Department of Mathematics, Samara University, Samara, Ethiopia.
Department of Mathematics, Faculty of Natural Sciences, Mizan Tepi University, Tepi, Ethiopia.
BMC Res Notes. 2024 Aug 16;17(1):226. doi: 10.1186/s13104-024-06877-7.
Nonlinear time-fractional partial differential equations (NTFPDEs) play a great role in the mathematical modeling of real-world phenomena like traffic models, the design of earthquakes, fractional stochastic systems, diffusion processes, and control processing. Solving such problems is reasonably challenging, and the nonlinear part and fractional operator make them more problematic. Thus, developing suitable numerical methods is an active area of research. In this paper, we develop a new numerical method called Yang transform Adomian decomposition method (YTADM) by mixing the Yang transform and the Adomian decomposition method for solving NTFPDEs. The derivative of the problem is considered in sense of Caputo fractional order. The stability and convergence of the developed method are discussed in the Banach space sense. The effectiveness, validity, and practicability of the method are demonstrated by solving four examples of NTFPEs. The findings suggest that the proposed method gives a better solution than other compared numerical methods. Additionally, the proposed scheme achieves an accurate solution with a few numbers of iteration, and thus the method is suitable for handling a wide class of NTFPDEs arising in the application of nonlinear phenomena.
非线性时分数阶偏微分方程 (NTFPDE) 在交通模型、地震设计、分数阶随机系统、扩散过程和控制处理等实际现象的数学建模中起着重要作用。解决这些问题具有相当的挑战性，非线性部分和分数阶算子使它们更成问题。因此，开发合适的数值方法是一个活跃的研究领域。本文提出了一种新的数值方法，即杨变换 Adomian 分解法 (YTADM)，通过混合杨变换和 Adomian 分解法来求解 NTFPDE。问题的导数在 Caputo 分数阶意义下进行考虑。在巴拿赫空间意义下讨论了所发展方法的稳定性和收敛性。通过求解四个 NTFPE 的例子验证了该方法的有效性、有效性和实用性。研究结果表明，与其他比较数值方法相比，该方法给出了更好的解。此外，该方案仅用少量的迭代次数就可以得到精确的解，因此该方法适用于处理非线性现象应用中出现的广泛的 NTFPDE 类。
