A LAPLACE DECOMPOSITION ALGORITHM APPLIED TO A CLASS OF NONLINEAR DIFFERENTIAL EQUATIONS

In this paper, a numerical Laplace transform algorithm which is based on the decomposition method is introduced for the approximate solution of a class of nonlinear differential equations. The technique is described and illustrated with some numerical examples. The results assert that this scheme is rapidly convergent and quite accurate by which it approximates the solution using only few terms of its iterative scheme.


Introduction
This paper presents a Laplace transform numerical scheme, based on the decomposition method, for solving nonlinear differential equations. The analysis will be adapted to the approximate solution of a class of nonlinear second-order initial-value problems, though the algorithm is well suited for a wide range of nonlinear problems. The numerical technique basically illustrates how the Laplace transform may be used to approximate the solution of the nonlinear differential equation by manipulating the decomposition method which was first introduced by Adomian [1,2]. The underlying idea of the technique is to assume an infinite solution of the form u = ∞ n=0 u n , then apply Laplace transformation to the differential equation. The nonlinear term is then decomposed in terms of Adomian polynomials and an iterative algorithm is constructed for the determination of the u n s in a recursive manner. The method is implemented for three numerical examples and the numerical results show that the scheme approximates the exact solution with a high degree of accuracy using only few terms of the iterative scheme. The main thrust of this technique is that the solution which is expressed as an infinite series converges fast to exact solutions.
The balance in this paper is as follows. In Section 2, the Laplace transform decomposition method will be presented as it applies to a class of secondorder nonlinear equations. In Section 3, the algorithm is implemented for three numerical examples.

Numerical Laplace transform method
In this paper, a Laplace transform decomposition algorithm is implemented for the solution of the following class of second-order nonlinear initial-value problems Here f(y) is a nonlinear operator and a(x) and b(x) are known functions in the underlying function space. The technique consists first of applying Laplace transformation (denoted throughout this paper by L) to both sides of (2.1), hence Applying the formulas on Laplace transform, we obtain Using the initial conditions (2.2), we have The Laplace transform decomposition technique consists next of representing the solution as an infinite series, namely, where the terms y n are to be recursively computed. Also the nonlinear operator f(y) is decomposed as follows: where A n = A n (y 0 ,y 1 ,y 2 , ..., y n ) are the so-called Adomian polynomials. The first few polynomials are given by (2.9) Substituting (2.7) and (2.8) into (2.6) results (2.10) Using the linearity of Laplace transform it follows that (2.11) Matching both sides of (2.11) yields the following iterative algorithm: (2.14) In general, Applying the inverse Laplace transform to (2.12) we get Substituting this value of y 0 into (2.13) gives Evaluating the Laplace transform of the quantities on the right-hand side of (2.17) then applying the inverse Laplace transform, we obtain the value of y 1 . The other terms y 2 ,y 3 ,... can be obtained recursively in a similar fashion using (2.15).

Numerical examples
The Laplace transform decomposition algorithm, described in Section 2, is applied to some special cases of the class of nonlinear initial-value problems given in (2.1) and (2.2).
Example 3.1. Consider the nonlinear problem whose closed form solution is Taking Laplace transform of both sides of (3.1) gives The initial conditions (3.2) imply Following the technique, if we assume an infinite series solution of the form (2.7) we obtain where the nonlinear operator f(y) = y 3 is decomposed as in (2.8) in terms of the Adomian polynomials. From (2.9) the first few Adomian polynomials Suheil A. Khuri 145 for f(y) = y 3 are given by Upon using the linearity of Laplace transform then matching both sides of (3.7), results in the iterative scheme In general, Operating with Laplace inverse on both sides of (3.9) gives Substituting this value of y 0 and that of A 0 = y 3 0 given in (3.8) into (3.10), we get The inverse Laplace transform applied to (3.15) yields Substituting (3.16) into (3.11) and using the value of A 1 given in (3.8) implies (3.17) Simplifying the right-hand side of (3.17) then applying the inverse Laplace transform, we obtain x 4 + 7 6 x 5 + 9 10 x 6 + 38 105 x 7 + 3 40 x 8 + 1 120 x 9 . (3.18) Higher iterates can be easily obtained using the computer algebra system Maple. For example, (3.20) Therefore, the approximate solution is x 16 + 211 3536000 x 17 + ··· .
(3.21) (3.22) In Table 3.2 we calculate the absolute and relative errors using this [5,5] Padé approximant of the infinite series solution obtained by the Laplace decomposition algorithm. In both cases the error is less than 0.75%. Clearly for large values of x, calculating the errors using the Padé approximant instead of the approximate infinite solution will lead to a drastic improvement in the degree of accuracy. The infinite series solution does not provide a good approximation for substantial values of x, however replacing the partial sum of the infinite series solution with its Padé approximant yields a very accurate rational solution.

