Ghafoor Abdul, Fiaz Muhammad, Shah Kamal, Abdeljawad Thabet
Institute of Numerical Sciences, Kohat University of Science and Technology, Kohat 26000, KP, Pakistan.
Department of Mathematics and Sciences, Prince Sultan University, Riyadh, 11586, Saudi Arabia.
Heliyon. 2024 Jul 3;10(13):e33842. doi: 10.1016/j.heliyon.2024.e33842. eCollection 2024 Jul 15.
This paper demonstrates, a numerical method to solve the one and two dimensional Burgers' equation involving time fractional Atangana-Baleanu Caputo derivative with a non-singular kernel. The numerical stratagem consists of a quadrature rule for time fractional derivative along with Haar wavelet (HW) approximations of one and two dimensional problems. The key feature of the scheme is to reduce fractional problems to the set of linear equations via collocation procedure. Solving the system gives the approximate solution of the given problem. To verify the effectiveness of the developed method five numerical examples are considered. Besides this, the obtained simulations are compared with some published work and identified that proposed technique is better. Moreover, computationally the convergence rate in spatiotemporal directions is presented which shows order two convergence. The stability of the proposed scheme is also described via Lax-Richtmyer criterion. From simulations it is obvious that the scheme is quite useful for the time fractional problems.
本文展示了一种数值方法，用于求解涉及具有非奇异核的时间分数阶阿坦加纳 - 巴莱努卡普托导数的一维和二维伯格斯方程。该数值策略包括用于时间分数阶导数的求积规则以及一维和二维问题的哈尔小波（HW）近似。该方案的关键特征是通过配置过程将分数阶问题简化为线性方程组。求解该系统可得到给定问题的近似解。为验证所开发方法的有效性，考虑了五个数值例子。除此之外，将获得的模拟结果与一些已发表的工作进行比较，发现所提出的技术更好。此外，给出了在时空方向上的计算收敛速率，显示出二阶收敛。还通过拉克斯 - 里希特迈尔准则描述了所提方案的稳定性。从模拟结果可以明显看出，该方案对于时间分数阶问题非常有用。
