Numerical approximation of the time-fractional regularized long-wave equation emerging in ion acoustic waves in plasma

Hasim Khan; Mohammad Tamsir; Manoj Singh; Ahmed Hussein Msmali; Mutum Zico Meetei; Hasim Khan; Mohammad Tamsir; Manoj Singh; Ahmed Hussein Msmali; Mutum Zico Meetei

doi:10.3934/math.2025261

AIMS Mathematics

2025, Volume 10, Issue 3: 5651-5670. doi: 10.3934/math.2025261

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Research article

Numerical approximation of the time-fractional regularized long-wave equation emerging in ion acoustic waves in plasma

Department of Mathematics, College of Science, Jazan University, P.O. Box 114, Jazan 45142, Saudi Arabia

Received: 08 November 2024 Revised: 06 February 2025 Accepted: 26 February 2025 Published: 13 March 2025
MSC : 35R11, 65M12

This work accomplished a novel approximate solution of the time-fractional regularized long-wave (TFRLW) equation. This equation is an appropriate mathematical model in physical sciences that designates the nature of ion acoustic waves in plasma and waves of shallow water. A cubic B-spline (CBS) collocation procedure was used for the spatial discretization, offering greater flexibility and accuracy compared to traditional spline methods. For time discretization, the finite difference method was used, ensuring computational efficiency, while the time-fractional derivative was settled by Caputo's definition. The Rubin-Graves linearization procedure was involved to handle the nonlinear term. To demonstrate the possessions of different constraints and variables on the displacement, the approximate solutions were shown in tabular as well as graphical forms. The method's unconditional stability was confirmed through a detailed von Neumann stability analysis, making it particularly robust for long-term simulations. The order of convergence was also estimated numerically. Three invariant capacities analogous to mass, momentum, and energy were assessed for further justification. Obtained solutions established the exactitude and efficiency of the anticipated method. Furthermore, unlike many existing methods, this approach can be tailored to handle the complexity of higher-order equations while maintaining stability and accuracy over large-scale problems.
- TFRLW equation,
- Caputo's fractional derivative,
- CBS collocation technique,
- stability analysis,
- numerical convergence
Citation: Hasim Khan, Mohammad Tamsir, Manoj Singh, Ahmed Hussein Msmali, Mutum Zico Meetei. Numerical approximation of the time-fractional regularized long-wave equation emerging in ion acoustic waves in plasma[J]. AIMS Mathematics, 2025, 10(3): 5651-5670. doi: 10.3934/math.2025261

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Abstract

This work accomplished a novel approximate solution of the time-fractional regularized long-wave (TFRLW) equation. This equation is an appropriate mathematical model in physical sciences that designates the nature of ion acoustic waves in plasma and waves of shallow water. A cubic B-spline (CBS) collocation procedure was used for the spatial discretization, offering greater flexibility and accuracy compared to traditional spline methods. For time discretization, the finite difference method was used, ensuring computational efficiency, while the time-fractional derivative was settled by Caputo's definition. The Rubin-Graves linearization procedure was involved to handle the nonlinear term. To demonstrate the possessions of different constraints and variables on the displacement, the approximate solutions were shown in tabular as well as graphical forms. The method's unconditional stability was confirmed through a detailed von Neumann stability analysis, making it particularly robust for long-term simulations. The order of convergence was also estimated numerically. Three invariant capacities analogous to mass, momentum, and energy were assessed for further justification. Obtained solutions established the exactitude and efficiency of the anticipated method. Furthermore, unlike many existing methods, this approach can be tailored to handle the complexity of higher-order equations while maintaining stability and accuracy over large-scale problems.

References

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