New development of HAM for approximating linear integro-differential equation of order two

In many problems of mathematical physics, theory of elasticity, visco-dynamics fluid and mixed problems of mechanics of continuous media reduces to the integral and intro-differential equation (Volterra or Fredholm type) first or second kind. In the early XXth (around 1910) century, to solve some problems in the field of mechanics Volterra had to solve integro-differential equations (IDEs) with variable boundaries. One of the popular methods for solving Volterra IDEs is the method of quadratures when problem of consideration is a linear problem. When IDEs are nonlinear or integral kernel are complicated (singular or nonlinear) then quadrature rule is not most suitable therefore other type of methods are needed to develop. In this note, we have used the standard and new-development of homotopy analysis method (ND-HAM) for the general second order linear integral-differential equations (IDEs) of the second kind. Both methods reduce IDEs into iterative sequence of algebraic integral equations with known integrant function. Gauss-Legendre quadrature formulas are applied for kernel integrations. Obtained system of algebraic equations are solved numerically. Moreover, numerical examples demonstrate the high accuracy of the proposed method. Comparisons with other methods are also provided. The results show that the proposed method is simple, effective and reliable.