Adomian Decomposition Method And Its Recent Progress

The Adomian decomposition method is a creative and effective method for exactly solving various kinds of problems modeled by differential, partial differential, delay-differential, integro-differential and stochastic equations. The decomposition method provides the solution in a rapidly convergent series with components that are elegantly computed.This thesis summarizes the main points of the decomposition method: the principle, the Adomian polynomials, the convergence results, the noise terms and the partial solution technique. Numerical comparison between the decomposition method and other conventional methods such as the Taylor method and the Runge-Kutta method is introduced, the numerical results demonstrate that the decomposition method is relatively efficient and easily implemented, and we even may derive analytical solutions some times. We illustrate the power of the decomposition method with the help of some concrete examples involved in real physical problems, such as wave equation, variable coefficient PDEs, and inverse heat conduction problems.Some modified forms of the decomposition method have been proposed when applied to special problems by many researchers. Special attention should be paid when we apply the decomposition method to two-point boundary value problems, it is necessary to solve a algebraic system;It should be noted that a modification of inverse operator should be made when dealing with singular boundary value problems;Laplace transform and Pade approximants can be coupled to enhance the efficiency of the decomposition method.However, there are some restrictions of the decomposition method. Fortunately, we have solved the problem recently, therefore, makes a more fundamental contribution to the field. Further, we developed an efficient modification of the decomposition methed, namely two-step decomposition method, that will facilitate the calculations. Some suggestions are proposed at the end of the thesis for the convenience of further research.