A Modification of Differential Transform Method for Solving Systems of Second Order Ordinary Differential Equations

The method of differential transform (DTM) is among the famous mathematical approaches for obtaining the differential equations solutions. This is due to its simplicity and efficient numerical performance. However, the major drawback of the DTM is obtaining a truncated series solution which is often a good approximation to the true solution of the equation in a specified region. In this study, a modification of DMT scheme known as MDTM is proposed for obtaining an accurate approximation of ordinary differential equations of second order. The scheme whose procedure is designed via DTM, the Laplace transforms and finally Pade approximation, gives a good approximate for the true solution of the equations in a large region. The proposed approach would be able to overcome the difficulty encountered using the classical DTM, and thus, can serve as an alternative approach for obtaining the solutions of these problems. Preliminary results are presented based on some examples which illustrate the strength and application of the defined scheme. Also, all the obtained results corresponded to exact solutions.


Introduction
Consider the order system of ODEs where for = 1,2,3, … , are nonlinear continuous functions of its argument [1]. The above system has been studied by numerous scientists and engineers due to its importance in vast applications including mathematical and physical models. Recently, much attention had been given to numerical solutions of these systems of ODEs. One of the efficient numerical methods for solving this system is the Differential transform method (DTM).
The DTM which constructs a polynomial analytical solution was first introduced by Zhou [3] as a new idea for solving differential equations (DEs). This form of polynomials is used as the approximate to exact solutions. The idea of DTM is based on the concept of Taylor's series [4][5][6], whose solution is usually obtained in series form. DTM is a reiterative approach for obtaining solutions of Taylor series DEs. Recently, the DTM has been extended to boundary value problems, difference equations, initial value problems.
However, the DTM has some drawbacks as stated in the abstracts [7]. To overcome this drawback, the study proposed an alternative scheme to improve the efficiency of DTM by modifying the series solution of systems of second order ODEs. The proposed scheme starts with the DMT obtained truncated series by applying the Laplace transform, followed by employing the Padé approximants to transform series into a meromorphic function, and lastly obtaining an analytic solution using the inverse Laplace transform. This obtained solution may be an improved approximation or periodic solution compared to the DTM truncated series solution. The proposed MDTM will be used without requiring any linearization or smallness assumptions [2].
The rest of the paper is arranged as follows. In this paper: Section 2 discusses brief overview and some fundamental results of DTM, Padé approximants and Laplace transform. The proposed scheme is applied to 472 A Modification of Differential Transform Method for Solving Systems of Second Order Ordinary Differential Equations various problems in order to illustrate its simplicity and effectiveness in Section 3. Lastly, the conclusion and discussion are in Section 4.

Preliminaries
This section presents some definitions of DTM and Padé approximants.

Differential Transform Method
Definition 2.1. [8] For a function ( ) that is analytical at 0 in the domain of interest, then (2.1) whose inverse differential transform is Let the DEs of ( ), ( ), and ℎ( ) be ( ), ( ), and ( ) respectively at 0 = 0. Then the known DTM operations are presented in Table 1.
A differential equation can be transformed into an equation of algebraic form in the -domain using differential transform and the Taylor series expansion finite-term to obtain ( ) as The series solution (2.3) possesses a rapid convergence rate but only in a small region. However, the convergence is often very slow in the wide region, and thus, yielding an inaccurate truncated result.
In the MDTS, we start by applying the Laplace transform to the obtained DMT truncated series, followed by employing the Padé approximants to transform the obtained series into a meromorphic function, and lastly obtaining an analytic solution using the inverse Laplace transform. This obtained solution may be an improved approximation or periodic solution compared to the DTM truncated series solution. For further reference on differential transform and DTM see [8][9][10][11][12][13]18].

Padé Approximation [14, 15]
The approximants of Padé refers to the ratio of two polynomials derived from Taylor series expansion coefficients function and ( ) and ( ) are known as the polynomials of degree at most and respectively. More so, are the formal power series determining the coefficients of the polynomials ( ) and ( ) by the equation. It is obvious the numerator and denominator of [ / ] remain unchanged even after multiplying by a constant, thus, the following normalization condition is imposed (0) = 1. (2.7) Lastly, ( ) and ( ) are required to have no common factors. Rewriting ( ) and ( ) coefficient as also, from (2.7) and (2.8), we can multiply (2.6) by ( ), thereby linearizing the coefficient equations.
To obtain the solutions of the above equations, we begin with all the unknown of the linear equations (2.9) ' . Once we obtain the values of the ' , then (2.10) will produce a formula for-all the defined unknown ' , and thus, concludes the solution.
If equation (2.9) and equation (2.10) are non-singular, then, the solution can be obtained directly as presented in equation (2.11) [15], which implied that (2.11) holds, but if the index of the lower sum exceeds the upper, then, it would be substituted by zero: From this step, the symbolic calculus software, MATLAB is often employed to obtain the Padé approximants diagonal matrix of orders including [2 /2], [4/4] or [6/6].

Remark
Note that typically the Padé approximant, obtained from a partial Taylor sum, is more accurate than the latter. However; the Padé, being a rational expression, has poles, which are not present in the original function. It is a simple algebraic task to expand the form of an [ , ] Padé in a Taylor series and compute the Padé coefficients by matching with the above [14].

Numerical Results
This section presents the solutions of systems of second order ODEs by reducing higher order DE's to system of first order DEs.