A comparative analysis of two computational schemes for solving local fractional Laplace equations. (1st September 2021)

• Record Type: Journal Article
• Authors: Dubey, Ved Prakash; Singh, Jagdev; Alshehri, Ahmed M.; Dubey, Sarvesh; Kumar, Devendra
• Is Part Of: Mathematical methods in the applied sciences. Volume 44:Number 17(2021)
• Page Start: 13540
• Page End: 13559
• Publication Date: 2021-09-01
• Subjects: local fractional differential operator -- local fractional Laplace equation -- local fractional Sumudu transform -- nondifferentiable solution; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.7642
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : In this paper, we implement the semi‐analytical schemes, namely, local fractional homotopy perturbation Sumudu transform method (LFHPSTM) and local fractional homotopy analysis Sumudu transform method (LFHASTM), for finding the approximate analytical solutions of local fractional Laplace equations under different initial conditions on Cantor sets. The novelty of the work lies in the application of these suggested schemes which were never used to solve the local fractional Laplace equation. The Laplace equation mainly contributes to electrostatistics, mechanical engineering, and theoretical physics. This equation also helps in formulation of the electrostatic problem of a rod under a torsion load. Moreover, it helps in handling potential field problems also. These aspects enhance the significance of getting a solution of the local fractional Laplace equation in fractal media. The suggested methods are copulation of local fractional homotopy perturbation method and local fractional homotopy analysis method with the local fractional Sumudu transform, respectively. The computational process indicates that the schemes are very efficient to obtain nondifferentiable solutions for given equations in a smooth way. Moreover, the solution analysis depicts that both the local fractional hybrid homotopy schemes are in a good agreement with each other. These methods depict the accuracy and efficiency of the implemented methods in view of correspondence with solutions obtained with other methods in previous works. The numerical simulations for obtained results are discussed for various values of order of a local fractional derivative. The 3D graphs on the Cantor set are constructed with the help of Matlab software.