Moving least squares and spectral collocation method to approximate the solution of stochastic Volterra–Fredholm integral equations
Introduction
Stochastic models are especially important to us because we encounter to real phenomena in nature that we had to use different types of stochastic models to mathematical modeling of them [30], [1]. Due to stochastic nature of these phenomena, they have some stochastic factors in their mathematical models and these random factors be caused complexity for solving these models exactly or numerically [29]. Thus, providing efficient methods to solve them have been attracted the attention of mathematicians and some numerical approaches have been introduced. Since NSVFIEs arise in many problems in the various sciences, it is very practical to study these models and try to improve the numerical methods to solve them. In recent years, some approximate methods have been provided to obtain the numerical solution of nonlinear stochastic integral equations driven by standard Brownian motion (sBm). For example, the combination of Legendre wavelet with Galerkin method has been employed to provide the numerical solution of some nonlinear stochastic Itô–Volterra integral equations [13]. An iterative method has been expanded to solve stochastic integral equations in [32] and its convergence analysis has been investigated in [31]. A traditional method based on dividing the whole of interval into subintervals and using the Lagrange interpolation technique at Chebyshev–Gauss–Radau collocation nodes has been presented in [26]. However, most of the numerical methods that have been used to solve these problems are based on polynomials or functions. For instance, delta functions [18], Chebyshev wavelets [25], Legendre wavelets [13], Fibonacci functions [20], Bernoulli polynomials [22], Bernstein polynomials [2], Euler polynomial [21], hat basis functions [19], and generalized hat basis functions [12] have been used to derive the numerical solutions of NSVFIEs.
The fundamental problem that this paper focuses on is as follows: where κ is a real fixed number, , are analytic known functions, and is an unknown function. Also, all these functions are stochastic processes defined on the probability space , and denotes a sBm. The interested researchers can refer to [15] to study the most important properties of sBm.
MLS method is a meshless method where have implemented a significant role in numerical analysis field [24], [33]. In recent decade, MLS method has been widely used to solve mathematical functional equations. Finding the approximate solution by using a small number of bases and collocation points and solving some small systems, instead of a large system, are the main reasons for the widespread use of this method. Also, this method does not depend on the problem area and does not require for separating the domain into smaller elements. So, it is an efficient mathematical tool to overcome the high dimensions and irregular domains. In [3], two dimensional (2D) nonlinear integral equations of the second kind defined on non-rectangular domains have been efficiently solved via this approach. This technique has been applied to solve 1D and 2D Fredholm and Volterra integral equations [23], nonlinear 1D integro-differential equations [8], 2D Schrödinger equation [7], potential problem [27], boundary integral equation [36], and boundary value problems [34].
In the present paper, a new method based on MLS and spectral collocation method has been suggested to obtain the numerical solution of NSVFIEs. In the process of using the proposed method, we encounter a nonlinear system of algebraic equations that is made of transforming the underlying problem. Therefore, we use a suitable method such as Newton's method to approximate this system and obtain the numerical solution of NSVFIEs. Finally, convergence and error analysis are checked theoretically and numerically to confirm accuracy and efficiency of the proposed method.
Section snippets
Stochastic calculus
Definition 1 The Itô integral [28] Suppose that , and be the class of functions such that satisfy in the following conditions: The function is measurable, where β is the Borel algebra. The function has been adapted respect to filter . .
The Itô integral of is defined as follows: where is a sequence of elementary functions such that:
Property 1 Itô formula and integration by parts [28] Suppose that is a continuous
MLS approximation
In recent years, meshless methods have received a lot of attention from authors due to their special and practical features and have been used in various fields [5], [6]. The MLS method as a practical numerical method has been introduced by Shepard [35] and Lancaster and Salkauskas [17]. The MLS method uses a local approximation, i.e., in the approximation process uses a number of nodes, and presents a trial function with unknown variable values in those nodes. The MLS method is a part of the
Description of the method
In this section, we implement the MLS method for solving NSVFIEs of the form (1). The nodal points are selected over the interval as . The distribution of nodes could be selected regularly or randomly. Chebyshev polynomials are used in a variety of contexts [5], [4]. In this paper, we have selected the roots of the as nodal points: In order to estimate the solution of the integral parts, we employ the shifted
Error analysis
The error estimation of the proposed method has been checked in this section. Consider denotes the MLS approximation of the exact solution . A main work after obtaining approximate solution is that we investigate the achieved approximate solution is good or bad. To do this, we need to estimate the error function as . Therefore, it is necessary to introduce a suitable procedure for estimating the error function when the exact solution is not available. In the following,
Test problems
In this section, our method has been used to solve three numerical examples to demonstrate the applicability, efficiency, and accuracy of the proposed method. We measure the accuracy of this method using the maximum error definition provided below: where and are the exact and approximate solutions of Eq. (1), respectively. In all examples, we considered scale number with scale number =3. Also, we have used Simpson's method to calculate the
Conclusion
In this paper, we surveyed an appropriate approach based on MLS and the spectral collocation method for solving the NSVFIEs. We used the shifted Chebyshev polynomials as basis functions and collocation method. So, we reduced the problem into a nonlinear system of algebraic equations and solved it by using Newton's method. This approach has several advantages which have described below:
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The most important advantage of the proposed method is that it is free from domain elements because our
Acknowledgements
The authors would like to state our appreciation to the editor and referees for their costly comments and constructive suggestions which have improved the quality of the current paper.
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