Numerical simulation of fractional-order two-dimensional Helmholtz equations

Naveed Iqbal; Muhammad Tajammal Chughtai; Nehad Ali Shah; Naveed Iqbal; Muhammad Tajammal Chughtai; Nehad Ali Shah

doi:10.3934/math.2023667

AIMS Mathematics

2023, Volume 8, Issue 6: 13205-13218. doi: 10.3934/math.2023667

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Numerical simulation of fractional-order two-dimensional Helmholtz equations

1.
Department of Mathematics, College of Science, University of Ha'il, Ha'il 2440, Saudi Arabia
2.
Department of Electrical Engineering, College of Engineering, University of Ha'il, Ha'il 55427, Saudi Arabia
3.
Department of Mechanical Engineering, Sejong University, Seoul 05006, Korea

Received: 08 December 2022 Revised: 06 March 2023 Accepted: 20 March 2023 Published: 03 April 2023
MSC : 33B15, 34A34, 35A20, 35A22, 44A10

In this paper, we investigate the exact solutions of several fractional-order Helmholtz equations using the homotopy perturbation transform method. We specify sufficient requirements for its convergence and provide error estimations. The homotopy perturbation transform method yields a quickly converging succession of solutions. Solutions for various fractional space derivatives are compared to present approaches and explained using figures. Appropriate parameter selection produces approximations identical to the exact answer. Test examples are provided to demonstrate the proposed approach's precision and competence. The results demonstrate that our system is appealing, user-friendly, dependable, and highly effective.
- fractional Helmholtz equations,
- Caputo operator,
- homotopy perturbation transform method,
- analytical solution
Citation: Naveed Iqbal, Muhammad Tajammal Chughtai, Nehad Ali Shah. Numerical simulation of fractional-order two-dimensional Helmholtz equations[J]. AIMS Mathematics, 2023, 8(6): 13205-13218. doi: 10.3934/math.2023667

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Abstract

In this paper, we investigate the exact solutions of several fractional-order Helmholtz equations using the homotopy perturbation transform method. We specify sufficient requirements for its convergence and provide error estimations. The homotopy perturbation transform method yields a quickly converging succession of solutions. Solutions for various fractional space derivatives are compared to present approaches and explained using figures. Appropriate parameter selection produces approximations identical to the exact answer. Test examples are provided to demonstrate the proposed approach's precision and competence. The results demonstrate that our system is appealing, user-friendly, dependable, and highly effective.

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