Abstract

The Adomian decomposition method together with some properties of nested integrals is used to provide a solution to a class of nonlinear ordinary differential equations and a coupled system.

1. Introduction

Most scientific problems and phenomena such as heat transfer occur nonlinearly. We know that only a limited number of these problems have a precise analytical solution [1–5]. In the 1980t’s, George Adomian (1923–1996) introduced a powerful method for solving nonlinear functional equations. His method is known as the Adomian decomposition method (ADM) [6]. This technique is based on the representation of a solution to a functional equation as series of functions. Each term of the series is obtained from a polynomial generated by a power series expansion of an analytic function. Although the abstract formulation of the Adomian method is very simple, the calculations of the polynomials and the verification of convergence of the function series in specific situations are usually a difficult task [7, 8].

We will see that if the nested integral properties are used properly in the Adomian decomposition method, the analytical solution to the initial value problem is easily obtained.

Nested integrals integrals which are evaluated several times on the same variable. In contrast, multiple integrals consist of a number of integrals evaluated with respect to different variables. Concretely, if is a continuous function defined on a (open) interval and , Also, (see [9]),

2. Solution Method

Consider the IVP where is a continuous function defined on an (open) interval , and .

In operator form, (2.1) becomes where . Then inverse of is, therefore, . Applying to both sides of 4 we find that where .

Adomiant’s technique consists in writing the solution of (1.2) as an infinite series and decomposing the nonlinear term as where each is an Adomian polynomial depending on , which is given by (see [6, 7]).

Substituting (2.4) and (2.5) into (2.3), we obtain This leads to the following recurrence scheme We then define the solution as The following algorithm will be used in order to calculate the Adomian polynomials (see [10])

Combining this with (2.8), one obtains where .

By using we find that Now, using property (1.1) in (2.13) yields Since One finally obtains In order to obtain , we again use (1.1) and (2.16) continuing in this fashion, we obtain The solution is given by By replacing (2.18) into (2.19), one obtains Or, in a more compact form, The latter equation can be written as Observe that in this case, Adomiant’s method yields an exact analytical solution. The analytical solution to this probleme can be obtained by performing the substitution , which leads to a Bernoulli differential equation whose solution is a straightforward exercise.

3. Examples

3.1. Example 1

Consider the nonlinear initial value problem In this case, and consequently .

Thus, the analytical solution is given by

3.2. Example 2

Consider the nonlinear initial value problem In this case, , and therefore . The analytical solution is where .

4. Application of the Method to Coupled Systems of ODE’s

Consider the coupled system Together with the initial conditions we shall obtain its solution by using the Dirichlet's integral formula (1.2) and the Adomian decomposition method.

Equation (4.1) in operator form takes the form where . The inverse of is . Applying to both sides of (4.3) and using (4.2), we find that In order to obtain , we apply the Adomian iterative scheme in (4.4).

Similarly, is obtained by applying the scheme to (4.5).

Replacing in (4.6), we find . In fact, Also, by replacing and (1.2) in (4.5), we obtain Applying (1.2) to the right hand side of the last equation, one finds that To obtain , (4.8) into (4.7), and (1.2), we have Continuing in this fashion, one arrives at the formula The solution is the given by Rearranging terms and writing as a single integral, we have This is easily recognized as where denotes convolution.

The analogous process gives Rearranging, we obtain Writting this as a single integral, we have And then, It is important to observe that the analytical solution of the IVP given by (4.1) and (4.2) is precisely In particular, let us consider the forced undamped system given by (Note that (4.21) is equivalent to the system formed by (4.1) and (4.2) with ).

There are two cases.

Case 1. . In this case, we obtain the solutions Let us observe that the solutions are bounded in this case

Case 2. . In this case, one obtain Observe that the solutions are unbounded in this case, and we have resonance.

5. Conclusion

The results obtained in this paper show that the Adomian decomposition method is a powerful technique for finding the theoretical solutions of nonlinear initial value problems and coupled systems if properties of nested integrals are used properly. If a solution in closed form is not found, the method always provides a convergent series which solve the problem, see [11].