Application Of Reduced Differential Transformation Method For Solving Fourth-order Parabolic Partial Differential Equations

Abstract

The purpose of this paper is to obtain the approximate solution of fourth-order parabolic partial differential equations by the reduced differential transform method (RDTM).This method provides the solution in the form of a convergent series with easily calculable terms. Comparing RDTM with some other methods in the literature shows present approach is very simple, effective, powerful and can be easily applied to other linear or nonlinear PDEs in science and engineering.Keywords: Reduced differential transform method (RDTM), differential transform method (DTM) fourthorder parabolic partial differential equations, initial value problems.

Three-Dimensional Fourth-Order Time-Fractional Parabolic Partial Differential Equations and Their Analytical Solution

“…In this study, we successfully employed the FRDTM to find the approximate and exact solutions of three-dimensional fourth-order time-fractional parabolic partial differential equations with variable coefficients taking the fractional derivative in Caputo sense. It is found that the results we obtained in Examples 1 and 2 when α � 1 are in excellent agreement with the exact solutions obtained by VIM [35], ADM [36], and RDTM [37]. Also, the illustrated examples demonstrate that the suggested technique can be observed as a fast, simple, and efficient tool for computing three-dimensional fourth-order time-fractional parabolic partial differential equations (PDEs).…”

“…As it can be seen from these tables, when the value of the fractional order α becomes close to 1, the obtained FRDTM solutions of Examples 1 and 2 are close to the exact solutions of its respective classical (nonfractional) equations obtained by different methods. In particular, when α � 1, the FRDTM solutions of the mentioned examples are in excellent agreement with the exact solutions of their corresponding classical (nonfractional) form, for details, see [35][36][37].…”

“…When α � 1, equation (20) reduces to the classical (or nonfractional) three-dimensional homogeneous parabolic partial differential equation [35][36][37].…”

“…e exact solution of the classical form of Example 1 is u(x, y, z, t) � (x + y + z − cos x − cos y − cos z)e − t as in [35][36][37].…”

Analytical Solution of Two-Dimensional Sine-Gordon Equation

“…Thirdly, this method can reduce the size of the calculations and can provide an analytic approximation, in many cases exact solutions, in rapidly convergent power series form with elegantly computed terms ( [37] and see the references therein). Moreover, the reduced differential transform method (RDTM) has an alternative approach of solving problems to overcome the demerit of discretization, linearization, or perturbations of well-known numerical and analytical methods such as Adomian decomposition, differential transform, homotopy perturbation, and variational iteration [37][38][39].…”