# A new strategy for the approximate solution of fourth-order parabolic partial differential equations with fractional derivative

Muhammad Nadeem (School of Mathematics and Statistics, Qujing Normal University, Qujing, China)
Zitian Li (School of Mathematics and Statistics, Qujing Normal University, Qujing, China)

ISSN: 0961-5539

Article publication date: 2 November 2022

Issue publication date: 20 January 2023

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## Abstract

### Purpose

This study aims to purpose the idea of a new hybrid approach to examine the approximate solution of the fourth-order partial differential equations (PDEs) with time fractional derivative that governs the behaviour of a vibrating beam. The authors have also demonstrated the physical representations of the problem in different fractional order.

### Design/methodology/approach

Mohand transform is a new technique that the authors use to reduce the order of fractional problems, and then the homotopy perturbation method can be used to handle the further series solution in the form of convergence. The formulation of Mohand transform and the homotopy perturbation method is known as Mohand homotopy perturbation transform (MHPT). The fractional order in this paper is considered in the Caputo sense.

### Findings

The results are formulated in the shape of iterative series and predict the solution close to the exact solution. This successive iteration demonstrates the authenticity and reliability of this scheme.

### Research limitations/implications

This paper presents the significance of MHPT such that, firstly, Mohand transform is coupled with homotopy perturbation method and, secondly, the fractional order a is used to show the physical behaviour of the graphical solution.

### Practical implications

This study presents the consistency and authenticity of the graphical solution with the exact solutions.

### Social implications

This study demonstrates that Mohand transform is capable to handle the fractional order problem without any constraints and assumptions.

### Originality/value

A new integral transform has been introduced without any restriction of variables that produces the results in a series form and confirms the validity of the proposed algorithm by graphical illustrations.

## Citation

Nadeem, M. and Li, Z. (2023), "A new strategy for the approximate solution of fourth-order parabolic partial differential equations with fractional derivative", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 33 No. 3, pp. 1062-1075. https://doi.org/10.1108/HFF-08-2022-0499

## Publisher

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Emerald Publishing Limited