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On the Approximation of fractal-fractional differential equations Using Numerical Inverse Laplace Transform Methods

Computer Modeling in Engineering & Sciences 2023年第6期135卷 2743-2765页

作者： Kamran Siraj Ahmad Kamal Shah Thabet Abdeljawad Bahaaeldin Abdalla Department of Mathematics Islamia College PeshawarKhyber Pakhtoon KhwaPeshawar25120Pakistan Department of Mathematics and Sciences Prince Sultan UniversityP.O.Box 66833Riyadh11586Saudi Arabia Department of Mathematics University of MalakandChakdara Dir(L)Khyber Pakhtunkhwa18000Pakistan Department of Medical Research China Medical UniversityTaichung40402Taiwan

Laplace transform is one of the powerful tools for solving differential equations in engineering and other science *** the Laplace transform for solving differential equations,however,sometimes leads to solutions in the Laplace domain that are not readily invertible to the real domain by ***,we need numerical inversionmethods to convert the obtained solution fromLaplace domain to a real *** this paper,we propose a numerical scheme based on Laplace transform and numerical inverse Laplace transform for the approximate solution of fractal-fractional differential equations with orderα,β.Our proposed numerical scheme is based on three main ***,we convert the given fractal-fractional differential equation to fractional-differential equation in Riemann-Liouville sense,and then into Caputo ***,we transformthe fractional differential equation in Caputo sense to an equivalent equation in Laplace *** the solution of the transformed equation is obtained in Laplace ***,the solution is converted into the real domain using numerical inversion of Laplace *** inversion methods are evaluated in this paper,and their convergence is also *** test problems are used to validate the inversion *** demonstrate our results with the help of tables and *** obtained results show that Euler’s and Talbot’s methods performed better than Stehfest’s method.

关键词： fractal-fractional differential equation power law kernel exponential decay kernel Mittag-Leffler kernel Laplace transform Euler’s method Talbot’s method Stehfest’s method

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