In the present research paper we derive results about existence and uniqueness of solutions and Ulam{Hyers and Rassias stabilities of nonlinear VOLTERRA{Fredholm delay integrodi erential EQUATIONS. Pachpatte's inequality and Picard operator theory are the main tools that are used to obtain our main results. We concluded this work with applications of obtained results and few illustrative examples.

Introduction: In this paper, we have introduced a new method for solving a class of the partial INTEGRO-DIFFERENTIAL equation with the singular kernel by using the finite difference method. One of the best subjects in the numerical analysis is a finite difference method (FDM). We used (FDM) to solve problems in mathematical physics, integral EQUATIONS, and engineering, such as electromagnetic potential, fluid flow, radiation heats transfer, laminar boundary-layer theory and mass transport, Abel integral EQUATIONS, and problem of mechanics or physics. Also in some physical problems such as fluid flow and heat transfer problems, the Laplace EQUATIONS and the Poisson EQUATIONS are describe by (FDM). In real life most phenomena are modelled by partial differential EQUATIONS. Material and methods: First, we employing an algorithm for solving the problem based on the Crank-Nicholson scheme with given conditions. Furthermore, we discrete the singular integral for solving of the problem. Also, the numerical results obtained here can be compared with the cubic B-spline method. Results and discussion: In addition, solving some examples demonstrates the validity and applicability of the approached method, so that the results are reported in the tables and their figures are shown. The high speed of the calculations, and the assurance of having an approximate solution are obtain by proving the stability of the method. Conclusion: The following conclusions were drawn from this research. Coefficients of the approximate function via Crank-Nicholson scheme are found very easily and therefore many calculations are reduced. The numerical results obtained here can be compared with the cubic B-spline method The assurance of having an approximate solution are obtain by proving the stability of the method.

In this study, a new application of Taylor expansion is considered to estimate the solution of VOLTERRA-Fredholm integral EQUATIONS (VFIEs) and systems of VOLTERRA-Fredholm integral EQUATIONS (SVFIEs). Our proposed method is based upon utilizing the nth-order Taylor polynomial of unknown function at an arbitrary point and employing integration method to convert VFIEs into a system of linear EQUATIONS with respect to unknown function and its derivatives. An approximate solution can be easily determined by solving the obtained system. Furthermore, this method leads always to the exact solution if the exact solution is a polynomial function of degree up to n. Also, an error analysis is given. In addition, some problems are provided to demonstrate the validity and applicability of the proposed method.