The Solutions of Legendre’s and Chebyshev’s Differential Equations by Using the Differential Transform Method (original) (raw)
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A Cumulative Study on Differential Transform Method
International Journal of Mathematical, Engineering and Management Sciences, 2019
Many real-world phenomena when modelled as a differential equation don't generally have exact solutions, so their numerical or analytic solutions are sought after. Differential transform method (DTM) is one of the analytical methods that gives the solution in the form of a power series. In this paper, a cumulative study is done on DTM and its evolution as an effective method to solve the gamut of differential equations.
n this paper, we will compare the Differential Transform Method (DTM) and Taylor Series Method (TSM) applied to the solution of linear and nonlinear ordinary differential equations. The comparison shows that the Differential Transform Method is reliable, efficient and easy to use from computational point of view. Although the both of methods provide the solution in an infinite series, the Differential Transform Method provides a fast convergent series of easily computable components and eliminates heavy computational work needed by Taylor Series Method.
Mathematics and Statistics, 2020
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 Padé 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.
Solution of differential–difference equations by using differential transform method
Applied Mathematics and Computation, 2006
In this work, we successfully extended differential transform method (DTM), by presenting and proving new theorems, to the solution of differential-difference equations (DDEs). Theorems are presented in the most general form to cover a wide range of DDEs, being linear or nonlinear and constant or variable coefficient. In order to show the power and the robustness of the method and to illustrate the pertinent features of related theorems, examples are presented.
Journal of Mathematics
In this study, we develop the differential transform method in a new scheme to solve systems of first-order differential equations. The differential transform method is a procedure to obtain the coefficients of the Taylor series of the solution of differential and integral equations. So, one can obtain the Taylor series of the solution of an arbitrary order, and hence, the solution of the given equation can be obtained with required accuracy. Here, we first give some basic definitions and properties of the differential transform method, and then, we prove some theorems for solving the linear systems of first order. Then, these theorems of our system are converted to a system of linear algebraic equations whose unknowns are the coefficients of the Taylor series of the solution. Finally, we give some examples to show the accuracy and efficiency of the presented method.
In this work, we studied that Power Series Method is the standard basic method for solving linear differential equations with variable coefficients. The solutions usually take the form of power series; this explains the name Power series method. We review some special second order ordinary differential equations. Power series Method is described at ordinary points as well as at singular points (which can be removed called Frobenius Method) of differential equations. We present a few examples on this method by solving special second order ordinary differential equations.
A Review: Differential Transform Method for Semi-Analytical Solution of Differential Equations
International Journal of Applied Mechanics and Engineering, 2020
In this article, the semi-analytical method known as the Differential Transform Method (DTM) for solving different types of differential equations is reviewed. First, basic definitions and formulas of DTM and Differential Transform-Padé approximation (DTM-Padé), which are used to increase the convergence and accuracy of DTM approximations, are discussed. Then both techniques of DTM and DTM-Padé, which have been successfully applied to partial differential equations, as well as the application of these methods in fluid mechanic and heat transfer are presented. In addition, the extension of DTM for integral differential equations and the fuzzy differential transformation method (FDTM) for fuzzy problems are discussed.
SOLVING DIFFERENTIAL EQUATIONS USING ADOMIAN DECOMPOSITION METHOD AND DIFFERENTIAL TRANSFORM METHOD
Pushpa Publishing House, 2018
Many problems in science and engineering fields can be described by differential equations. In the early 1980's, an American applied mathematician George Adomian developed a powerful decomposition methodology for practical solution of differential equations known today as the Adomian decomposition method (ADM). The ADM is a powerful method which provides an efficient means for the analytical and numerical solution of differential equations which model realworld physical problems. The differential transform method (DTM) was first proposed by Zhou in 1986. The DTM is used to find coefficients of the Taylor series of the function by solving the induced recursive equation from the given differential equation. Recently there T. P. Ungani and E. Matabane 324 has been a big debate among researchers on which method is the best method to solve nonlinear differential equations. The DTM is clearly documented and well understood for solving ordinary differential equations. In this paper, we apply the ADM and clearly document how the DTM can be used to solve both ordinary differential equations (ODE's) and partial differential equations (PDE's).
2018
Integral transform method is most useful technique for solving differential equation of Mathematics. The natural transform is derived from the Fourier Integral. In this research paper, natural transform converges to Laplace and Sumudu transform. For inverse natural transform, we have Bromwich contour integral and Heaviside’s expansion formula. Here, we are using natural decomposition method (NDM) to find the exact solution of different types of non-linear ordinary differential equation which is based on natural transform method (NTM) and Adomian Decomposition Method (ADM).