A computational study of time-fractional gas dynamics models by means of conformable finite difference method

Majeed A. Yousif; Juan L. G. Guirao; Pshtiwan Othman Mohammed; Nejmeddine Chorfi; Dumitru Baleanu; Majeed A. Yousif; Juan L. G. Guirao; Pshtiwan Othman Mohammed; Nejmeddine Chorfi; Dumitru Baleanu

doi:10.3934/math.2024969

AIMS Mathematics

2024, Volume 9, Issue 7: 19843-19858. doi: 10.3934/math.2024969

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A computational study of time-fractional gas dynamics models by means of conformable finite difference method

1.
Department of Mathematics, College of Education, Zakho University, Duhok 42001, Iraq
2.
Department of Applied Mathematics and Statistics, Technical University of Cartagena, Hospital de Marina, Cartagena 30203, Spain
3.
Department of Mathematics, College of Education, University of Sulaimani, Sulaymaniyah 46001, Iraq
4.
Research and Development Center, University of Sulaimani, Sulaymaniyah 46001, Iraq
5.
Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
6.
Department of Computer Science and Mathematics, Lebanese American University, Beirut 11022801, Lebanon
7.
Institute of Space Science-Subsidiary of INFLPR, R76900 Magurele-Bucharest, Romania

Received: 21 February 2024 Revised: 19 May 2024 Accepted: 27 May 2024 Published: 19 June 2024
MSC : 26A33, 35R11, 76M20, 76N10

This paper introduces a novel numerical scheme, the conformable finite difference method (CFDM), for solving time-fractional gas dynamics equations. The method was developed by integrating the finite difference method with conformable derivatives, offering a unique approach to tackle the challenges posed by time-fractional gas dynamics models. The study explores the significance of such equations in capturing physical phenomena like explosions, detonation, condensation in a moving flow, and combustion. The numerical stability of the proposed scheme is rigorously investigated, revealing its conditional stability under certain constraints. A comparative analysis is conducted by benchmarking the CFDM against existing methodologies, including the quadratic B-spline Galerkin and the trigonometric B-spline functions methods. The comparisons are performed using $ {L}_{2} $ and $ {L}_{\infty } $ norms to assess the accuracy and efficiency of the proposed method. To demonstrate the effectiveness of the CFDM, several illustrative examples are solved, and the results are presented graphically. Through these examples, the paper showcases the capability of the proposed methodology to accurately capture the behavior of time-fractional gas dynamics equations. The findings underscore the versatility and computational efficiency of the CFDM in addressing complex phenomena. In conclusion, the study affirms that the conformable finite difference method is well-suited for solving differential equations with time-fractional derivatives arising in the physical model.
- conformable fractional derivative,
- finite difference method,
- stability,
- time-fractional gas dynamics models
Citation: Majeed A. Yousif, Juan L. G. Guirao, Pshtiwan Othman Mohammed, Nejmeddine Chorfi, Dumitru Baleanu. A computational study of time-fractional gas dynamics models by means of conformable finite difference method[J]. AIMS Mathematics, 2024, 9(7): 19843-19858. doi: 10.3934/math.2024969

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Abstract

This paper introduces a novel numerical scheme, the conformable finite difference method (CFDM), for solving time-fractional gas dynamics equations. The method was developed by integrating the finite difference method with conformable derivatives, offering a unique approach to tackle the challenges posed by time-fractional gas dynamics models. The study explores the significance of such equations in capturing physical phenomena like explosions, detonation, condensation in a moving flow, and combustion. The numerical stability of the proposed scheme is rigorously investigated, revealing its conditional stability under certain constraints. A comparative analysis is conducted by benchmarking the CFDM against existing methodologies, including the quadratic B-spline Galerkin and the trigonometric B-spline functions methods. The comparisons are performed using $ {L}_{2} $ and $ {L}_{\infty } $ norms to assess the accuracy and efficiency of the proposed method. To demonstrate the effectiveness of the CFDM, several illustrative examples are solved, and the results are presented graphically. Through these examples, the paper showcases the capability of the proposed methodology to accurately capture the behavior of time-fractional gas dynamics equations. The findings underscore the versatility and computational efficiency of the CFDM in addressing complex phenomena. In conclusion, the study affirms that the conformable finite difference method is well-suited for solving differential equations with time-fractional derivatives arising in the physical model.

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