Jena P, Mohapatra S N, Mishra S R
Department of Mathematics, Siksha 'O'Anusandhan Deemed to Be University, Bhubaneswar, Odisha 751030 India.
Int J Appl Comput Math. 2022;8(4):212. doi: 10.1007/s40819-022-01384-4. Epub 2022 Aug 3.
The variable fractional dimensions differential and integral operator overrides the phenomenon of the constant fractional order. This leads to exploring some new ideas in the proposed direction due to its varied applications in the recent era of science and engineering. The present papers deal with the replacement of the constant fractional order by variable fractional order in various fractal-fractional differential equations. An advanced numerical scheme is developed with the help of Lagrange three-point interpolation and further, it is employed for the solution of the proposed differential equations. However, the properties of these new operators are presented in detail. Finally, the error analysis is also conducted for the numerical scheme deployed. The results are validated by the suitable choice of applications to real-life problems. The well- known multi-step-Adams-Bashforth numerical scheme for classical differential equations is recovered when the non-integer order is one.
可变分数维微分和积分算子克服了常分数阶的现象。由于其在最近的科学和工程时代的各种应用，这导致在提出的方向上探索一些新想法。本文讨论了在各种分形-分数阶微分方程中用可变分数阶代替常分数阶的问题。借助拉格朗日三点插值法开发了一种先进的数值格式，并将其用于求解所提出的微分方程。然而，详细介绍了这些新算子的性质。最后，还对所采用的数值格式进行了误差分析。通过对实际问题进行适当的应用选择来验证结果。当非整数阶为1时，可以恢复经典微分方程中著名的多步亚当斯-巴什福斯数值格式。
