A generalized differential transform method for linear partial differential equations of fractional order

In this letter we develop a new generalization of the two-dimensional differential transform method that will extend the application of the method to linear partial differential equations with space- and time-fractional derivatives. The new generalization is based on the two-dimensional differential transform method, generalized Taylor’s formula and Caputo fractional derivative. Several illustrative examples are given to demonstrate the effectiveness of the present method. The results reveal that the technique introduced here is very effective and convenient for solving linear partial differential equations of fractional order

Reduced Differential Transform Method for (2+1) Dimensional type of the Zakharov-Kuznetsov ZK(n,n) Equations

In this paper, reduced differential transform method (RDTM) is employed to approximate the solutions of (2+1) dimensional type of the Zakharov-Kuznetsov partial differential equations. We apply these method to two examples. Thus, we have obtained numerical solution partial differential equations of Zakharov-Kuznetsov. These examples are prepared to show the efficiency and simplicity of the method

Status of the differential transformation method

Further to a recent controversy on whether the differential transformation method (DTM) for solving a differential equation is purely and solely the traditional Taylor series method, it is emphasized that the DTM is currently used, often only, as a technique for (analytically) calculating the power series of the solution (in terms of the initial value parameters). Sometimes, a piecewise analytic continuation process is implemented either in a numerical routine (e.g., within a shooting method) or in a semi-analytical procedure (e.g., to solve a boundary value problem). Emphasized also is the fact that, at the time of its invention, the currently-used basic ingredients of the DTM (that transform a differential equation into a difference equation of same order that is iteratively solvable) were already known for a long time by the "traditional"-Taylor-method users (notably in the elaboration of software packages --numerical routines-- for automatically solving ordinary differential equations). At now, the defenders of the DTM still ignore the, though much better developed, studies of the "traditional"-Taylor-method users who, in turn, seem to ignore similarly the existence of the DTM. The DTM has been given an apparent strong formalization (set on the same footing as the Fourier, Laplace or Mellin transformations). Though often used trivially, it is easily attainable and easily adaptable to different kinds of differentiation procedures. That has made it very attractive. Hence applications to various problems of the Taylor method, and more generally of the power series method (including noninteger powers) has been sketched. It seems that its potential has not been exploited as it could be. After a discussion on the reasons of the "misunderstandings" which have caused the controversy, the preceding topics are concretely illustrated.Comment: To appear in Applied Mathematics and Computation, 29 pages, references and further considerations adde

Analytical Solutions of Heat Conduction Problems

The following thesis deals with the analytical methods which are in vogue for solving problems to the area of heat conduction. There have been discussed two methods, an old method known as the HEAT BALANCE INTEGRAL METHOD, and a relatively newer method christened as the DIFFERENTIAL TRANSFORMATION METHOD. The latter is dealt with first, as it is easier of the two. Dealing involves the basic idea of the method used, followed by the general theorems adopted. Two problems follow, illustrating the ease of use of this method, along with a comparison with the solutions of the problem using the numerical methods. The former method, on the other hand, is more of an assumptive method, where one has to guess a temperature profile for proceeding. This is, nonetheless, a very accurate method, albeit a long one. Similar comparisons have been made for this method, like the ones made for the DT method. The reader may use either method with ease, as it was for the simplification of the problem that these methods were developed

The non-standard finite difference scheme for linear fractional PDEs in fluid mechanics

AbstractA non-standard finite difference scheme is developed to solve the linear partial differential equations with time- and space-fractional derivatives. The Grunwald–Letnikov method is used to approximate the fractional derivatives. Numerical illustrations that include the linear inhomogeneous time-fractional equation, linear space-fractional telegraph equation, linear inhomogeneous fractional Burgers equation and the fractional wave equation are investigated to show the pertinent features of the technique. Numerical results are presented graphically and reveal that the non-standard finite difference scheme is very effective and convenient for solving linear partial differential equations of fractional order

Comparison between numerical methods for generalized Zakharov system

In this paper, two numerical methods has been applied to numerically solve the generalized Zakharov system (GZS). The spectral collocation method, which based on two dimensional Legendre polynomials (LCM) and the well-known differential transform method (DTM). Both of the proposed methods have high accuracy and have been successfully compared with Adomian decomposition method.Publisher's Versio

A Hybrid Method with RDTM for Solving the Biological Population Model

In this paper, we establish an analytical solution to the non-linear biological population model using a hybrid method that combines a reduced differential transform method with a resummation method based on Yang transform and a Padé approximant. The proposed method significantly improves the approximate solution series and broadens the convergence field, It is also dependent on a few straightforward steps, and does not depend on a perturbation parameter or produce secular terms. Three examples are given to test the effectiveness, accuracy, and performance of the suggested method. The results and graph demonstrate that the suggested method is successful and more accurate than other methods. In addition, PYRDTM is a useful tool with great potential for solving nonlinear BPM. Keywords: Biological population model, Yang transform, RDTM, Padé approximation, accuracy. DOI: 10.7176/MTM/12-2-01 Publication date:September 30th 2022

Comparison Differential Transformation Technique with Adomian Decomposition Method for Dispersive Long-wave Equations in (2+1)-Dimensions

In this paper, we will introduce two methods to obtain the numerical solutions for the system of dispersive long-wave equations (DLWE) in (2+1)-dimensions. The first method is the differential transformation method (DTM) and the second method is Adomian decomposition method (ADM). Moreover, we will make comparison between the solutions obtained by the two methods. Consequently, the results of our system tell us the two methods can be alternative ways for solution of the linear and nonlinear higher-order initial value problems

Numerical Simulation of One-Dimensional Shallow Water Equations

In this study, a relatively new semi-analytic technique, the reduced differential transform method is employed to obtain high accurate solutions of the famous coupled partial differential equations with physical interests namely the variable-depth shallow water equations with source term. The solutions are calculated in the form of a convergent power series with easily computable components. The Reduced differential transform method is easy to apply, reduces the size of computations, and produces an approximate solution without any discretization or perturbation. The results show the accuracy and efficiency of the reduced differential transform method in comparison to other existing methods