Iterative Method for Solving a Nonlinear Fourth Order Integro-Differential Equation

Abstract

This study presents the execution of an iterative technique suggested by Temimi and Ansari (TA) method to approximate solutions to a boundary value problem of a 4th-order nonlinear integro-differential equation (4th-ONIDE) of the type Kirchhoff which appears in the study of transverse vibration of hinged shafts. This problem is difficult to solve because there is a non-linear term under the integral sign, however, a number of authors have suggested iterative methods for solving this type of equation. The solution is obtained as a series that merges with the exact solution. Two examples are solved by TA method, the results showed that the proposed technique was effective, accurate, and reliable. Also, for greater reliability, the approximate solutions were compared with the classic Runge-Kutta method (RK4M) where good agreements were observed. For more accuracy the maximum error remainder was found, and the absolute error was computed between the semi-analytical method and the numerical method RK4M.  Mathematica^® 11 was used as a program for calculations.