(3.29)
Following the Laplace transform decomposition method, if we match both sides of (3.27) we obtain the iterative scheme and the general iterative step is (3.33) The inverse Laplace transform applied to (3.30) results (3.35) Consequently, (3.36) Using this value of y 1 into (3.32) yields Hence, (3.39) The following higher iterates are obtained using Maple: (3.41) The infinite series solution becomes, upon using six iterations,  (3.43) Table 3.3 gives the absolute and relative errors of the infinite series approximation using six iterations of the Laplace transform decomposition technique. The error is less than 0.025%. As in Example 3.1, it was noticed that for large values of x, replacing the infinite series solution (3.42) with its Padé approximant (3.43) will improve the error. Example 3.3. Consider the following nonlinear problem: The exact solution is y = 2 e 8x e 8x + 15 where the nonlinear operator f(y) = y 3 is decomposed as in (2.8) in terms of the Adomian polynomials, which for this case the first few are given in (3.8). Matching both sides of (3.49), the components of y can be defined Suheil A. Khuri 151 as follows: and the general term is (3.53) The terms y n can be obtained in a recursive manner. Taking the inverse Laplace transform of (3.50) gives Substituting this value of y 0 into (3.51), and using that A 0 = y 3 0 from (3.8), we obtain (3.55) It follows that Consequently y 2 = 3.046875x 2 . (3.58) The next higher iterates are obtained using Maple, The series solution is therefore The [4,4] Padé approximant of this approximate solution is y 0.5 + 2.940330021x + 9.262070767x 2 + 13.01811008x 3 + 9.597056446x 4 1 + 2.130660041x + 4.440416382x 2 − 6.684988403x 3 + 19.98685852x 4 .
(3.61)   Simplifying this equation and using the initial conditions (3.63) yields Assuming an infinite series solution of the form (2.7) we get where the nonlinear operator f(y) = y 3 is decomposed in terms of the Adomian polynomials which for this case are given in (3.8). Upon using Suheil A. Khuri 153 the linearity of Laplace transform then matching both sides of (3.67), results in the iterative algorithm In general, Consider the case where α = β = k = 1. Operating with Laplace inverse on both sides of (3.68) gives (3.72) Substituting this value of y 0 and that of A 0 = y 3 0 given in (3.8) into (3.69), we obtain (3.73) so L y 1 = 8/s − 12/(s + 1)+6/(s + 2)−1/(3 + s) s 2 + s . (3.74) The inverse Laplace transform applied to (3.74) yields (3.75) Substituting the value of A 1 given in (3.8) into (3.70) implies Substituting the values of y 0 and y 1 given in (3.72) and (3.75) into (3.76) then applying the inverse Laplace transform, we obtain (3.77) (3.79) Suheil A. Khuri 155 Table 3.5 shows the absolute and relative errors that result from comparing the approximate solution obtained from the Laplace transform decomposition algorithm using four iterations, and the numerical solution of the damped Duffing's equation evaluated using Maple solve commands. The error is less than 0.001%. In all the previous four examples, it was observed that increasing the number of iterates will improve the accuracy of the solution.