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In this paper, stochastic integral equations characterized by fractional Brownian motion have been studied. The fractional stochastic integral equation has been solved by second kind Chebyshev wavelets. The convergence and error analysis have been discussed for the efficiency of the discussed method. In addition, two illustrative examples have been solved to examine the efficiency and accuracy of the proposed scheme. [ABSTRACT FROM AUTHOR]

Purpose: This paper aims to study fractional Brownian motion and its applications to nonlinear stochastic integral equations. Bernstein polynomials have been applied to obtain the numerical results of the nonlinear fractional stochastic integral equations. Design/methodology/approach: Bernstein polynomials have been used to obtain the numerical solutions of nonlinear fractional stochastic integral equations. The fractional stochastic operational matrix based on Bernstein polynomial has been used to discretize the nonlinear fractional stochastic integral equation. Convergence and error analysis of the proposed method have been discussed. Findings: Two illustrated examples have been presented to justify the efficiency and applicability of the proposed method. The corresponding obtained numerical results have been compared with the exact solutions to establish the accuracy and efficiency of the proposed method. Originality/value: To the best of the authors' knowledge, nonlinear stochastic Itô–Volterra integral equation driven by fractional Brownian motion has been for the first time solved by using Bernstein polynomials. The obtained numerical results well establish the accuracy and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Purpose: A novel technique based on Bernoulli wavelets has been proposed to solve two-dimensional Fredholm integral equation of second kind. Bernoulli wavelets have been created by dilation and translation of Bernoulli polynomials. This paper aims to introduce properties of Bernoulli wavelets and Bernoulli polynomials. Design/methodology/approach: To solve the two-dimensional Fredholm integral equation of second kind, the proposed method has been used to transform the integral equation into a system of algebraic equations. Findings: Numerical experiments shows that the proposed two-dimensional wavelets technique can give high-accurate solutions and good convergence rate. Originality/value: The efficiency of newly developed two-dimensional wavelets technique has been validated by different illustrative numerical examples to solve two-dimensional Fredholm integral equations. [ABSTRACT FROM AUTHOR]

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 [ 0 , 1 ]. 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. [ABSTRACT FROM AUTHOR]

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 ) $[0, 1)$. 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. [ABSTRACT FROM AUTHOR]

In this article, Bernoulli wavelet method has been developed to solve nonlinear fuzzy Hammerstein–Volterra integral equations with constant delay. This type of integral equation has a particular case the fuzzy variant of a mathematical model from epidemiology. Bernoulli wavelets have been generated by dilation and translation of Bernoulli polynomials. The properties of Bernoulli wavelets and Bernoulli polynomials are first presented. The present wavelet method reduces these integral equations to a system of nonlinear algebraic equations and again these algebraic systems have been solved numerically by Newton's method. Convergence analysis of the present method has been discussed in this article. Also the results obtained by present wavelet method have been compared with that of by B-spline wavelet method. Some illustrative examples have been demonstrated to show the applicability and accuracy of the present method. [ABSTRACT FROM AUTHOR]

This paper presents an efficient numerical method for finding solutions of the nonlinear Fredholm integral equations system of second kind based on Bernstein polynomials basis. The numerical results obtained by the present method have been compared with those obtained by B-spline wavelet method. This proposed method reduces the system of integral equations to a system of algebraic equations that can be solved easily any of the usual numerical methods. Numerical examples are presented to illustrate the accuracy of the method. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a new approach based on linear B-spline wavelet method has been developed to approximate the solutions of system of linear Fredholm integral equations of second kind. Compactly supported linear semi-orthogonal B-spline scaling functions and wavelet functions together with their dual functions are applied to approximate the solutions of linear Fredholm integral equations system of second kind. This method reduces the system of integral equations to a linear system of algebraic equations that can be solved easily with any of the usual methods. The numerical results obtained by the present method have been compared with those obtained by adaptive method based on Trapezoidal rule. Numerical examples are presented to illustrate the accuracy of the method. [Copyright &y& Elsevier]

• A novel numerical approach for solving SIVIE has been proposed first time ever. • Operational matrices are used to transform the SIVIE into a system of algebraic equations. • Resultant system of algebraic equations has been solved by the collocation method. • This technique has the advantage of being simple to implement. • Error estimate, convergence, and stability analysis are also established. In this paper, a numerical method is implemented to solve the stochastic Itô-Volterra integral equations. In this approach, operational matrices have been applied to reduce the stochastic Itô-Volterra integral equations to linear algebraic equations. Then collocation method is applied to solve the algebraic equations. The error, convergence, and stability analysis of the proposed method are discussed. Also, the steps of the proposed method have been presented in the form of an algorithm. Numerical examples are introduced to confirm the efficiency and reliability of the proposed scheme. [ABSTRACT FROM AUTHOR]