In this article, a numerical method based on  improvement of block-pulse functions (IBPFs) is discussed for solving the system of linear Volterra and Fredholm integral equations. By using IBPFs and their operational matrix of integration, such systems can be reduced to a linear system of algebraic equations. An efficient error estimation and associated theorems for the proposed method are also ...

We present some open problems pertaining to the approximation theory involved in the solution of the Nonlinear Schrödinger (NLS) equation. For this important equation, any Initial Value Problem (IVP) can be theoretically solved by the Inverse Scattering Transform (IST) technique whose main steps involve the solution of Volterra equations with structured kernels on unbounded domains, the solutio...

This paper presents a numerical method for solving the stochastic nonlinear volterra-fredholm integral equation (SNVFIE) driven by a standard Brownian motion (SBM). The method is illustrated via a stochastic operational matrix (SOM) based on the triangular functions (TFs) in combination with the collocation method. With using this approach, the SNVFIE reduces to a stochastic nonlinear system of...

Random or stochastic integral equations are important in the study of many physical phenomena in life sciences, engineering, and technology 1–13 . Currently there are two basic versions of stochastic integral equations being studied bymathematical statisticians and probabilists namely, those integral equations involving Ito-Doob type of stochastic integrals and those which can be formed as prob...

In this paper, an efficient method is presented for solving two dimensional Fredholm and Volterra integral equations of the second kind. Chebyshev polynomials are applied to approximate a solution for these integral equations. This method transforms the integral equation to algebraic equations with unknown Chebyshev coefficients. The high accuracy of this method is verified through some numeric...

Some existence and uniqueness theorems are established for weakly singular Volterra and Fredholm-Volterra integral equations in C[a, b]. Our method is based on fixed point theorems which are applied to the iterated operator and we apply the fiber Picard operator theorem to establish differentiability with respect to parameter. This method can be applied only for linear equations because otherwi...

In this paper, an iterative scheme for extracting approximate solutions of two dimensional Volterra-Fredholm integral equations is proposed. Considering some conditions on the kernel of the integral equation obtained by discretization of the integral equation, the convergence of the approximate solution to the exact solution is investigated. Several examples are provided to demonstrate the effi...

Integro-differential equations involving Volterra and Fredholm operators (VFIDEs) are used to model many phenomena in science engineering. Nonlocal boundary conditions more effective, some cases necessary, because they accurate measurements of the true state than classical (local) initial conditions. Closed-form solutions always desirable, not only efficient, but also can be valuable benchmarks...

Several numerical methods for approximating the solution of Hammerstein integral equations are known. For Fredholm-Hammerstein integral equations, the classical method of successive approximations was introduced in [16]. A variation of the Nystrom method was presented in [11]. A collocation-type method was developed in [9]. In [3], Brunner applied a collocation-type method to nonlinear Volterra...