Abstract
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.
Highlights
Consider the n order system of ODEs⎧φ1 x, y1(x), y1′(x) ... , y1(n)(x), y2(x), y2′(x) ... , y2(n)(x), yN(x), yN′(x) ... , yN(n)(x) = 0 ⎪φ2 x, y1(x), y1′(x) ... , y1(n)(x), y2(x), y2′(x) ... , y2(n)(x), yN(x), yN′(x) ... , yN(n)(x) = 0 ⎨ ⋮⎩⎪φN x, y1(x), y1′(x) ... , y1(n)(x), y2(x), y2′(x) ... , y2(n)(x), yN(x), yN′(x) ... , yN(n)(x) = 0 where φi for i = 1,2,3, ... , N are nonlinear continuous functions of its argument [1]
The Differential transform method (DTM) which constructs a polynomial analytical solution was first introduced by Zhou [3] as a new idea for solving differential equations (DEs)
The DTM has been extended to boundary value problems, difference equations, initial value problems
Summary
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. 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.
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