A Comparative Study of Numerical Methods for Solving the Riccati Equation

Abstract

Sometimes using several numerical methods to solve a problem may give similar results. It is noticeable that applying different numerical methods to solve a problem may provide just the same results. For example, using the ADM and the successive  approximation method for linear integral equations the ADM and the power series method for differential equations and the Jacobi
iterative method for system of linear equation, we will get just the same results. Since the beginning of the 1980s, the ADM has been applied to a wide class of functional equations. The ADM has been demonstrated to provide accurate and computable solutions for a wide class of linear or nonlinear equations involving differential operators by representing nonlinear terms in Adomian's polynomials. This procedure requires neither linearization nor perturbation. The LTDA is an approach based on the ADM, which is to be considered an effective method in solving many problems. In general, a rapidly convergent series solutions can be obtained. Since the use of the Laplace transform replaces differentiation with simple algebraic operations on the transform, the algebraic equation is then solvable by decomposition. The LTDA approximates the exact solution with a high degree of accuracy using only few terms of the iterative scheme. Many authors have applied Numerical method to solve Bratu's equation, the Duffing equation, the Klein-Gordon equation, some nonlinear differential equations, the nonlinear