Numerical Solution of Nonlinear Stochastic Itô–Volterra Integral Equations Driven by Fractional Brownian Motion Using Block Pulse Functions

This paper presents a valid numerical method to solve nonlinear stochastic Itô–Volterra integral equations (SIVIEs) driven by fractional Brownian motion (FBM) with Hurst parameter H ∈ 1 / 2 , 1 . On the basis of FBM and block pulse functions (BPFs), a new stochastic operational matrix is proposed. The nonlinear stochastic integral equation is converted into a nonlinear algebraic equation by this method. Furthermore, error analysis is given by the pathwise approach. Finally, two numerical examples exhibit the validity and accuracy of the approach.

Bernoulli Wavelets Operational Matrices Method for the Solution of Nonlinear Stochastic Itô-Volterra Integral Equations

This article gives an effective strategy to solve nonlinear stochastic Itô-Volterra integral equations (NSIVIE). These equations can be reduced to a system of nonlinear algebraic equations with unknown coefficients, using Bernoulli wavelets, their operational matrix of integration (OMI), stochastic operational matrix of integration (SOMI) and these equations can be solved numerically. Error analysis of the proposed method is given. Moreover, the results obtained are compared to exact solutions with numerical examples to show that the method described is accurate and precise.

A stochastic operational matrix method for numerical solutions of multi-dimensional stochastic Itô–Volterra integral equations

In this article, hybrid Legendre block-pulse functions are implemented in determining the approximate solutions for multi-dimensional stochastic Itô–Volterra integral equations. The block-pulse function and the proposed scheme are used for deriving a methodology to obtain the stochastic operational matrix. Error and convergence analysis of the scheme is discussed. A brief discussion including numerical examples has been provided to justify the efficiency of the mentioned method.

Wavelets method for solving nonlinear stochastic Itô–Volterra integral equations

In this paper, a new computational method based on the Chebyshev wavelets (CWs) is proposed for solving nonlinear stochastic Itô–Volterra integral equations. In this way, a new stochastic operational matrix (SOM) for the CWs is obtained. By using these basis functions and their SOM, such problems can be transformed into nonlinear systems of algebraic equations which can be simply solved. Moreover, a new technique for computation of nonlinear terms in such problems is presented. Further error analysis of the proposed method is also investigated and the efficiency of this method is shown on some concrete examples. The obtained results reveal that the proposed method is very accurate and efficient.

A stochastic operational matrix method for numerical solutions of mixed stochastic Volterra–Fredholm integral equations

In this paper, we have studied space-time Brownian motion and its applications to mixed type stochastic integral equations. Approximate solutions of mixed stochastic integral equations have been obtained by using two-dimensional (2D) second kind Chebyshev wavelets (CWs). Furthermore, some examples have been presented to justify the efficiency of 2D second kind CWs.

Application of Bernoulli wavelet method for estimating a solution of linear stochastic Itô-Volterra integral equations

Purpose The purpose of this paper is to develop a new method based on operational matrices of Bernoulli wavelet for solving linear stochastic Itô-Volterra integral equations, numerically. Design/methodology/approach For this aim, Bernoulli polynomials and Bernoulli wavelet are introduced, and their properties are expressed. Then, the operational matrix and the stochastic operational matrix of integration based on Bernoulli wavelet are calculated for the first time. Findings By applying these matrices, the main problem would be transformed into a linear system of algebraic equations which can be solved by using a suitable numerical method. Also, a few results related to error estimate and convergence analysis of the proposed scheme are investigated. Originality/value Two numerical examples are included to demonstrate the accuracy and efficiency of the proposed method. All of the numerical calculation is performed on a personal computer by running some codes written in MATLAB software.

Stochastic operational matrix of Chebyshev wavelets for solving multi-dimensional stochastic Itô–Volterra integral equations

In this paper, the numerical solutions of multi-dimensional stochastic Itô–Volterra integral equations have been obtained by second kind Chebyshev wavelets. The second kind Chebyshev wavelets are orthonormal and have compact support on [Formula: see text]. The block pulse functions and their relations to second kind Chebyshev wavelets are employed to derive a general procedure for forming stochastic operational matrix of second kind Chebyshev wavelets. The system of integral equations has been reduced to a system of nonlinear algebraic equations and solved for obtaining the numerical solutions. Convergence and error analysis of the proposed method are also discussed. Furthermore, some examples have been discussed to establish the accuracy and efficiency of the proposed scheme.

Numerical Solutions of Stochastic Volterra–Fredholm Integral Equations by Hybrid Legendre Block-Pulse Functions

In this article, the numerical solutions of stochastic Volterra–Fredholm integral equations have been obtained by hybrid Legendre block-pulse functions (BPFs) and stochastic operational matrix. The hybrid Legendre BPFs are orthonormal and have compact support on$[0, 1)$. The numerical results obtained by the above functions have been compared with those obtained by second kind Chebyshev wavelets. Furthermore, the results of the proposed computational method establish its accuracy and efficiency.