On novel nondifferentiable exact solutions to local fractional Gardner's equation using an effective technique

• Published in: Mathematical methods in the applied sciences Vol. 44; no. 6; pp. 4673 - 4685
• Main Author: Ghanbari, Behzad
• Format: Journal Article
• Language: English
• Published: Freiburg Wiley Subscription Services, Inc 01.04.2021
• Subjects: Basis functions; cantor set; efficient technique; Exact solutions; Fractional calculus; Gardner's equation; local fractional derivative; numerical simulation; Partial differential equations; Thermal expansion
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.7060

One of the most interesting branches of fractional calculus is the local fractional calculus, which has been used successfully to describe many fractal problems in science and engineering. The main purpose of this contribution is to construct a novel efficient technique to retrieve exact fractional solutions to local fractional Gardner's equation defined on Cantor sets by an effective numerical methodology. In the framework of this technique, first a set of elementary functions are defined on the contour set. Taking these functions into account, the general structure of the exact solution for the equation is suggested as a specific combination of the defined basis functions. By determining the unknown coefficients in this expansion and by placing them in the original equation, specific forms of solutions to the fractional equation are determined. Several interesting numerical simulations of the achieved results are also presented in the article to give a better understanding of the dynamic behavior of these results. The results obtained in this research confirm that the method used is very simple and efficient in terms of application. Moreover, it is accurate and free of any errors in terms of calculation. Therefore, it can be employed to handle other partial differential equations including local fractional derivatives.