In this work, we conduct a comparative study among the combine Laplace transform and modified Adomian decomposition method (LMADM) and two traditional methods for an analytic and approximate treatment of special type of nonlinear Volterra integro-differential equations of the second kind. The nonlinear part of integro-differential is approximated by Adomian polynomials, and the equation is reduced to a simple equations. The proper implementation of combine Laplace transform and modified Adomian decomposition method can extremely minimize the size of work if compared to existing traditional techniques. Moreover, three particular examples are discussed to show the reliability and the performance of method. Manuscript profile

In this study, a new and efficient approach is presented for numerical solution ofFredholm integro-differential equations (FIDEs) of the second kind on unbounded domainwith degenerate kernel based on operational matrices with respect to generalized Laguerrepolynomials(GLPs). Properties of these polynomials and operational matrices of integration, differentiation are introduced and are ultilized to reduce the (FIDEs) to the solution ofa system of linear algebraic equations with unknown generalized Laguerre coefficients. Inaddition, two examples are given to demonstrate the validity, efficiency and applicability ofthe technique. Manuscript profile