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.

In this paper‎, ‎a numerical scheme based on shifted Vieta-Lucas polynomials is utilised to solve mentioned equation‎. ‎The main characteristic of the presented method is to approximate Brownian motion with help of the Gauss-Legendre quadrature‎, ‎which makes calculations easier‎. ‎Another characteristic of this method are employed suitable collocation points to convert the stochastic equation under the study into a system of algebraic equations by using the operational matrices‎. ‎So that‎, ‎Newton's method is applied to solve them‎. The convergence analysis and error bound of the suggested method are well established‎. ‎Additionally‎, ‎the proofs related to the existence and uniqueness of the solutions for the equations under investigation have been provided‎. ‎In order to illustrate the effectiveness‎, ‎compatibility and plausibility of the proposed technique‎, ‎four numerical examples are presented.

In this work, we describe a new method for the calculation of the numerical solution of the mixed singular Integro-differential equation. The method is mainly based on a finite element approximation. For this purpose, we obtain the variational form of the problem under consideration. The result-ing system has been approximated numerically by linear B-spline function. The method of convergence and its order of convergence are established. Finally, to investigate the accuracy of the proposed method some test ex-amples are presented. The numerically obtained results are compared with another method for the validity of our results and shown by some tables and figures.

In this work we constructed a numerical scheme to approximate the Volterra Integro-differential equations of convolution type using Laplace transform. The solution of the problem is recovered using inverse Laplace transform as contour integral in the complex plane. The integral is then approximated along a suitable contour using the trapezoidal rule with equal step size. The solution accuracy depends on optimal contour of integrations to compute accurately the inverse Laplace transform. For better accuracy two types of contour parabolic and hyperbolic are used which are available in the literature. The performance of the numerical scheme is tested for different examples. The actual error well agree with the corresponding error estimates of the proposed numerical scheme for both parabolic as well as hyperbolic contours.

In this research, fourth-order Improved Runge-Kutta method with three stages for solving fuzzy Volterra Integro-differential (FVID) equations of the second kind under the concept of generalized Hukuhara differentiability is proposed. The advantage of the proposed method in this study compared with the same order classic Runge-Kutta method is, Improved Runge-Kutta (IRK) method uses a fewer number of stages in each step which causes less computational cost in total. Here, the integral part is approximated by applying the combination of Lagrange interpolation polynomials and Simpson’s rule. The numerical results are compared with some existing methods such as the fourth-order Runge-Kutta (RK) method, variational iteration method (VIM), and homotopy perturbation method (HPM) to prove the efficiency of IRK method. Based on the obtained results, it is clear that the fourth-order Improved Runge-Kutta method with higher accuracy and less number of stages which leads the less computational cost is more efficient than other existing methods for solving FVID equations.

In this paper, a novel hybrid method based on learning algorithm of fuzzy neural network and Newton-Cotes methods with positive coefficient for the solution of linear Fredholm Integro-differential equation of the second kind with fuzzy initial value is presented. Here neural network is considered as a part of large field called neural computing or soft computing. We propose a learning algorithm from the cost function for adjusting fuzzy weights. This paper is one of the first attempts to derive learning algorithms from fuzzy neural networks with real input, fuzzy output, and fuzzy weights. Finally, we illustrate our approach by numerical examples.