Moving least squares and spectral collocation method to approximate the solution of stochastic Volterra–Fredholm integral equations

doi:10.1016/j.apnum.2020.11.013

Applied Numerical Mathematics

Volume 161, March 2021, Pages 275-285

https://doi.org/10.1016/j.apnum.2020.11.013 Get rights and content

Abstract

In this article, an idea based on moving least squares (MLS) and spectral collocation method is used to estimate the solution of nonlinear stochastic Volterra–Fredholm integral equations (NSVFIEs). The main advantage of the suggested approach is that in some parts where interpolation and integration are necessary, this approach does not require any meshes. Therefore, it is independent of the geometry of the domains, and this advantage helps us to solve the problems on irregular domains with relatively fewer computations. Another advantage of our proposed method is that with a small number of points and base functions, we were able to obtain the results with acceptable accuracy, and this is very attractive and practical. Applying the proposed method leads to the conversion of the problem into a system of algebraic equations. It is worth noting, some examples and error estimations have been provided to illustrate the accuracy and applicability of this technique. Also, we present a convergence analysis of the proposed method.

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: $ψ (x) = ξ (x) + \int_{0}^{x} ϒ_{1} (x, t) N_{1} (t, ψ (t)) d t + κ \int_{0}^{1} ϒ_{2} (x, t) N_{2} (t, ψ (t)) d t + \int_{0}^{x} ϒ_{3} (x, t) N_{3} (t, ψ (t)) d B (t), 0 \leq x \leq 1,$ where κ is a real fixed number, $ξ (x)$ , $ϒ_{1} (x, t), ϒ_{2} (x, t), ϒ_{3} (x, t), N_{1} (t, ψ (t)), N_{2} (t, ψ (t)), N_{3} (t, ψ (t))$ are analytic known functions, and $ψ (x)$ is an unknown function. Also, all these functions are stochastic processes defined on the probability space $(Ω, F, P)$ , and $B (t)$ 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 $0 \leq α \leq σ$ , and $Λ = Λ (α, σ)$ be the class of functions $g (ξ, γ) : [0, 1] \times Ω \to R^{n}$ such that satisfy in the following conditions:

(i)
The function $g (ξ, γ)$ is $β \times F$ measurable, where β is the Borel algebra.
(ii)
The function $g (ξ, γ)$ has been adapted respect to filter $F_{ξ}$ .
(iii)
$E [\int_{α}^{σ} g^{2} (ξ, γ) d ξ] < \infty$ .

The Itô integral of

g (ξ, γ) \in Λ (α, σ)

is defined as follows:

\int_{α}^{σ} g (ξ, γ) d B (ξ) (w) = \lim_{n \to \infty} \int_{α}^{σ} φ_{n} (ξ, γ) d B (ξ) (w),

where

φ_{n}

is a sequence of elementary functions such that:

E [\int_{α}^{σ} {(g - φ_{n})}^{2} d ξ] \to 0, n \to \infty .

Property 1 Itô formula and integration by parts

[28] Suppose that $g (s, w) = g (s)$ 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 $n + 1$ nodal points $x_{i}$ are selected over the interval $[0, 1]$ as $0 \leq x_{1} < x_{2} < \dots < x_{n + 1} \leq 1$ . 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 $n + 1$ roots of the $T_{n + 1}^{⁎} (x)$ as nodal points: $x_{i} = \frac{1}{2} (1 - \cos (\frac{(2 i + 1) π}{2 n + 2})), i = 0, 1, \dots, n .$ 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 $\hat{ψ} (x)$ denotes the MLS approximation of the exact solution $ψ (x)$ . 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 $e_{n} (t) = ψ (t) - \hat{ψ} (t)$ . 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: ${‖ e ‖}_{\infty} = \max_{x \in [0, 1]} | ψ (x) - \hat{ψ} (x) |,$ where $ψ (x)$ and $\hat{ψ} (x)$ are the exact and approximate solutions of Eq. (1), respectively. In all examples, we considered $d_{i} =$ scale number $\times | x - x_{i} |$ 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:

•
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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Moving least squares and spectral collocation method to approximate the solution of stochastic Volterra–Fredholm integral equations

Abstract

Introduction

Section snippets

Stochastic calculus

MLS approximation

Description of the method

Error analysis

Test problems

Conclusion

Acknowledgements

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Numerical solution of Itô–Volterra integral equation by least squares method

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