A New Algorithm of Geometric Mean for Solving High-Order Fredholm Integro-differential Equations

The Fredholm Integro-differential equations (IDEs) of the second kind appear in many scientific applications. Mathematical methods for the solution of the Fredholm IDEs have been developed over the last decade. In this article, we introduce a new variant of Geometric Mean iterative (MGM) method to solve the Fredholm fourth order IDEs of the second kind. As is typical with the IDEs, the problem is first transformed into a dense algebraic system which is derived from finite difference and three-point composite closed Newton-Cotes approximation schemes. For the solution of such system, the MGM method under the standard Geometric Mean iterative method is developed. Based on three criteria of a number of iterations, CPU time and the root mean square error (RMSE) for various mesh sizes, numerical simulation has been carried out to compare the validity and applicability of the proposed method with some existing methods such as the Gauss-Seidel, the Arithmetic Mean and the standard Geometric Mean iterative methods. The proposed method is verified to be stable and has the optimal convergence order to solve this types of IDEs. To demonstrate the fast and smooth convergence of the proposed method, we use two examples of the IDEs. The numerical experiments confirm that the proposed method gives a better performance comparing to other mentioned methods. It is computationally stable, valid and accurate, and its most significant features are simplicity, fast and smooth convergence with desirable accuracy.