Compact finite difference scheme is applied for a Partial integro-differential equation with a weakly singular kernel. The product trapezoidal method is applied for discretization of the integral term. The order of accuracy in space and time is O (h4, k2-a), where 0<a<1. Stability and convergence in 2 L norm are discussed through energy method. Numerical examples are provided to confirm the theoretical prediction and to show that the combination of the compact finite difference approximation and product trapezoidal method give an efficient method for solving a Partial integro-differential equation.

Operational risk is one of the identified risks in organizations, especially in banks, and Basel committees on banking supervision pay special attention to it. In this paper, the mathematical model based on the advanced operational risk model is considered to calculate the probability of survive as a Partial Volterra integro-differentail equation. This equation has been solved numerically by applying the finite differences method with trapezoidal rule to estimate its integral part and the effect of changing the model parameters on the output of the problem has been investigated. In addition, the stability and convergence of the method are discussed and its numerical results are presented.

In this paper, we apply a numerical scheme for the solution of a second order Partial integro-differential equation with a weakly singular kernel. In the time direction, the backward Euler method time-stepping is used to approximate the differential term and the cubic B-splines is applied to the space discretization. Detailed discrete schemes, the convergence and the stability of the method is demonstrated. Next, the computational efficiency and accuracy of the method are examined by the numerical results.

In this paper, first the properties of one and two-dimensional differential transforms are presented. Next, by using the idea of differential transform, we will present a method to find an approximate solution for a Volterra integro-Partial differential equations. This method can be easily applied to many linear and nonlinear problems and is capable of reducing computational works. In some particular cases, the exact solution may be achieved. Finally, the convergence and efficiency of this method will be discussed with some examples which indicate the ability and accuracy of the method.

In this paper, an expansion method based on orthonormal polynomials is presented to find the numerical solution of Partial fractional Fredholm integro-differential equations (PFFIDEs). A PFFIDE is converted to a system of linear algebraic equations, which is solved for the expansion coefficients of approximate solution based on orthonormal polynomials. An estimation error is discussed and some illustrative examples are given to demonstrate the validity and applicability of the proposed method.

In this paper, mixed spectral method is applied to solve the fractional fourth order Partial integro-differential equations together with weak singularity. Eigenfunctions of the fourth order self-adjoint positive-definite differential operator are used for the discretization of spatial variable and its derivatives. Also, shifted Legendre polynomials are applied to the discretization of time variable. Numerical results are presented for some problems to demonstrate the usefulness and accuracy of this approach. The method is easy to apply and produces very accurate numerical results.

This article is devoted to investigate the anlysis for the solution for a class of linear and nonlinear Caputo fractional FredholmVolterra integro-differential equations with multiple delays. The convergence and stability analysis for these equations are investigated.

Abstract. In this paper, we introduce a hybrid approach based on modified neural networks and optimization teqnique to solve ordinary differential equation. Using modified neural net- work makes that training points should be selected over the  open interval (a, b) without training the network in the range of first and end points. Therefore, the calculating volume involving computational error is reduced. In fact, the training points depending on the distance [a, b] selected for training neural networks are converted to similar points in the open interval (a, b) by using a new approach, then the network is trained in these similar areas. In comparison with existing similar neural networks proposed model provides solutions with high accuracy. Numerical examples with simulation results illustrate the effectiveness of the proposed model.