A Numerical Method for Partial Differential Algebraic Equations Based on Differential Transform Method (original) (raw)
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International journal of scientific research and management, 2022
This paper presents a study using a novel linearization technique based on the Differential transformation method (DTM) to seek analytical solutions if it exists and approximate solutions where closed form solutions are not available. The effectiveness and accuracy of this procedure is verified by solving six problems comprising both initial and boundary value problems by a combination of DTM and Laplace transform method. The resulting solution is then treated with Padé approximation to obtain a better approximation that converges to the exact solution. Simulated results of the study reveal the proposed technique is reliable, accurate and computationally convenient even with few iterations. The result obtained is in good agreement with existing literature.
Using the Differential Transform Method to Solve Non-Linear Partial Differential Equations
Journal of Advances in Mathematics and Computer Science
In this work, we reviewed the two-dimensional differential transform, and introduced the differential transform method (DTM). As an application, we used this technique to find approximate and exact solutions of selected non-linear partial differential equations, with constant or variable coefficients and compared our results with the exact solutions. This shows that the introduced method is very effective, simple to apply to linear and nonlinear problems and it reduces the size of computational work comparing with other methods.
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.
Application Of The Differential Transform Method To Differential-Algebraic Equations With Index 2
2008
In this paper, we have used the differential transform method to solve differential-algebraic equations with index 2. Two kind of differentialalgebraic equations have been considered and solved numericaly, then we compared numerical and analytical solution of the given equations. Examples were presented to show the ability of the method for differentialalgebraic equations. We use MAPLE computer algebra system to solve given problems [4].
2012
This paper proposes another use of the Differential transform method (DTM) in obtaining approximate solutions to nonlinear partial differential equations (PDEs). The idea here is that a PDE can be converted to an ordinary differential equation (ODE) upon using a wave variable, then applying the DTM to the resulting ODE. Three equations, namely, Benjamin-Bona-Mahony (BBM), Cahn-Hilliard equation and Gardner equation are considered in this study. The proposed method reduces the size of the numerical computations and use less rules than the usual DTM method used for multi-dimensional PDEs. The results show that this new approach gives very accurate solutions.
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.
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.
The combined Laplace transform-differential transform method for solving linear non-homogeneous PDEs
2012
In this work, a combined form of the Laplace transform method (LTM) with the differential transform method (DTM) will be used to solve non-homogeneous linear partial differential equations (PDEs). The combined method is capable of handling non-homogeneous linear partial differential equations with variable coefficient. The aim of using the Laplace transform is to overcome the deficiency that is caused by unsatisfied boundary conditions in using differential transform method. Illustrative examples will be examined to support the proposed analysis.