scholarly journals The Fornberg-Whitham Equation Solved by the Differential Transform Method

2020 ◽  
pp. 85-95
Author(s):  
Helena Nayar ◽  
Patrick Azere Phiri
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Differential Transform Method ◽  
Time Dependent ◽  
Partial Differential ◽  
Transform Method ◽  
Differential Transform ◽  
Whitham Equation ◽  
Temporal Variable

The Differential Transform Method is a powerful analytical method that can solve nonlinear partial differential equations. Yet, the method cannot be used to solve time-dependent partial differential equations that involve more than one partial derivative with respect to the temporal variable t when they are of the same order, as in the case of the Fornberg-Whitham type equations. In this paper, a new theorem is devised to overcome the aforementioned problem ofthe method, and it has been successfully applied to solve the Fornberg-Whitham equation. The other equations belonging to this group of equations, such as the Camassa-Holm equation and the Degasperi-Procesi equation, may also be solved by this approach.

scholarly journals A fuzzy solution of nonlinear partial differential equations

2021 ◽  
Vol 5 (1) ◽  
pp. 51-63
Author(s):  
Mawia Osman ◽  
◽  
Zengtai Gong ◽  
Altyeb Mohammed Mustafa ◽  
◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Infinite Series ◽  
Nonlinear Partial Differential Equations ◽  
Differential Transform Method ◽  
Series Expansions ◽  
Partial Differential ◽  
Transform Method ◽  
Differential Transform ◽  
Fuzzy Solution

In this paper, the reduced differential transform method (RDTM) is applied to solve fuzzy nonlinear partial differential equations (PDEs). The solutions are considered as infinite series expansions which converge rapidly to the solutions. Some examples are solved to illustrate the proposed method.

scholarly journals Nonlinear Klein-Gordon and Schrödinger Equations by the Projected Differential Transform Method

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Younghae Do ◽  
Bongsoo Jang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Taylor Series ◽  
Nonlinear Partial Differential Equations ◽  
Differential Transform Method ◽  
Schrödinger Equations ◽  
Partial Differential ◽  
Transform Method ◽  
Differential Transform ◽  
Klein Gordon

The differential transform method (DTM) is based on the Taylor series for all variables, but it differs from the traditional Taylor series in calculating coefficients. Even if the DTM is an effective numerical method for solving many nonlinear partial differential equations, there are also some difficulties due to the complex nonlinearity. To overcome difficulties arising in DTM, we present the new modified version of DTM, namely, the projected differential transform method (PDTM), for solving nonlinear partial differential equations. The proposed method is applied to solve the various nonlinear Klein-Gordon and Schrödinger equations. Numerical approximations performed by the PDTM are presented and compared with the results obtained by other numerical methods. The results reveal that PDTM is a simple and effective numerical algorithm.

scholarly journals A Numerical Computation of a System of Linear and Nonlinear Time Dependent Partial Differential Equations Using Reduced Differential Transform Method

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
Brajesh Kumar Singh ◽  
Mahendra
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Differential Transform Method ◽  
Time Dependent ◽  
Test Problems ◽  
Partial Differential ◽  
Transform Method ◽  
Differential Transform ◽  
Linear And Nonlinear ◽  
Reduced Differential Transform Method

This paper deals with an analytical solution of an initial value system of time dependent linear and nonlinear partial differential equations by implementing reduced differential transform (RDT) method. The effectiveness and the convergence of RDT method are tested by means of five test problems, which indicates the validity and great potential of the reduced differential transform method for solving system of partial differential equations.

THE MODIFIED DIFFERENTIAL TRANSFORM METHOD FOR SOLVING OFF-CENTERED STAGNATION FLOW TOWARD A ROTATING DISC

2010 ◽  
Vol 07 (04) ◽  
pp. 655-670 ◽  
Author(s):  
ESMAEEL ERFANI ◽  
MOHAMMAD MEHDI RASHIDI ◽  
AMIR BASIRI PARSA
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Differential Transform Method ◽  
Stagnation Flow ◽  
Rotating Disc ◽  
Partial Differential ◽  
Transform Method ◽  
Differential Transform

The similarity solution for the steady stagnation flow toward an off-centered rotating disc gives a system of nonlinear partial differential equations. These nonlinear differential equations are analytically solved by applying a newly developed method called DTM–Padé technique (the combination of the differential transform method (DTM) and the Padé approximation). This technique is extended to give solutions for nonlinear differential equations with boundary conditions at infinity. Graphical results are presented to investigate influence of the rotation ratio α on the radial velocity, azimuthal velocity, and the induced velocity. In order to show the effectiveness of the DTM–Padé technique, the results obtained from the DTM–Padé technique are compared with available solutions obtained using shooting method to generate the numerical solution. The obtained results demonstrate the reliability of the algorithm and the DTM–Padé technique is an attractive method in solving the systems of nonlinear partial differential equations.

Analytical Approach to Time-Fractional Partial Differential Equations in Fluid Mechanics

2011 ◽  
Vol 347-353 ◽  
pp. 463-466
Author(s):  
Xue Hui Chen ◽  
Liang Wei ◽  
Lian Cun Zheng ◽  
Xin Xin Zhang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fluid Mechanics ◽  
Approximate Solutions ◽  
Differential Transform Method ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Transform Method ◽  
Differential Transform ◽  
Generalized Differential

The generalized differential transform method is implemented for solving time-fractional partial differential equations in fluid mechanics. This method is based on the two-dimensional differential transform method (DTM) and generalized Taylor’s formula. Results obtained by using the scheme presented here agree well with the numerical results presented elsewhere. The results reveal the method is feasible and convenient for handling approximate solutions of time-fractional partial differential equations.

scholarly journals Using the Differential Transform Method to Solve Non-Linear Partial Differential Equations

2020 ◽  
pp. 34-43
Author(s):  
Fadwa A. M. Madi ◽  
Fawzi Abdelwahid
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Variable Coefficients ◽  
Differential Transform Method ◽  
Partial Differential ◽  
Transform Method ◽  
Linear Partial Differential Equations ◽  
Differential Transform ◽  
Non Linear

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.

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