Differential Transformation Method (DTM) for Approximate Solution of Ordinary Differential Equation (ODE)

This paper deals with large deflection problem of a cantilever beam with a constant section under the action of a transverse tip load. The differential transformation method (DTM) is used to solve the nonlinear differential equation governing the problem. An approach treats trigonometric nonlinearity is used in DTM. The results obtained from DTM are compared with those results obtained by the finite difference method and they agree well.

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Differential Transformation Method

Advances in Mechatronics and Mechanical Engineering - Numerical and Analytical Solutions for Solving Nonlinear Equations in Heat Transfer ◽

10.4018/978-1-5225-2713-8.ch006 ◽

2017 ◽

pp. 140-172

Keyword(s):

Heat Transfer ◽

Differential Equation ◽

Nonlinear Equation ◽

Nonlinear Differential Equation ◽

Transformation Method ◽

Differential Transformation Method ◽

Differential Transformation

In this chapter, a new linearization procedure based on Differential Transformation Method (DTM) will be presented. The procedure begins with solving nonlinear differential equation by DTM. The effectiveness of the procedure is verified using a heat transfer nonlinear equation. The simulation result shows the significance of the proposed technique.

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ON A NEW DIFFERENTIAL TRANSFORMATION METHOD FOR SOLVING NONLINEAR DIFFERENTIAL EQUATIONS

Asian-European Journal of Mathematics ◽

10.1142/s1793557113500575 ◽

2013 ◽

Vol 06 (04) ◽

pp. 1350057 ◽

Cited By ~ 1

Author(s):

Abdelhalim Ebaid

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Nonlinear Term ◽

Nonlinear Differential Equations ◽

Mathematical Structure ◽

Transformation Method ◽

Differential Transformation Method ◽

Differential Transformation ◽

Computational Budget ◽

Symbolic Formula

The main difficulty in solving nonlinear differential equations by the differential transformation method (DTM) is how to treat complex nonlinear terms. This method can be easily applied to simple nonlinearities, e.g. polynomials, however obstacles exist for treating complex nonlinearities. In the latter case, a technique has been recently proposed to overcome this difficulty, which is based on obtaining a differential equation satisfied by this nonlinear term and then applying the DTM to this obtained differential equation. Accordingly, if a differential equation has n-nonlinear terms, then this technique must be separately repeated for each nonlinear term, i.e. n-times, consequently a system of n-recursive relations is required. This significantly increases the computational budget. We instead propose a general symbolic formula to treat any analytic nonlinearity. The new formula can be easily applied when compared with the only other available technique. We also show that this formula has the same mathematical structure as the Adomian polynomials but with constants instead of variable components. Several nonlinear ordinary differential equations are solved to demonstrate the reliability and efficiency of the improved DTM method, which increases its applicability.

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On the Multistage Differential Transformation Method for Analyzing Damping Duffing Oscillator and Its Applications to Plasma Physics

Mathematics ◽

10.3390/math9040432 ◽

2021 ◽

Vol 9 (4) ◽

pp. 432

Author(s):

Noufe H. Aljahdaly ◽

S. A. El-Tantawy

Keyword(s):

Approximate Solution ◽

Plasma Physics ◽

Duffing Oscillator ◽

Transformation Method ◽

Duffing Equation ◽

Differential Transformation Method ◽

Differential Transformation ◽

Plasma Species ◽

Unmagnetized Plasma ◽

Residual Errors

The multistage differential transformation method (MSDTM) is used to find an approximate solution to the forced damping Duffing equation (FDDE). In this paper, we prove that the MSDTM can predict the solution in the long domain as compared to differential transformation method (DTM) and more accurately than the modified differential transformation method (MDTM). In addition, the maximum residual errors for DTM and its modification methods (MSDTM and MDTM) are estimated. As a real application to the obtained solution, we investigate the oscillations in a complex unmagnetized plasma. To do that, the fluid govern equations of plasma species is reduced to the modified Korteweg–de Vries–Burgers (mKdVB) equation. After that, by using a suitable transformation, the mKdVB equation is transformed into the forced damping Duffing equation.

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Study of Properties of Differential Transform Method for Solving the Linear Differential Equation

Asian Journal of Engineering and Applied Technology ◽

10.51983/ajeat-2019.8.2.1138 ◽

2019 ◽

Vol 8 (2) ◽

pp. 50-56

Author(s):

Nandita Das

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Linear Equations ◽

Transformation Method ◽

Differential Transformation Method ◽

Transform Method ◽

Differential Transformation ◽

Differential Transform ◽

Linear Differential ◽

Non Linear Equations

The differential transformation method (DTM) is an alternative procedure for obtaining an analytic Taylor series solution of differential linear and non-linear equations. However, the proofs of the properties of equation have been long ignored in the DTM literature. In this paper, we present an analytical solution for linear properties of differential equations by using the differential transformation method. This method has been discussed showing the proof of the equation which are presented to show the ability of the method for linear systems of differential equations. Most authors assume the knowledge of these properties, so they do not bother to prove the properties. The properties are therefore proved to serve as a reference for any work that would want to use the properties without proofs. This work argues that we can obtain the solution of differential equation through these proofs by using the DTM. The result also show that the technique introduced here is accurate and easy to apply.

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A multistage differential transformation method for approximate solution of Hantavirus infection model

Communications in Nonlinear Science and Numerical Simulation ◽

10.1016/j.cnsns.2011.05.023 ◽

2012 ◽

Vol 17 (1) ◽

pp. 1-8 ◽

Cited By ~ 26

Author(s):

Ahmet Gökdoğan ◽

Mehmet Merdan ◽

Ahmet Yildirim

Keyword(s):

Approximate Solution ◽

Transformation Method ◽

Differential Transformation Method ◽

Hantavirus Infection ◽

Infection Model ◽

Differential Transformation

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AN APPLICATION OF DIFFERENTIAL TRANSFORMATION TO STABILITY ANALYSIS OF HEAVY COLUMNS

International Journal of Structural Stability and Dynamics ◽

10.1142/s0219455406001988 ◽

2006 ◽

Vol 06 (03) ◽

pp. 317-332 ◽

Cited By ~ 19

Author(s):

Y. H. CHAI ◽

C. M. WANG

Keyword(s):

Differential Equation ◽

Transformation Method ◽

Differential Transformation Method ◽

Recursive Equation ◽

Transformation Technique ◽

Critical Buckling Load ◽

Differential Transformation ◽

Governing Differential Equation ◽

Support Conditions ◽

The Stability

This paper uses a recently developed technique, known as the differential transformation, to determine the critical buckling load of axially compressed heavy columns of various support conditions. In solving the problem, it is shown that the differential transformation technique converts the governing differential equation into an algebraic recursive equation, which must be solved together with the differential transformation of the boundary conditions. Although a fairly large number of terms are required for convergence of the solution, the differential transformation method is nonetheless efficient and fairly easy to implement. The method is also shown to be very accurate when compared with a known analytical solution. The stability of heavy columns is further examined using approximate formulae currently available in the literature. In this case, the differential transformation method offers a reference for assessing the accuracy of the approximate buckling formulae.

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