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

Rashid Jan1, Hassan Khan2,3, Poom Kumam4,5,*, 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 this Special Issue: Recent Advances in Fractional Calculus Applied to Complex Engineering Phenomena)

Computers, Materials & Continua 2021, 68(3), 3185-3201.


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.


Cite This Article

R. Jan, H. Khan, P. Kumam, F. Tchier, R. Shah et al., "The investigation of the fractional-view dynamics of helmholtz equations within caputo operator," Computers, Materials & Continua, vol. 68, no.3, pp. 3185–3201, 2021.


This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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