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ARTICLE

The Investigation of the Fractional-View Dynamics of Helmholtz Equations Within Caputo Operator

by Rashid Jan¹, Hassan Khan^2,3, Poom Kumam^4,5,*, Fairouz Tchier⁶, Rasool Shah², Haifa Bin Jebreen⁶

1 Department of Mathematics and Statistics, Bacha Khan University, Charsadda, 24420, Pakistan
2 Department of Mathematics, Abdul Wali Khan University, Mardan, 23200, Pakistan
3 Department of Mathematics, Near East University TRNC, Mersin, 10, Turkey
4 Center of Excellence in Theoretical and Computational Science (TACS-CoE) & Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, 10140, Bangkok, Thailand
5 Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
6 Mathematics Department, King Saud University, Riyadh, Saudi Arabia

* Corresponding Author: Poom Kumam. Email:

(This article belongs to the Special Issue: Recent Advances in Fractional Calculus Applied to Complex Engineering Phenomena)

Computers, Materials & Continua 2021, 68(3), 3185-3201. https://doi.org/10.32604/cmc.2021.015252

Received 12 November 2020; Accepted 24 December 2020; Issue published 06 May 2021

Abstract

It is eminent that partial differential equations are extensively meaningful in physics, mathematics and engineering. Natural phenomena are formulated with partial differential equations and are solved analytically or numerically to interrogate the system’s dynamical behavior. In the present research, mathematical modeling is extended and the modeling solutions Helmholtz equations are discussed in the fractional view of derivatives. First, the Helmholtz equations are presented in Caputo’s fractional derivative. Then Natural transformation, along with the decomposition method, is used to attain the series form solutions of the suggested problems. For justification of the proposed technique, it is applied to several numerical examples. The graphical representation of the solutions shows that the suggested technique is an accurate and effective technique with a high convergence rate than other methods. The less calculation and higher rate of convergence have confirmed the present technique’s reliability and applicability to solve partial differential equations and their systems in a fractional framework.

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