Numerical Investigation Of Two Kinds Of Fourth-order Integro-differential Equations

Many problems in natural science and engineering and other areas can be modeled byintegro-differential equations. With the advent and fast development of computation technology,integro-differential equations have been widely and effectively used in many areas. However, inmost cases, the analytical solutions of the integro-differential equations cannot be expressedexplicitly. Therefore, it is meaningful to investigate the numerical solutions of theintegro-differential equations.In this paper, we consider firstly the numerical solution of the fourth-orderintegro-differential equation modeling the suspension bridge. The finite differenceapproximation, finite element approximation and Legendre-Galerkin spectral approximation ofthe equation are developed. Newton type iterative method and a simple iterative method arepresented to deal with the integral term in the equation respectively. Error analysis of theabove method are also carried out. Numerical examples are presented to show the feasibility ofthe methods.Secondly, we consider the numerical solution of another fourth-order integro-differentialequation, which was established in the study of transverse vibrations of a hinged beam. Acompact finite difference scheme and a Legendre-Galerkin spectral method are presented toapproximate the equation. Newton type iterative algorithm and a simple iterative algorithm arepresented to deal with the integral term respectively. Error analysis of the methods are alsocarried out. The feasibility of the methods are confirmed by some numerical examples.