A class of singular integro-differential equations in Lebesgue spaces are studied. There are many applications of the singular integro-differential equations discussed in this paper. An example in modeling the stress distribution of an elastic medium with holes is discussed in the paper. Direct numerical schemes using a collocation method and a mechanical quadrature rule designed for the singul...

HIV evolution model that describes the dynamics of concentrations of uninfected and infected cells is considered in the paper. The introduction of dimensionless variables and parameters leads to the initial boundary value problem for singularly perturbed system of partial integro-differential equations. By Tikhonov-Vasil’eva boundary functions method asymptotic solution with boundary layer is c...

We compare piecewise linear and polynomial collocation approaches for the numerical solution of a Fredholm integro-differential equations modelling neural networks. Both approaches combine the use of Gaussian quadrature rules on an infinite interval of integration with interpolation to a uniformly distributed grid on a bounded interval. These methods are illustrated by numerical experiments on ...

An analytic technique, the homotopy analysis method (HAM), is applied to obtain the approximate analytical solutions of fourth-order integro-differential equations. The homotopy analysis method (HAM) is one of the most effective method to obtain the exact and approximate solution and provides us with a new way to obtain series solutions of such problems. HAM contains the auxiliary parameter ħ, ...

The main purpose of this paper is to consider Adomian's decomposition method in non-linear Volterra integro-differential equations. The advantages of this method, compared with the recent numerical techniques (in particular the implicitly linear collocation methods) , and the convergence of Adomian's method applied to such nonlinear integro-differential equations are discussed. Finally, by ...

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...

We describe a symbolic framework for treating linear boundary problems with a generic implementation in the Theorema system. For ordinary differential equations, the operations implemented include computing Green’s operators, composing boundary problems and integrodifferential operators, and factoring boundary problems. Based on our factorization approach, we also present some first steps for s...