This study aims to investigate a stochastic Volterra integral equation driven by fractional Brownian motion with Hurst parameter $H\in (\frac 12, 1)$. We employ the Wong-Zakai approximation to simplify this intricate problem, transforming the stochastic integral equation into an ordinary integral equation. Moreover, we consider the convergence and the rate of convergence of the Wong-Zakai approximation for this kind of equation.

The main purpose of this paper is to propose a high order numerical method based on the finite difference methods for solving nonlinear Itˆo stochastic Volterra integral equations (SVIEs) of the second kind. To develop the method, a fourth-order implicit finite difference method and the explicit Milstein method are implemented for the discretization of non-stochastic and stochastic integral parts, respectively. To solve the original SVIEs, the proposed method has the deterministic fourth-order and strong stochastic first-order accuracy. The convergence analysis of the proposed method is proved. The finite difference method under consideration requires solving a 2×2 system of equations at each step for one-dimensional SVIE. Therefore, the proposed method is very simple to implement and does not require a lot of computational cost. Some numerical examples are prepared to indicate the verity and efficiency of the new method. Moreover, the comparative numerical results show that this method is more accurate than those existing methods given in the literature.

In this paper, we present an efficient method for determining the solution of the stochastic second kind Volterra integral equations (SVIE) by using the Taylor expansion method. This method transforms the SVIE to a linear stochastic ordinary differential equation which needs specified boundary conditions. For determining boundary conditions, we use the integration technique. This technique gives an approximate simple and closed form solution for the SVIE. Expectation of the approximating process is computed. Some numerical examples are used to illustrate the accuracy of the method.

Voltaire integral equations as the output of problems in basic sciences and engineering have a special application in advancing the solution of complex problems. One of the most widely used types, which consists of a random process under external motion, is the equations of random Volta integral. Solving this type of equation has always been a challenge for researchers. On the other hand, with the development of artificial intelligence and the presentation of fuzzy artificial neural network method as a model inspired by the process of thinking and analysis in the human brain, advanced models of algorithms have been designed. Some of these learning algorithms have been used in fuzzy artificial neural networks to solve equations. In this paper, using this method and designing a learning algorithm, the random equations of random Voltaire type is investigated. The method presented in this article, in addition to being more accurate than the previous methods, posseses more speed for solving problem. This topic provides an acceptable level of confidence for researchers when dealing with such issues.

A Legendre wavelet method is presented for numerical solutions of stochastic Volterra-Fredholm integral equations. The main characteristic of the proposed method is that it reduces stochastic Volterra-Fredholm integral equations into a linear system of equations. Convergence and error analysis of the Legendre wavelets basis are investigated. The efficiency and accuracy of the proposed method was demonstrated by some non-trivial examples and comparison with the block pulse functions method.

The purpose of this paper is to analyze the solvability of a class of stochastic functional integral equations by utilizing the measure of non-compactness with Petryshyn’s fixed point theorem in a Banach space. The results obtained in this paper cover numerous existing results concluded under some weaker conditions by many authors. An example is given to support our main theorem.

In the present work, a new stochastic algorithm is proposed to solve multiple dimensional Fredholm integral equations of the second kind. The solution of the integral equation is described by the Neumann series expansion. Each term of this expansion can be considered as an expectation which is approximated by a continuous Markov chain Monte Carlo method. An algorithm is proposed to simulate a continuous Markov chain with probability density function arisen from an importance sampling technique. Theoretical results are established in a normed space to justify the convergence of the proposed method. The method has a simple structure and it is a good candidate for parallelization because of the fact that many independent sample paths are used to estimate the solution. Numerical results are performed in order to confirm the efficiency and accuracy of the present work.

This article proposes an efficient method based on the Fibonacci functions for solving nonlinear stochastic Ito-Volterra integral equations. For this purpose, we obtain stochastic operational matrix of Fibonacci functions. We use the proposed basis function in combination with stochastic operational matrix. This problem is then reduced into a system of nonlinear equations which can be solved by Newton's method. Also, the existence, uniqueness, and convergence of the proposed method are discussed. Furthermore, in order to show the accuracy and reliability of the proposed method, the new approach is applied to some practical problems.

In this paper, using the concept of measure of noncompactness, we introduce a new extended contraction of operators on a Banach space and obtain some generalizations of Darbo’s fixed-point theorem. In the following, as an application of the obtained results, we deal with the solvability of a system of integral equations of Stochastic type in Banach space. Our results generalize and extend a lot of comparable results in the literature. Finally, a concrete example is also included, which demonstrates the applicability of the obtained results.