# Preface

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

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# Adomian Decomposition Method

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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1. John P. Boyd, Padé approximant algorithm for solving nonlinear ordinary differential equation boundary value problems on an unbounded domain, Computers in Physics, 1997, Vol. 11, 299 (1997); https://doi.org/10.1063/1.168606
2. Jun-Sheng Duan, Randolph Rach, Abdul-Majid Wazwaz, Temuer Chaolu, Zhong Wang, A new modified Adomian decomposition method and its multistage form for solving nonlinear boundary value problems with Robin boundary conditions, Applied Mathematical Modelling, 37, 2013, 8687--8708.
3. Kim, W. and Chun, C., A Modified Adomian Decomposition Method for Solving Higher-OrderSingular Boundary Value Problems, 2010,
4. Randolph Rach, Abdul‐Majid Wazwaz, Jun‐Sheng Duan, A reliable modification of the Adomian decomposition method for higher‐order nonlinear differential equations, Kybernetes, Vol. 42 Issue: 2, 2013, pp.282--308, https://doi.org/10.1108/03684921311310611
5. Abdul-Majid Wazwaz, Randolph Rach, Jun-Sheng Duan, Adomian decomposition method for solving the Volterra integral form of the Lane–Emden equations with initial values and boundary conditions, Applied Mathematics and Computation, 219, 2013, 5004--5019.
6. Holmquist, Sonia, An examination of the effectiveness of the Adomian Decomposition Method in fluid dynamic applications, University of Central Florida, Theses and Dissertations, Doctoral Dissertation (Open Access), 2007.

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