Analysis of local fractional coupled Helmholtz and coupled Burgers' equations in fractal media

Ved Prakash Dubey; Jagdev Singh; Ahmed M. Alshehri; Sarvesh Dubey; Devendra Kumar; Ved Prakash Dubey; Jagdev Singh; Ahmed M. Alshehri; Sarvesh Dubey; Devendra Kumar

doi:10.3934/math.2022450

AIMS Mathematics

2022, Volume 7, Issue 5: 8080-8111. doi: 10.3934/math.2022450

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Research article

Analysis of local fractional coupled Helmholtz and coupled Burgers' equations in fractal media

1.
Faculty of Mathematical and Statistical Sciences, Shri Ramswaroop Memorial University, Barabanki-225003, Uttar Pradesh, India
2.
Department of Mathematics, JECRC University, Jaipur-303905, Rajasthan, India
3.
Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Sciences, King Abdulaziz University, Jeddah, 21589, Saudi Arabia
4.
Department of Physics, L.N.D. College (B.R. Ambedkar Bihar University, Muzaffarpur), Motihari-845401, Bihar, India
5.
Department of Mathematics, University of Rajasthan, Jaipur-302004, Rajasthan, India

Received: 02 November 2021 Revised: 11 January 2022 Accepted: 20 January 2022 Published: 24 February 2022
MSC : 26A27, 26A30, 26A33, 28A80, 35R11

In this paper, we present a computational algorithm, namely, local fractional natural homotopy analysis method (LFNHAM) to explore the solutions of local fractional coupled Helmholtz and local fractional coupled Burgers' equations (LFCHEs and LFCBEs). This work also investigates the uniqueness and convergence of the solution of a general local fractional partial differential equation (LFPDE) obtained by the suggested method in view of theory of fixed point and Banach spaces. Furthermore, the error analysis of the LFNHAM solution is also discussed. Moreover, the numerical simulations are presented for each of the local fractional coupled equations on the Cantor set. The computational procedure clearly illustrates the validity and reliability of the proposed method for achieving the solutions of local fractional coupled Helmholtz and coupled Burgers' equations. The proposed method also minimizes the computational work unlike other conventional methods while still giving extremely precise results. The implemented combination supplies a more general solution as compared to other methods and assimilates their consequences as a special case. In addition, the acquired solutions are also in excellent match with previously determined solutions.
Citation: Ved Prakash Dubey, Jagdev Singh, Ahmed M. Alshehri, Sarvesh Dubey, Devendra Kumar. Analysis of local fractional coupled Helmholtz and coupled Burgers' equations in fractal media[J]. AIMS Mathematics, 2022, 7(5): 8080-8111. doi: 10.3934/math.2022450

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Abstract

In this paper, we present a computational algorithm, namely, local fractional natural homotopy analysis method (LFNHAM) to explore the solutions of local fractional coupled Helmholtz and local fractional coupled Burgers' equations (LFCHEs and LFCBEs). This work also investigates the uniqueness and convergence of the solution of a general local fractional partial differential equation (LFPDE) obtained by the suggested method in view of theory of fixed point and Banach spaces. Furthermore, the error analysis of the LFNHAM solution is also discussed. Moreover, the numerical simulations are presented for each of the local fractional coupled equations on the Cantor set. The computational procedure clearly illustrates the validity and reliability of the proposed method for achieving the solutions of local fractional coupled Helmholtz and coupled Burgers' equations. The proposed method also minimizes the computational work unlike other conventional methods while still giving extremely precise results. The implemented combination supplies a more general solution as compared to other methods and assimilates their consequences as a special case. In addition, the acquired solutions are also in excellent match with previously determined solutions.

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