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

Abstract

In this study, the fractional reduced differential transform method (FRDTM) is employed to solve three-dimensional fourth-order time-fractional parabolic partial differential equations with variable coefficients. The fractional derivative used in this study is in the Caputo sense. A few important lemmas which are essential to solve the problems using the proposed method are proved. The novelty of this method is that it uses appropriate initial conditions and finds the solution to the problems without any discretization, linearization, perturbation, or any restrictive assumptions. Two numerical examples are considered in order to validate the efficiency and reliability of the method. Furthermore, the FRDTM solution when α = 1 is compared with other analytical methods available in the existing literature. Computational results are shown in tables and graphs. The obtained results revealed that the method is capable and simple to solve fractional partial differential equations. The software used for the calculations in this study is Mathematica 7.

Analysis of Age-Structured Mathematical Model of Malaria Transmission Dynamics via Classical and ABC Fractional Operators

“…As far as the researchers are aware, all these models employ integer-order derivatives in their differential equations. Fractional calculus, a branch of applied mathematics that extends integer calculus to noninteger orders, has found numerous applications in diverse fields such as engineering, control networks, physical systems, and mathematical modeling [15][16][17][18][19][20].…”

Analytical Solution of One-Dimensional Nonlinear Conformable Fractional Telegraph Equation by Reduced Differential Transform Method

“…These representations have offered good results in the modeling of real-world problems [8][9][10]. Recently, investigators have paid attention to studying the solution of partial differential equations of fractional order [11][12][13]. For instance, Singh [14] studied the new aspects of the blood alcohol model associated with the Hilfer fractional operator.…”

“…One of the greatest methods found by researchers is the reduced differential transform method (RDTM). What distinguishes this method is that it provides us with analytical approximations, which in many cases are exact solutions, in a rapidly convergent power series form with elegantly computed terms ( [12,35] and see the references therein).…”

“…Paper [40] presents the RDTM for finding the exact and approximate analytical solution for the conformable time-fractional Swift-Hohenberg (S-H) equations with and without dispersive terms. For extra information, see [12,35,[41][42][43] and the references therein.…”