Khater Mostafa M A, Attia Raghda A M, Lu Dianchen
Department of Mathematics, Faculty of Science, Jiangsu University, Zhenjiang 212013, China.
Department of Basic Science, Higher Technological Institute 10th of Ramadan City, El Sharqia 44634, Egypt.
Entropy (Basel). 2019 Apr 15;21(4):397. doi: 10.3390/e21040397.
This study investigates the solitary wave solutions of the nonlinear fractional Jimbo-Miwa (JM) equation by using the conformable fractional derivative and some other distinct analytical techniques. The JM equation describes the certain interesting (3+1)-dimensional waves in physics. Moreover, it is considered as a second equation of the famous Painlev'e hierarchy of integrable systems. The fractional conformable derivatives properties were employed to convert it into an ordinary differential equation with an integer order to obtain many novel exact solutions of this model. The conformable fractional derivative is equivalent to the ordinary derivative for the functions that has continuous derivatives up to some desired order over some domain (smooth functions). The obtained solutions for each technique were characterized and compared to illustrate the similarities and differences between them. Profound solutions were concluded to be powerful, easy and effective on the nonlinear partial differential equation.
本研究利用一致分数阶导数和其他一些不同的分析技术，研究了非线性分数阶Jimbo-Miwa（JM）方程的孤立波解。JM方程描述了物理学中某些有趣的（3 + 1）维波。此外，它被认为是著名的可积系统Painlev'e层级的第二个方程。利用分数阶一致导数的性质将其转化为整数阶常微分方程，以获得该模型的许多新颖精确解。对于在某个域上具有直至某个期望阶数的连续导数的函数（光滑函数），一致分数阶导数等同于普通导数。对每种技术得到的解进行了表征和比较，以说明它们之间的异同。得出的深刻解对于非线性偏微分方程来说是强大、简便且有效的。
