In this paper, an effective numerical method is introduced for the treatment of nonlinear two-dimensional Volterra-Fredholm integro-differential equations. Here, we use the so-called two-dimensional block-pulse functions.First, the two-dimensional block-pulse operational matrix of integration and differentiation has been presented. Then, by using this matrices, the nonlinear two-dimensional Vol...

*Correspondence: [email protected] School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, P.R. China Abstract In this paper, we study some new retarded Volterra-Fredholm type integral inequalities on time scales, which provide explicit bounds on unknown functions. These inequalities generalize and extend some known inequalities and can be used as tools in the qualitative theory ...

In this paper, we study polynomial spline collocation methods applied to a particular class of integral-algebraic equations of Volterra type. We analyse mixed systems of second and first kind integral equations. Global convergence and local superconvergence results are established.

In this paper, we present a numerical method to solve the nonlinear Volterra-Fredholm integral equations by new basis functions (NFs). This is demonstrated by using a complementary pair of new orthogonal basis functions set (NFs) derived from the well-known block pulse functions set. Finally, we give some numerical examples, and the efficiency of this method will be shown by comparison with som...

Abstract In this paper, the representation of the exact solution to the nonlinear Volterra-Fredholm integral equations will be obtained in the reproducing kernel space. The exact solution is represented in the form of series. Its approximate solution is obtained by truncating the series and a new numerical approximate method is obtained. The error of the approximate solution is monotone deceasi...

In this paper, a numerical procedure for solving a class of nonlinear VolterraFredholm integral equations is presented. The method is based upon the globally defined sinc basis functions. Properties of the sinc procedure are utilized to reduce the computation of the nonlinear integral equations to some algebraic equations. Illustrative examples are included to demonstrate the validity and appli...

We propose a new method, called the textit{the weighted space method}, for the study of the generalized Hyers-Ulam-Rassias stability. We use this method for a nonlinear functional equation, for Volterra and Fredholm integral operators.

As one of themost important subjects of mathematics, differential and integral equations are widely used to model a variety of physical problems. Perturbation methods have been used in search of approximate analytical solutions for over a century [1–3]. Algebraic equations, integral-differential equations, and difference equations could be solved by these techniques approximately. However, a ma...