# Preface

This section discusses an application of the Adomian decomposition method to boundary value problems.

The algorithm for solving boundary value problems for differential equations on infinite or semi-infinite intervals include four steps. The first step is to compute high-order Taylor series expansions using an algebraic manipulation language such as Maple or Mathematica. These expansions will contain one or more unknown parameters z which will be determined by the boundary condition at infinity. The second step is to convert the Taylor expansions into diagonal Padé approximants. The boundary condition that u(x) decays to zero at infinity becomes the condition that the coefficient of the highest power of x in the numerator polynomial must be zero. The third step is to solve this equation for the free parameter z. The final step is to evaluate each of the multiple solutions of this equation for physical plausibility and convergence (as N increases). Methods for nonlinear problems are almost always iterative and need a first guess to initialize the iteration. The Padé algorithm is unusual in that it is a direct method that requires no a priori information about the solution.

Application of the Adomian decomposition method in boundary value problems is based on inverse operator, denerated by the second derivative and subject to Dirichlet boundary conditions:
$y(x) = A + \left( x-a \right) \frac{B-A}{b-a} + \frac{1}{2} \int_0^x \left( x-t \right) f(t)\,{\text d}t - \frac{1}{2} \int_x^b \left( x-t \right) f(t)\,{\text d}t + \frac{1}{2} \,\frac{x-b}{a-b} \int_a^b \left( a-t \right) f(t) \,{\text d}t - \frac{1}{2} \,\frac{x-a}{b-a} \int_a^b \left( b-t \right) f(t) \,{\text d}t.$

Example: Consider the nonlinear differential operator:

$L \left[ \cdot \right] = \frac{1}{r} \,\frac{\text d}{{\text d}r} \left[ r \left( - \frac{\text d}{{\text d}r} \left[ \cdot \right] \right)^n \right] .$
Its inverse is
$L^{-1} \left[ \cdot \right] = - \int_h^r {\text d}r \sqrt[n]{\frac{1}{r} \int_0^r r \left[ \cdot \right] {\text d}r}$
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