This section provides an illustration of application of the Adomian
decomposition method (ADM for short) to second order singular differential equations.
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This equation is named after astrophysicists Jonathan Homer Lane (1819--1880) and Robert Emden (1862--1940). The Lane--Emden equation has been used to model several phenomena in mathematical physics and astrophysics such as thermal explosions, stellar structure, the thermal behavior of a spherical cloud of gas, isothermal gas spheres, and thermionic currents. Writing the above equation in the operator form
We will discuss more general equations later. Let us consider
the initial conditions
\[
y\left( x_0 \right) = a, \qquad y' \left( x_0 \right) = b ,
\]
where x_{0} is an arbitrary real number other than zero because
the origin is a singular point for this differential equation. Without any loss of generality, we can assume that x_{0} is positive. The driving function g(x) is assumed to be known and f(x, y)
is the nonlinearity term. In preparation to apply the Adomian decomposition method, it is convenient to introduce the differential operator
where \( \texttt{D} = {\text d}/{\text d}x \) is the
derivative operator and \( \texttt{I} \) is the identity operator. Since the inverse operator is known to be
we are ready to apply the ADM to solve the given initial value problem.
Suppose that we want to incorporate a constant term to our differential
operator. This task is accomplished with the differential operator:
Applying the inverse operator formular to zero function, we obtain
\[
y(x) = \frac{\sin \pi x}{x} .
\]
■
Example:
Consider the initial value problem
\[
y'' +
\]
We can rewrite the Duffing equation in the operator form:
\[
L\left[ y \right] \equiv \texttt{D}^2 y = N \left[ y \right] \equiv
\varepsilon \,y^3 - k\, y.
\]
Physically, the resilience of the oscillator is directly proportional to
N[y]. When k ≥ 0 and ε < 0, there is one
unique equilibrium point y = 0 where the nonlinear term vanishes.
However, when k ≤ 0 and ε > 0, there are two
equilibrium points: y = 0 and
\( y = \sqrt{|k/\varepsilon |} . \)
From the physical points of view, the sum of the kinetic and
potential energy of the oscillator keeps the same, therefore it is clear that
the oscillation motion is periodic, no matter k is positive or
negative. Thus, from physical points of view, it is easy to know that
y(t) is periodic, even if we do not directly solve the Dufing
equation.
■
where p, q are some real numbers, g(x) is a given
input function,
\( \texttt{D} = {\text d}/{\text d}x \) is the
derivative operator and \( \texttt{I} \) is the
identity operator. One of the ways to determine a solution f the inhomogeneous
differential equation, is its factorization. If the following system of
algebraic equations
\[
\begin{split}
p &= m +2n , \\
q &= n \left( m+n-1 \right) ,
\end{split}
\]
has two distinct real solutions (i.e., \( (p-1)^2 - 4q > 0 \) ), then the operator L admits factorization:
The Adomian decomposition method and its modifications can be considered as an analytic approximation method, which does not require accepting a priori assumptions in our cybernetics models that drastically alter the outcomes so that they do not faithfully replicate reality. It permits us to solve both nonlinear initial value problems and boundary value problems without unphysical restrictive assumptions such as required by linearization,
perturbation, ad hoc assumptions, guessing the initial term or a set of basis functions, and so forth, most of which would change the physical behavior of the problem.
Consider an ordinary differential equation written in operator form:
\[
L\left[ y \right] = L_0 \left[ y \right] + N\left[ y \right] + g(x) ,
\]
where \( L\left[ y \right] = \texttt{D}^2 + a\,\texttt{D}
+ b\, \texttt{I} \) with
\( \texttt{D} = {\text d}/{\text d}x \)
being the derivative operator and \( \texttt{I} \)
being the identity operator, is the linear differential operator acting on a
function y(x),
\( L_0 \left[ y \right] = p(x)\,\texttt{D} + q(x)\,
\texttt{I}\) is a linear differential operator of order 1,
N[y] represents an analytic nonlinear operator, and
g(x) is a specified analytic input function (usually called the
driving term).
The linear operator is usually included in the nonlinear one N[y]
so we will drop it. The Adomian method does not react whether lower order
terms as \( a\,\texttt{D}
+ b\, \texttt{I} \) are included in L or not. Including these
terms depends whether you can obtain the explicit formula for the inverse
L^{-1} or not.
Very few nonlinear problems have simple, closed-form solutions. In most
cases, solutions of nonlinear differential equations can be expressed
only by means of an infinite series.
Application of the Adomian decomposition method is based on the assumption
that the unknown solution is represented as an infinite sum:
\[
y(x) = \sum_{n\ge 0} u_n (x) .
\]
The crusual part of the method is decomposition of
the nonlinear term by an infinite series of the Adomian polynomials:
where λ is a grouping parameter. In computational practice, we
truncate the decomposition series after n = M for some finite M because the higher order solution components u_{n} for n > M do not contribute to the accuracy of calculations.
Mathematica code for evaluating Adomian polynomials for the one-variable Adomian polynomials:
AP1[f_, M_]:=
Module[{F}, Subscript[F, 0] = f[Sum[Subscript[u, k] * s^k, {k, 0, M}] ];
For [i = 0, i <'= M, i++ , A[i] = Collect[Expand[1/i! * (Subscript[ F, i]/. s -> 0)],
Derivative[_][f ][_]];
Subscript[F, i + 1] = D[Subscript[F, i], s]];
Table[A[i], {i, 0, M}]]
Another Mathematica code for evaluating Adomian polynomials for the one-variable Adomian polynomials:
Example: Move to chapter 4, application4
Consider the Lane--Emden type
equation with an exponential nonlinearity of the
solution with respect to the variable base
Example:
The system of partial differential equations, called KdV equations, cna be
reduced by appropriate substitution to the following ordinary differential
equation
\[
v' = k\,v''' - v\,v' .
\]
■
ADM for singular differential equations
Duan, J.-S., Rach, R., Baleanu, 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.
Rach, R., Wazwaz, A.-M., and Duan, J.-S., “A reliable modification of the
Adomian decomposition method for higher-order nonlinear differential equations”,
Kybernetes, 2013, Vol. 42, No 2, pp. 282--308, doi: 10.1108/03684921311310611
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