Analytical Aspect of Fourth-Order Parabolic Partial Differential Equations with Variable Coefficients

Najeeb Khan,Muhammad Afzal,Asmat Ara,Azam Khan

doi:10.3390/mca15030481

Abstract

In this work, the homotopy analysis method (HAM) is applied to solve the fourth-order parabolic partial differential equations. This equation practically arises in the transverse vibration problems. The proposed iterative scheme finds the solution without any discritization, linearization, or restrictive assumptions. Some applications are given to verify the reliability and efficiency of the method. The convergence control parameter h in the HAM solutions has provided a convenient way of controlling the convergence region of series solutions. It is also shown that the solutions that are obtained by Adomian decomposition method (ADM) and variational iteration method (VIM) are special cases of the solutions obtained by HAM.

Highlights

Liao [1] employed the basic idea of the homotopy in topology to propose method for nonlinear problems, namely, homotopy analysis method (HAM) [2,3]
The HAM always provides us with a family of solution expressions in the auxiliary parameter; the convergence region and rate of each solution might be determined conveniently by the auxiliary parameter
The computations associated with the examples in this work were performed by using MATHEMATICA

Summary

INTRODUCTION

Liao [1] employed the basic idea of the homotopy in topology to propose method for nonlinear problems, namely, homotopy analysis method (HAM) [2,3] This method has many advantages over the classical methods; mainly, it is independent of any small or large quantities. The fourth-order parabolic partial differential equations with variable coefficients will be approached analytically. Wazwaz [4,5] and Biazar et al [7] studied fourth-order parabolic partial differential equations with variable coefficients by ADM and VIM. Differentiating (1) m times with respect to the embedding parameter p and setting p=0 and dividing by m!, we will have the so-called mth order deformation equation in the form: L[um (Ω) − χm um−1 (Ω)] = h [r. For the sake of comparison, we take the same examples as used in [4,5,7]

EXEMPLIFICATION

FINAL REMARKS