Preface

This section illustrates application of the modified decomposition method (MDM for short) in some systems of nonlinear ordinary differential equations.

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Glossary

Modified Decomposition Method

A large variety of linear and nonlinear Partial Differential Equations (PDEs) subject to certain initial conditions can be solved with the aid of Modified Decomposition Method (MDM), Invented by Randolph Rach together with his colleagues. Since the MDM is a simplified version of more general Adomian Decomposition method (ADM), it is based on constructing and using Adomian polynomials. These polynomials, in turn, are evaluated through differentiation of slope function that depends on many variables. Therefore, we need to present the main formula for differentiating composition of function, known as the multivariate Faà di Bruno formula (fdB for short), discovered by the Italian priest Francesco Faà di Bruno (1825--1888). However, this formula was known more than 50 year before Faà di Bruno to the French mathematician Louis François Antoine Arbogast (1759--1803).

Before we formulate te main Faà di Bruno formula, we need to introduce some notations. Let \( \maathbb{N}_0 \) denote the set of nonnegative integers, \( {\bf \nu} = [\nu_1 , \nu_2 , \ldots , \nu_d ] \in \maathbb{N}_0^d \) and \( {\bf z} = ( z_1 , z_2 , \ldots , z_d ) \in \mathbb{R}^d . \) We define

\[ \begin{split} \left\vert {\bf \nu} \right\vert &= \nu_1 + \nu_2 + \cdots + \nu_d , \qquad {\bf \nu}! = \left( \nu_1 ! \right) \left( \nu_2 ! \right) \cdots \left( \nu_d ! \right) , \\ {\bf z}^{\nu} &= z_1^{\nu_1} \cdot z_2^{\nu_2} \cdots z_d^{\nu_d} , \\ \texttt{D}^0 &= \mbox{identity operator} \quad \texttt{D}^{\bf \nu} = \frac{\partial^{\bf \nu}}{\partial x_1^{\nu_1} \cdots \partial x_d^{\nu_d}} \quad \mbox{for} \quad \left\vert {\bf \nu \right\vert > 0. \end{split} \]

Theorem (Multivariate Faà di Bruno Formula): Let \( f(y_1 , \ldots , y_m ), \quad g_1 \left( x_1 , \ldots , x_d \right) , \ \ldots , \ g_m \left( x_1 , \ldots , x_d \right) \) be real-valued smooth functions and

\[ h\left( x_1 , \ldots , x_d \right) = f\left[ g_1 \left( x_1 , \ldots , x_d \right) , \ldots , g_m \left( x_1 , \ldots , x_d \right) \right] \]
be the composition of these functions. Let the point x0 be a given point and all function have sufficient number of derivatives when required. Then the partial derivative \( \texttt{D}_{\bf x}^{\bf \nu} h({\bf x}) \) exists in a neighborhood of the given point x0 and can be explicitly expressed as below:
\[ h_{\bf \nu} ({\bf x}) = \sum_{1 \le |{\bf \lambda}| \le |{\bf \nu}}}\, f_{\bf \lambda} \left[ g({\bf x}) \right] \sum_{p({\bf \nu}, {\bf {\lambda}} \, \left( {\bf \nu}! \right) \prod_{j=1}^q \frac{\left[ \right]^{{\bf k}_j}}{\left( {\bf k}! \right) \left[ m_j ! \right]^{|k_j }}} , \]
where
\[ p\left( {\bf \nu}, {\bf \lambda} \right) = \left\{ \left( k_1 , \ldots , k_q; m_1 , \ldots , m_q \right) : \, \left\vert k_i \right\vert &ge 0, \quad \sum_{i=1}^q k_i = \lambda \mbox{ and } \sum_{i=1}^q \left\vert k_i \right\vert m_i = {\bf \nu}\right\} . \]
In the above, m1, ... , mq is a complete listing of all vectors m≤ν with
\[ \left\vert m \right\vert > 0, \quad q = q({\bf \nu}) = \left[ \prod_{s=1}^q \left( \nu_s +1 \right) \right] -1, \]
the vectors k ∈ ℕ0m and vectors m ∈ ℕ0d.     ⧫

 

  5. Wazwaz, A.-M., The modified decomposition method applied to unsteady flow of gas through a porous medium, Applied Mathematics and Computation, 2001, Vol. 118, Issue 2/3, pp. 123--132.
  6. Zitoun, F.B., Solutions of linear and nonlinear Partial DifferentialEquations with initial conditions and multivariate Faà di Bruno formula, HAL archives-ouvertes, 2014, hal-00845788

 

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