary conditions of the reaction‐diffusion equations. Based on the reproducing kernel space, a new algorithm for solving the reaction–diffusion equations with initial condition and nonlocal boundary conditions is presented. Some examples are displayed to demonstrate the validity and applicability of the proposed method. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

The exact solution of a linear integral equation with weakly singular kernel

A space , which is proved to be a reproducing kernel space with simple reproducing kernel, is defined. The expression of its reproducing kernel function is given. Subsequently, a class of linear Volterra integral equation (VIE) with weakly singular kernel is discussed in the new reproducing kernel space. The reproducing kernel method of linear operator equation Au=f, which request the image space of operator A is and operator A is bounded, is improved. Namely, the request for the image space is weakened to be L²[a,b], and the boundedness of operator A is also not required. As a result, the exact solution of the equation is obtained. The numerical experiments show the efficiency of our method.