Décomposition d'Adomian

The Adomian decomposition method (ADM) is a semi-analytical method for solving ordinary and partial nonlinear differential equations. The method was developed from the 1970s to the 1990s by George Adomian, chair of the Center for Applied Mathematics at the University of Georgia. It is further extensible to stochastic systems by using the Ito integral. The aim of this method is towards a unified theory for the solution of partial differential equations (PDE); an aim which has been superseded by the more general theory of the homotopy analysis method. The crucial aspect of the method is employment of the "Adomian polynomials" which allow for solution convergence of the nonlinear portion of the equation, without simply linearizing the system. These polynomials mathematically generalize to a Maclaurin series about an arbitrary external parameter; which gives the solution method more flexibility than direct Taylor series expansion. Adomian method is well suited to solve Cauchy problems, an important class of problems which include initial conditions problems. An example of initial condition problem for an ordinary differential equation is the following: To solve the problem, the highest degree differential operator (written here as L) is put on the left side, in the following way: with L = d/dt and . Now the solution is assumed to be an infinite series of contributions: Replacing in the previous expression, we obtain: Now we identify y0 with some explicit expression on the right, and yi, i = 1, 2, 3, ..., with some expression on the right containing terms of lower order than i. For instance: In this way, any contribution can be explicitly calculated at any order. If we settle for the four first terms, the approximant is the following: A second example, with more complex boundary conditions is the Blasius equation for a flow in a boundary layer: With the following conditions at the boundaries: Linear and non-linear operators are now called and , respectively.

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The Adomian Decomposition Method for Nonlinear Partial Differential Equations

Natalie Sauerwald

The goal of this report is to study the method introduced by Adomian known as the Adomian Decomposition Method (ADM), which is used to find an approximate solution to nonlinear partial differential equations (PDEs) as a series expansion involving the recursive solution of linear PDEs. We first describe the method, giving two specific examples with different nonlinearities and show exactly how the method works for these problems. Some analytical convergence results are then given, along with numerical solutions to the examples demonstrating these convergence results. A discussion of parameters inside of these nonlinearities follows, both for polynomial nonlinearities and for the more complicated hyperbolic sine nonlinearity problem. Finally, we compare the ADM with the Picard method, pointing out some important differences and demonstrating them by solving the given examples with both methods and comparing the results.

2013