ABSTRACT
There is a great deal of interest in solving a wide variety of stochastic and deterministic problems by the application of decomposition method proposed and developed by George Adomian [1]. This method aims at an unified treatment of linear/nonlinear, ordinary/partial differential equations for initial and boundary value problems. It has been found that accurate and easily computed quantitative solutions can be obtained for nonlinear systems without the assumption of small nonlinearity or computer intensive methods. Partial differential equations can be solved more efficiently with less computation [2,3,4]. To quote Adomian: Among these are the Navier-Stokes equation, the N-body problem, and the Yukawa-coupled Klein Gordan-Schrödinger equations. . .. This chapter presents an elementary exposition of this method. The emphasis is given on solving secondorder ordinary differential equations and integral equations. To begin with, a simple second-order differential equation with constant coefficients is chosen and then specific examples are chosen mainly on the basis of their wide area of applications in physics, namely, Airy equation, Hermite equation, Gauss-Hermite equation, Hypergeometric equation, Volterra integral equation, and Laguerre equation. The method is also extended to the nonlinear equations and partial differential equations. The method is used to illustrate a few problems in optics, namely, the ray equation in planar waveguide and to important research problems in nonlinear optics, i.e., ray tracing through the crystalline lens. In visual optics a major area of investigation is realistic eye models (a brief review of these models is given in [5]). In particular, Siedlecki et al. have developed an eye model using Kooijman’s model [6] as a starting point. A major modification is the assumption of a radial refractive index distribution for the crystalline lens. Here they have modeled the lens variation as a decreasing exponention with distance from the optic axis. In the present model the GRIN variation is used to show the applicability of the decomposition method in tracing rays through inhomogeneous media [7]. Nonlinear Schrödinger (NLS) equation is the fundamental wave equation in nonlinear optics and it describes the propagation of optical soliton through nonlinear media [8,9,10]. We show the applicability of the decomposition method to the nonlinear Schrödinger equation and the higher order nonlinear Schrödinger equation. In the appendix, Mathematica codes for Adomian solutions of the wave equations discussed in this chapter are given.
