Preface
This section provides an illustration of application of the Adomian decomposition method (ADM for short) to second order differential equations. This method is one of the most effective methods for constructing analytic approximate solutions of nonlinear differential equations.
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Glossary
Adomian Decomposition Method
The Adomian decomposition method (ADM for short) and its modifications can be considered as an quantitative rather than merely qualitative 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 decomposes the solution into a series of components to be determined by recursion, and then decomposes the analytic nonlinearity into the series of the Adomian polynomials. The ADM is applicable in both nonlinear initial value problems and boundary value problems without nonphysical 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. It will be very naive to assume that the Adomian decomposition scheme can solve any nonlinear problem. Nevertheless, the ADM is the first candidate for a trial.
Nonlinear differential equation models have been extensively applied in the majority of applications. However, very few nonlinear problems have simple, closed-form solutions. In most cases, solutions of nonlinear differential equations can be expressed only by means of special functions or infinite series. Before we start exposition of the Adomian decomposition method, it makes sense to look at some simple examples that show how solutions to the differential equations are sensitive to numerical values.
Consider an ordinary differential equation written in operator form:
As a rule, the linear operator L_{0} is included in the nonlinear one N[y] so we will drop it. The Adomian procedure 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.
The Adomian decomposition method assumes that the unknown solution can be represent by an infinite sum:
Mathematica code for evaluating Adomian polynomials for the one-variable Adomian polynomials:
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}]]
Module[{c, n, k, j, der}, Table[c[n, k], {n, 1, M}, {k, 1, n}];
der = Table [ D[f [Subscript[u, 0]], {Subscript[u, 0], k}], {k, 1, M}];
A[0] = f [Subscript[u, 0]];
For[n = 1, n <= M, n++ , c[n, 1] = Subscript[u, n];
For[k = 2, k <= n, k++, c[n, k] = Expand[1/n * Sum[(j + 1) * Subscript[u, j + 1] * c[n-1-j, k-1],{j, 0, n-k}]]];
A[n] = Take[der, n].Table[c[n, k], {k, 1, n}]];
Table[A[n], {n, 0, M}] ]
Once we know how to determine Adonian's polynomials, we proceed according to the main streamline. First, we find the initial term by solving nonhomogeneous initial value problem for the linear differential equation:
We demonstrate the Adomian decomposition method in a series of examples.
Example: Consider the initial value problem for undamped Duffing equation:
Example: Consider the second-order nonlinear differential equation with the exponential input
Example: Consider the second-order nonlinear differential equation with the exponential input
We input in Mathematica notebook:
Example: The system of partial differential equations, called KdV equations, can be reduced by appropriate substitution to the following ordinary differential equation
- 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.
- J.-S. Duan, J.-S., Rach, R., and Wazwaz, A.-M., A new modified Adomian decomposition method for higher-order nonlinear dynamical systems, CMES: Computer Modeling in Engineering & Sciences, 2013, Vol. 94, No. 1, pp. 77--118.
- Elsaid, A., Adomian polynomials: a powerful tool for iterative methods of series solution of nonlinear equations, Journal of Applied Analysis and Computation, 2012, Vol. 2, No 4, pp. 381--394.
- 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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