Numerical approximation for solving time-fractional Benjamin-Bona-Mahony-Burger model via cubic B-spline functions

Muserat Shaheen; Muhammad Abbas; Miguel Vivas-Cortez; M. R. Alharthi; Y. S. Hamed; Muserat Shaheen; Muhammad Abbas; Miguel Vivas-Cortez; M. R. Alharthi; Y. S. Hamed

doi:10.3934/math.2025624

AIMS Mathematics

2025, Volume 10, Issue 6: 13855-13879. doi: 10.3934/math.2025624

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Numerical approximation for solving time-fractional Benjamin-Bona-Mahony-Burger model via cubic B-spline functions

1.
Department of Mathematics, University of Sargodha, 40100 Sargodha, Pakistan
2.
Faculty of Exact, Natural and Environmental Sciences, Pontificia Universidad Católica del Ecuador, FRACTAL (Fractional Research in Analysis, Convexity and Their Applications Laboratory), Av 12 de octubre 1076 y Roca, Apartado Quito 17-01-2184, Ecuador
3.
Department of Mathematics and Statistics, College of Science, Taif University, P. O. Box 11099, Taif 21944, Saudi Arabia

Received: 05 March 2025 Revised: 19 May 2025 Accepted: 23 May 2025 Published: 17 June 2025
MSC : 34G20, 35A20, 35A22, 35R11

Many years of research have gone into spline functions, and they are now used in countless computational tasks. Splines have a lot of useful properties that make them an excellent tool for numerical problem solving, which account for their never-ending applications. The piecewise continuous functions known as spline functions yield smooth outcomes. The numerical solution to the nonhomogeneous time-fractional Banjamin-Bona-Mahony-Burger problem was presented in this study. The objective of the study was to obtain accurate numerical results by applying the Atangana-Baleanu fractional derivative with the help of the forward difference scheme for integer-order time derivative while the $ \theta $-weighted scheme with the collaboration of cubic B-spline functions was used for the spatial derivatives. The stability of the proposed scheme was analyzed and proved to be unconditionally stable. The convergence analysis was also studied, and it was of the second order $ O(h^2 + (\Delta s)^2) $. The proposed scheme was applicable and accurate, as demonstrated by numerical examples and their conceivable outcomes. The proposed scheme provided accuracy compared to other numerical techniques because it yielded numerical solutions in $ C^2 $ continuous piecewise form at each knot in the domain.
- BBM-Burger equation,
- cubic B-spline functions,
- 2nd order accuracy,
- finite difference techniques,
- stability,
- convergence,
- conclusion
Citation: Muserat Shaheen, Muhammad Abbas, Miguel Vivas-Cortez, M. R. Alharthi, Y. S. Hamed. Numerical approximation for solving time-fractional Benjamin-Bona-Mahony-Burger model via cubic B-spline functions[J]. AIMS Mathematics, 2025, 10(6): 13855-13879. doi: 10.3934/math.2025624

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Abstract

Many years of research have gone into spline functions, and they are now used in countless computational tasks. Splines have a lot of useful properties that make them an excellent tool for numerical problem solving, which account for their never-ending applications. The piecewise continuous functions known as spline functions yield smooth outcomes. The numerical solution to the nonhomogeneous time-fractional Banjamin-Bona-Mahony-Burger problem was presented in this study. The objective of the study was to obtain accurate numerical results by applying the Atangana-Baleanu fractional derivative with the help of the forward difference scheme for integer-order time derivative while the $ \theta $-weighted scheme with the collaboration of cubic B-spline functions was used for the spatial derivatives. The stability of the proposed scheme was analyzed and proved to be unconditionally stable. The convergence analysis was also studied, and it was of the second order $ O(h^2 + (\Delta s)^2) $. The proposed scheme was applicable and accurate, as demonstrated by numerical examples and their conceivable outcomes. The proposed scheme provided accuracy compared to other numerical techniques because it yielded numerical solutions in $ C^2 $ continuous piecewise form at each knot in the domain.

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