Fractional calculus analysis: investigating Drinfeld-Sokolov-Wilson system and Harry Dym equations via meshless procedures

Muhammad Nawaz Khan; Imtiaz Ahmad; Mehnaz Shakeel; Rashid Jan; Muhammad Nawaz Khan; Imtiaz Ahmad; Mehnaz Shakeel; Rashid Jan

doi:10.3934/mmc.2024008

Mathematical Modelling and Control

2024, Volume 4, Issue 1: 86-100. doi: 10.3934/mmc.2024008

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Research article

Fractional calculus analysis: investigating Drinfeld-Sokolov-Wilson system and Harry Dym equations via meshless procedures

1.
Mathematics in Applied Sciences and Engineering Research Group, Scientific Research Center, Al-Ayen University, Nasiriyah 64001, Iraq
2.
Institute of Informatics and Computing in Energy, Universiti Tenaga Nasional, Kajang, Selangor, Malaysia
3.
Department of Mathematics, Women University Mardan, Mardan, Pakistan
4.
Institute of Energy Infrastructure, Department of Civil Engineering, College of Engineering, Universiti Tenaga Nasional, Putrajaya Campus, Jalan IKRAM-UNITEN, Kajang 43000, Selangor, Malaysia
5.
Mathematics Research Center, Near East University North Cyprus, Mersin 99138, Turkey

Received: 27 August 2023 Revised: 02 December 2023 Accepted: 15 December 2023 Published: 29 March 2024

In this study, we present two meshless schemes, namely the radial basis function (RBF) method and the polynomial method, for the numerical investigation of the time-fractional Harry Dym equation and the Drinfeld-Sokolov-Wilson system. In both methods, the temporal derivatives are estimated using the Caputo operator, while the spatial derivatives are approximated either through radial basis functions or polynomials. Additionally, a collocation approach is employed to convert the system of equations into a system of linear equations that is easier to solve. The accuracy of the methods is assessed by calculating the $ L_{\infty} $ error norm, and the outcomes are displayed through tables and figures. The simulation results indicate that both methods exhibit strong performance in handling the fractional partial differential equations (PDEs) under investigation.
- meshless procedures,
- polynomial method,
- RBF method,
- Drinfeld-Sokolov-Wilson equation,
- Harry Dym equation
Citation: Muhammad Nawaz Khan, Imtiaz Ahmad, Mehnaz Shakeel, Rashid Jan. Fractional calculus analysis: investigating Drinfeld-Sokolov-Wilson system and Harry Dym equations via meshless procedures[J]. Mathematical Modelling and Control, 2024, 4(1): 86-100. doi: 10.3934/mmc.2024008

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Abstract

In this study, we present two meshless schemes, namely the radial basis function (RBF) method and the polynomial method, for the numerical investigation of the time-fractional Harry Dym equation and the Drinfeld-Sokolov-Wilson system. In both methods, the temporal derivatives are estimated using the Caputo operator, while the spatial derivatives are approximated either through radial basis functions or polynomials. Additionally, a collocation approach is employed to convert the system of equations into a system of linear equations that is easier to solve. The accuracy of the methods is assessed by calculating the $ L_{\infty} $ error norm, and the outcomes are displayed through tables and figures. The simulation results indicate that both methods exhibit strong performance in handling the fractional partial differential equations (PDEs) under investigation.

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