On the numerical solution of fractional stochastic integro-differential equations via meshless discrete collocation method based on radial basis functions
Introduction
Fractional stochastic integro-differential equation is a generalization of the fractional Fokker–Planck equation which describes the random walk of a particle [1]. This model has the following formwhere g(t), k1(t, s) and k2(t, s) for t, s ∈ [0, 1] are given functions, f(t) is the unknown function which should be approximated, B(t) denotes standard Brownian motion process defined on the probability space (Ω, A, P), Dα denotes fractional derivative operator of order α and is the Itô integral. Furthermore, Lebesgue and Itô integrals in Eq. (1) are well defined.
Integro-differential equations have strong physical background and also have many practical applications in scientific fields such as population and polymerrheology. These problems are often dependent on a noise source that is ignored due to the lack of powerful computing devices and are modeled by deterministic integro-differential equations. In recent years by increasing computational power, it has become apparent that real phenomena, classically modeled by deterministic integro-differential equations, can be more satisfactorily modeled by stochastic integro-differential equations. Numerical solution of various kinds of stochastic integral equations due to randomness has its own difficulties. In many situations, such equations can be rarely solved in analytic form and therefore design and analysis of special algorithms to solve them is an important topic of numerical analysis. Recently, operational matrix methods are extensively used to solve stochastic integral equations [2], [3] and Stratonovich integral equations [4], [5]. Special algorithms for solving fractional stochastic integro-differential equation are presented in some papers which are published recently. Asgari [6] provides a numerical method to solve fractional stochastic integro-differential equation by using block-pulse functions. Mirzaee and Samadyar [7] applied the operational matrix method based on orthonormal Bernstein polynomials to solve fractional stochastic integro-differential equations. Also, spectral collocation method [8] and Galerkin method [9], [10] have been used to solve this equation. However, there are still very few papers discussing the numerical solution of fractional stochastic integro-differential equations.
In this paper, we develop a new meshless approach based on radial basis functions to solve fractional stochastic integro-differential equations. Unlike traditional methods such as finite difference method, finite element method and finite volume method, in meshless method we should discrete the domain of the problem. The meshless techniques are introduced to avoid mesh generation. In a meshless method, a set of scattered data is applied instead of meshing the domain of the problem. So, meshfree methods are often better suited to cope with changes in the geometry of the domain than classical discretization methods. Also, meshless method to solve problems of high dimensions is more convenient than other methods. Some of the meshfree methods are the element-free Galerkin method, kernel methods, moving least squares method [11], [12], meshless local Petrov–Galerkin methods [13], radial point interpolation method [14], etc.
This paper is organized as follows. In Section 2, we review some essential preliminaries on fractional calculus, stochastic calculus and RBFs which are used in the next sections. In Section 3, we present a new numerical method to solve fractional stochastic integro-differential equations of the form (1). Our method is based on the approximation of the solutions of these equations by using RBFs. For this aim, we utilize Gaussian and Thin plate spline RBFs to approximate the solution of Eq. (1) and then use the quadrature formula to approximate the appeared integrals. Our main discussion on stability and convergence analysis of the numerical method is presented in Sections 4 and 5, respectively. The efficiency of the proposed method has been checked by various examples in Section 6. Finally, conclusion of the present paper is given in Section 7.
Section snippets
The preliminaries and basic definitions
In this section, we introduce the notations, definitions and some of the fundamental assumptions that are necessary in understanding the present work.
The proposed method
In this section, we solve Eq. (1) by using meshfree method based on strictly positive definite RBFs and strictly conditionally positive definite RBFs. We consider GAs and TPSs in cases strictly positive definite RBFs and strictly conditionally positive definite RBFs, respectively. Some of the most important advantages of this method are listed in the following
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Meshless method does not require any discretization and so it is independent of the geometry of the domains. Thus, many problems on the
Stability of the proposed method
As stated in previous sections, the coefficient matrix of the interpolation by RBFs is non-singular. This matrix, in spite of the fact that it is non-singular, is in the most cases ill-condition, i.e., this matrix has a very large condition number. So, a small perturbation in the input of the problem causes a large change in output. Suppose that λmax and λmin denote the largest and smallest eigenvalues of coefficient matrix A, respectively. So, the condition number of matrix A corresponding to
Convergence of the proposed method
In the current section, we present an error estimate of the proposed method based on strictly positive definite RBFs and . A similar way can be used to provide an error estimate for the proposed method based on strictly conditionally positive definite RBFs and various values of α.
The quality of data distribution is measured by fill distance parameter which is defined in the following definition. Definition 5.1 Fill distance parameter of a set of scattered data for a bounded domain D is
Numerical example
In this section, to illustrate efficiency and accuracy of the suggested method, it was applied to solve some fractional stochastic integro-differential equations. We use GAs and TPSs for solving these examples. As we mentioned in previous section, accuracy and stability of the proposed method depend on shape parameter. Although we can obtain more accurate results by selecting the smaller value for shape parameter, usually, the condition number of coefficients matrix grows rapidly and so the
Conclusion
A meshless discrete collocation method based on RBFs combined with quadrature integration method has been applied to solve fractional stochastic integro-differential equations. We utilized strictly positive definite RBFs such as GA and conditionally positive definite RBFs such as TPS to construct shape function. Then by using fractional derivative formulas for GA and TPS RBFs and appropriate quadrature rule, the solution of fractional stochastic integro-differential equation is converted to the
Acknowledgments
The authors would like to express our very great appreciation to editor and anonymous reviewers for their valuable comments and constructive suggestions which have helped to improve the quality and presentation of this paper.
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