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 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.

In this study, we give results on the existence and uniqueness of solutions for generalized fractional integrodifferential equations with a nonlocal terminal condition. We have proved the existence of solutions to the problem proposed using the Schauder fixed point theorem and the uniqueness of its solutions is proved using the Banach fixed point theorem. At the end, we discussed the examples to support our results.

During the past decade researchers have investigated vibrations of piezoelectric and piezomagnetic nanobeams. In addition to displacement fields, usually the electromagnetic fields are also considered one dimensional because the small ratio of height to length of the beam geometry. This makes the implication of electromagnetic boundary conditions on the upper and lower faces of the beam rigorous. In this study free vibration of Timoshenko beam having Magneto-Electro-Elastic properties laying on Elastic foundation has been studied considering one dimensional displacement fields and two dimensional electromagnetic fields. State equations which are a Integro-differential couple has been discretized using a modified radial point interpolation function. Frequency response and the mode shapes have been queried for Dirichlet and Neumman boundary conditions with two conditions open circuit and closed circuit. . Obtained frequencies have been compared with results of the corresponding fully one dimensional problem.