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

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.

Precision and efficiency of an interpolation approach to weakly singular integral equations

“…Furthermore, a class of novel fractional-dimensional reproducing kernel spaces based on Caputo fractional derivatives is discussed in Li et al (2022) to provide analytical and numerical solutions for the two-sided space-fractional super-diffusive model with variable coefficients, which is likewise difficult to solve numerically. Also, to examine the behaviour of the vibrating beam model governed by the fourth-order partial differential equations (PDEs) with a time-fractional derivative, the authors in Nadeem and Li (2022) examined the solution of the system numerically by using the new hybrid approach, the Mohand homotopy perturbation transform. Moreover, in article (He and Latifizadeh, 2020), the authors discussed the general numerical algorithm for nonlinear problems by the variational iteration method (VIM).…”

Numerical Analysis of Time-Fractional Porous Media and Heat Transfer Equations Using a Semi-Analytical Approach

“…Shah et al [20] utilized the Laplace decomposition transform method to compute series-type solutions under fuzzy fractional PDEs. Nadeem and Li [21] proposed a significant scheme based on the Mohand transform and the HPS to derive the solution of fourth-order parabolic PDEs with fractional derivatives. Merdan [22] used the fractional variational iteration method for finding the approximate analytical solutions of the nonlinear fractional Klein-Gordon equation with the Riemann-Liouville sense.…”