In this paper, we propose a numerical method based on the generalized hat functions (GHFs) and improved hat functions (IHFs) to find numerical solutions for Stochastic Volterra-Fredholm integral equation. To do so, all known and unknown functions are expanded in terms of basic functions and replaced in the original equation. The operational matrices of both basic functions are calculated and embeded in the equation to achieve a linear system of equations which give the expansion coefficients of the solution. We prove that the rate of the convergence is O(h2) and O(h4) for these two different bases under some conditions. Two examples are solved and the results are compared with those of block pulse functions method (BPFs) to show the accuracy and reliability of the methods.

In this paper, a linear combination of quadratic modified hat functions is proposed to solve Stochastic Itô – Volterra integral equation with multi-Stochastic terms. All known and unknown functions are expanded in terms of modified hat functions and replaced in the original equation. The operational matrices are calculated and embedded in the equation to achieve a linear system of equations which gives the expansion coefficients of the solution. Also, under some conditions the error of the method is O(h3). The accuracy and reliability of the method are studied and compared with those of block pulse functions and generalized hat functions in some examples.

In this paper, we introduce an efficient method based on Haar wavelet to approxi- mate a solution for the two-dimensional linear Stochastic Fredholm integral equation. We also give an example to demonstrate the accuracy of the method.

In this paper, modified hat functions and improved hat functions are proposed to solve Stochastic Ito ̂-Volterra integral equations with multi Stochastic terms. A linear system of equations are achieved by replacing the vector and matrix coefficients and operational matrices in the equation which is easy to solve with mathematical softwares. Also, under some conditions the error of these methods are o(h^3) and o(h^4 ) . The accuracy and reliability of these two methods are studied by solving and comparing the answers with block pulse functions and hat functions.

In this paper, a numerical efficient method based on two-dimensional block-pulse functions (BPFs) is proposed to approximate a solution of the two-dimensional linear Stochastic Volterra-Fredholm integral equation. Finally the accuracy of this method will be shown by an example.

In this article we investigate the two-term Abel’s integral equations. We will do this in two different ways and show that such equation is reducible to an integro-differential equation of Volterra type.

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.

Let MnMn be the set of all nn-by-nn real matrices, and let RnRn be the set of all nn-by-11 real (column) vectors. An nn-by-nn matrix R=[rij]R=[rij] with nonnegative entries is called row Stochastic, if ∑ nk=1rik∑ k=1nrik is equal to 1 for all ii, (1≤ i≤ n)(1≤ i≤ n). In fact, Re=eRe=e, where e=(1, … , 1)t∈ Rne=(1, … , 1)t∈ Rn. A matrix R∈ MnR∈ Mn is called integral row Stochastic, if each row has exactly one nonzero entry, +1+1, and other entries are zero. In the present paper, we provide an algorithm for constructing integral row Stochastic matrices, and also we show the relationship between this algorithm and majorization theory.

In this paper, we intend to solve special kind of ordinary differential equations which is called Heun equations, by converting to a corresponding Stochastic differential equation (S.D.E.). So, we construct a Stochastic linear equation system from this equation which its solution is based on computing fundamental matrix of this system and then, this S.D.E. is solved by numerically methods. Moreover, its asymptotic stability and statistical concepts like expectation and variance of solutions are discussed. Finally, the attained solutions of these S.D.E.s compared with exact solution of corresponding differential equations.

A new six order method developed for the approximation Fredholm integral equation of the second kind. This method is based on the quintic spline functions (QSF). In our approach, we , rst formulate the Quintic polynomial spline then the solution of integral equation approximated by this spline. But we need to develop the end conditions which can be associated with the quntic spline. To avoid the reduction accuracy, we formulate the end condition in such a way to obtain the band matrix and also to obtain the same order of accuracy. The convergence of the method is discussed by using matrix algebra. Finally, four test problems have been used for numerical illustration to demonstrate the practical ability of the new method.