This section discusses a simplified version of the Adomian decomposition method
first concept of which was proposed by Randolf Rach in 1989 that was crystallized later in a paper published with his colleagues G. Adomian and R.E. Meyers. That is way this technique is frequently referred to as the Rach--Adomian--Meyers modified decomposition method (MDM for short). Initially, this method was applied to power series expansions, which was based on the nonlinear transformation of series by the Adomian--Rach Theorem. Similar to the Runge--Kutta methods or spline method, the MDM can be implemented in numerical integration of differential equations by one-step methods. In case of polynomials or power series, it shows the advantage in speed and accuracy of calculations when at each step the Adomian decomposition method allows one to perform explicit evaluations.
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The modified decomposition method (MDM) is both a decelerated Adomian decomposition method and a developed power series method that nicely and accurately treats any analytic nonlinearity in differential equations. In case of autonomous homogeneous or nonhomogeneous equations, the method is seems to be superior over other methods. Therefore, we start demonstration of this method in initial value problems for first order differential equations of the form
where \( L \left[ y(t) \right] = y' (t) + a (t)\, y (t) = {\text d}y/{\text d}t + a (t)\, y (t) \) is the linear differential operator,
\( R \left[ y(t) \right] = b (t)\,y(t) , \) is
a linear operator, and
\( N \left[ y(t) \right] \) is a nonlinear term, g is the system input and y is the system output. The operator can be rewritten as \( L = \texttt{D} + a(t)\,\texttt{T} \) , where \( \texttt{D} = {\text d}/{\text d}t \) is the derivative operator and \( \texttt{I} \) is the identical operator. Since the unbounded derivative operator has a one-dimensional kernel (the set of functions that are annihilated by the derivative operator), its inverse, calling antiderivative, is a multivariable operator depending on some constant. Similar property is valid for L. In order to knock out a single inverse operator L^{-1}, we consider consider L on a space of functions that satisfy an initial condition; more over, it is assumed that an explicit formula is available.
Furthermore, we emphasize that the choice for L and concomitantly its
inverse L^{-1} are determined by the particular equation to be
solved. Hence, the choice for L is nonunique, e.g. for cases of differential
equations with singular coefficients, a different form for the linear
operator L can be chosen. As a rule, the highest derivative is taken
as the linear operator L. Later we consider more general equation including coefficients depending on independent variable.
The MDM and its modifications and generalizations have
been extensively applied in physics, chemistry, mechanics, hydrology,
engineering, economics, biology, epidemiology, etc. Some references can be found in the following link.
where A_{n} are the Adomian polynomials in terms of solution coefficients. Of course, the above formula is nothing more than Taylor's series expansion of the nonlinear function N[y]. However, the main difference of the MDM expansion from the power series is that in the former the coefficients depend recursively on the previous coefficients whereas Taylor's coefficients all depend on the infinitesimal behavior of the slope function, but not on previously found coefficients. It is convenient to make a shift in independent variable x = t - t_{0} to transfer the initial point from t_{0} into the origin:
Actually, Mathematica can find any Adomian polynomail for polynomial nonlinearities with a dedicated command Coefficient or CoefficientList; so sometimes the above formula is not needed when there is an access to the computer algebra system. Recall that if N is not a polynomail, then Mathematica cannot provide you an answer, and you have to use a special script to determine coefficients.
Mathematica code for evaluating Adomian polynomials (Jun-Sheng Duan, An efficient algorithm for the multivariable Adomian polynomials, Applied Mathematics and Computation, 2010, 217, pp. 2456--2467):
Before we go to more sofisticated examples, we demonstrate the MDM on the following familiar one. Let \( u = e^x = \sum_{n\ge 0} \frac{x^n}{n!} = \sum_{n\ge 0} c_n x^n . \) Raising it to the third power, we get
Using the derivative of the arccotangent function,
\( {\text d}\,\mbox{arccot}(x)/{\text d}x = -\left( 1 + x^2 \right)^{-1} , \) we calculate the next Adomian's polynomial:
where f is a known function having required number of derivatives (generally speaking, all derivatives). Assuming that the given differential equation has a convergent Maclaurin series solution
\[
y(x) = \sum_{n\ge 0} a_n x^n ,
\]
we expand the nonlinear term into Adomian's series
From the initial condition, we find immediately that a_{0} = 1, and for all other coefficients, we get the recurrence relation by equating the coefficients of like powers of x:
Application of the bounded operator to the given differential equation yields
the functional equation for which the fixed point theorem can be applied:
The Adomian decomposition method (ADM) states
that the dependent variable y(x) and the nonlinear term
f(y) = y² should be written as the following
infinite series
We consider the following initial value problem for the nonlinear differential equation with holomorphic input function g(x) and holomorphic coefficients
Adomian, G., Rach, R., Transformation of series, Applied Mathematics Letters
Volume 4, Issue 4, 1991, Pages 69-71; https://doi.org/10.1016/0893-9659(91)90058-4
Adomian, G., Rach, R., Nonlinear transformation of series—part II, Computers & Mathematics with Applications
Volume 23, Issue 10, May 1992, Pages 79-83; https://doi.org/10.1016/0898-1221(92)90058-P
Andrianov, I.V., Olevskii, V.I., Tokarzewski, S., A modified Adomian's decomposition method, Journal of Applied Mathematics and Mechanics, 1998, Volume 62, Issue 2, Pages 309--314; https://doi.org/10.1016/S0021-8928(98)00040-9
Duan, J.-S., Rach, R., Wazwaz, A.-M., Solution of the model of beam-type
micro- and nano-scale electrostatic actuators by a new modified
Adomian decomposition method for nonlinear boundary value problems,
International Journal of Non-Linear Mechanics, 2013, Volume 49, 2013, Pages
159--169.
Rach, R., Adomian, G., Meyers, R., A modified decomposition method, Computers & Mathematics with Applications, 1992,
Volume 23, Issue 1, January 1992, Pages 17-23; https://doi.org/10.1016/0898-1221(92)90076-T
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