This section discusses applications of the Adomian decomposition method (ADM
for short) to first order nonlinear differential equations with singularities
(application of the ADM for regular nonlinear differential equations was done
previously in section).
The ADM calculates the solutions of nonlinear equations as infinite series in
which each term can be determined in principle. The crucial part of the ADM is
the decomposition of the nonliearity as the series with respect to generalized
polynomials called Adomian’s polynomials. The series-solution is convergent
towards an accurate solution, and in some cases (at least for homogeneous equations) it gives the exact solution. To surpress noise terms that appear in Adomian's solutions for inhomogeneous equations, Abdul-Majid Wazwaz (A reliable modification of Adomian decomposition method, Applied Mathematics and
Computation, 1999, Vol. 102 No. 1, pp. 77-86.) proposed an ADM modification based to division of the input into two parts that are incorporated in the first two terms of ADM series solution.
Since the calculations of the Adomian’s polynomials are
based on the Faà di Bruno's formula, the number of terms in this formula grows exponentially; so this task cannot be accomplished on an electrical computer, generally speaking. Therefore, a truncated version of the ADM is usually used for approximations.
There are known four kinds of Adomian polynomials (see: Duan, J.-S., Rach, R., Băleanu, D., and Wazwaz, A.-M.,
A review of the Adomian decomposition method and its applications to fractional differential equations, Communications in Fractional Calculus,
2012, Vol. 3, No. 2, pp. 73--99.) and that certain circumstances could benefit from a different choice; however for most problems the classical Adomian polynomials are satisfactory. A special class of them constitute the accelerated Adomian polynomials that were first introduced by George Adomian in his 1989 book, and were detailed and intensively used in 2008 Randolph Rach article.
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where p and x_{0} are some positive numbers, while
y_{0} is an arbitrary constant, but it is kept fixed for the given initial value problem. Here g(x) is a
forcing (or driving) function (input) and f(x, y) is a nonlinear
operator. It
is convenient to rewrite the given singular differential equation in operator
form. Here we have two options to do this: either to isolate the derivative
operator \( \texttt{D} = {\text d}/{\text d}x \) or
to combine it together with the linear term. So we set two linear differential
operators:
\[
L\left[ y \right] = \frac{{\text d} y}{{\text d}x} + \frac{p}{x}\, y =
\left( \texttt{D} + \frac{p}{x}\,\texttt{I} \right) y\qquad\mbox{and} \qquad
L_0 \left[ y \right] = \frac{{\text d} y}{{\text d}x} = \texttt{D}\, y ,
\]
where \( \texttt{I} \) is the identity operator.
Correspondingly, we represent the given differential equation in two forms:
These two inverse operators involve y_{0} =
y(x_{0}), which plays the role of arbitrary constant.
Since our objective is education, we demonstrate three different approaches how to solve the given initial value problem based on the Adomian decomposition method. In the main streamline of the ADM, the solution y is represented as the infinite sum of series
\[
y(x) = u_0 (x) + \sum_{n\ge 1} u_n (x) ,
\]
and the nonlinear function is decomposed as follows:
\[
f(x,y) = \sum_{n\ge 0} A_n (x) ,
\]
where the n-th Adomian polynomial of \( u_0 , u_a ,
\ldots , u_n \) are calculated by the formula
Upon substitution of the Adomian decomposition series for the solution
y(x) and the series of Adomian polynomials tailored to the
nonlinearity term f(x, y) and applying the inverse operator
L^{-1}, we obtain
In order to apply the fixed point theorem, we need the homogeneous initial conditions as well as the homogeneous equation. This is achieved by solving the initial value problem for u_{0}:
The generalized decomposition method follows the same procedure, but instead of Adomian polynomials it calls for another polynomials---called the accelerated Adomian polynomails
All other terms are solutions of the homogeneose initial value problem
\[
\dot{u}_{m+1} = A_{m}, \qquad u(0) = 0,
\]
where A_{m}(u_{0}, … , u_{m}) are the classical Adomian polynomials that correspond to the given slope function \( f(y) = \left( 2t-1 \right) e^{y-2t} . \) The first one is
Option 2: using accelerated Adomian polynomials:
The first two initial terms u_{0} = 2, \( u_1 = \int_1^t A_0 (s)\,{\text d}s = 1 - t\, e^{2-2t} \) and the Adomian polynomail A_{0} remains the same. The accelerated Adomian polynomails are
Note that we use the Newton's dot notation for the derivative with respect to time variable t.
This problem has an explicit solution (a very rare case):
where p is a real constant,
\( \texttt{D} ≝ {\text d}/{\text d}x \) is the
derivative operator, and
\( \texttt{I} \) is the identity operator.
The kernel (= set of solutions L y = 0; in other words, the kernel of a linear operator includes all functions that are annihilated by L) of the linear unbounded differential operator L is an one-dimensional space. To find
its inverse L^{-1}, we need to solve the differential equation
Since this differential equation involves the derivative operator
\( \texttt{D} ≝ {\text d}/{\text d}x , \) the above equation has infinite many solutions depending on arbitrary constant. Such family of solutions is called the general solution. From mathematical point of view, L^{-1} is not an operator because it assigns infinite many outputs to every input. To single out one of them, we consider the differential operator \( L \left[ x, \texttt{D} \right] \) on a set of smooth functions \( C^1 (a,b) \) on some open interval \( (a,b) \) that have a specific value at one of its point: y(x_{0}) = y_{0}. Then \( L \left[ x, \texttt{D} \right] : C^1 \to C^0 , \) and it has a unique inverse. Recall that we denote by C¹ the set of functions having continuous derivatives in a specified interval.
We start with a simple case when the differential operator has the only one singlar point at the origin:
Multiplying both side of the differential equation
\( y; + p\,y/x = w \)
by an integrating
factor (x^{p} in our case), we obtain an exact differential
equation
where x_{0} is an arbitrary number other than zero because the
origin is a singular point for this differential equation. Here y_{0} plays the role of arbitrary constant to single out one inverse operator. So the singular
point x_{0} = 0 divides the real axis ℝ into two parts,
and the solution of the above differential equation
\( y' +p\,y/x = f \) exists in either of these two
parts. In future, we assume that x is positive and the initial
condition is also specified at a positive point x_{0} > 0.
The situation can be extended for arbitrary function p(x); so we
consider the first order differential operator:
where f is a holomorphic function in some two-dimensional domain and function 𝑎(x) has finite number of poles of first order. This means that there are finite number of m points x_{k}, k = 1, 2, ⃛, m such that
\( \left( x - x_k \right) a(x) \) is holomorphic.
Example:
Consider the following singular differential equation:
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) should be written as the following infinite series
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