Similarity Solutions to Nonlinear Diffusion/Harry Dym Fractional Equations

Chao Yue,Hanhui Dong,Kun Li,Guijuan Liu

doi:10.1155/2021/6670533

Abstract

By using scalar similarity transformation, nonlinear model of time-fractional diffusion/Harry Dym equation is transformed to corresponding ordinary fractional differential equations, from which a travelling-wave similarity solution of time-fractional Harry Dym equation is presented. Furthermore, numerical solutions of time-fractional diffusion equation are discussed. Again, through another similarity transformation, nonlinear model of space-fractional diffusion/Harry Dym equation is turned into corresponding ordinary differential equations, whose two similarity solutions are also worked out.

Highlights

Nonlinear partial differential equations arise in many fields of engineering, physics, and applied mathematics
In Ref. [10], the authors presented and discussed a fractional nonlinear partial differential equation by use of similarity reductions and recovered some interesting results associated with Harry Dym-type equations
In order to obtain the numerical solutions of time-fractional diffusion equation we consider the similarity transformation u where UðξÞ, ξ, p, and q are constants to be determined

Summary

Introduction

Nonlinear partial differential equations arise in many fields of engineering, physics, and applied mathematics. Many methods have been used to study and analyze fractional differential equations, in which the Lie-group analysis method is an effective tool to investigate symmetries of ordinary and partial differential equations. [10], the authors presented and discussed a fractional nonlinear partial differential equation by use of similarity reductions and recovered some interesting results associated with Harry Dym-type equations. [11], the fractional nonlinear space-time wave-diffusion equation was discussed and solved by the similarity method utilizing fractional derivatives in the Caputo, Riesz-Feller, and Riesz senses. We shall treat a nonlinear model of time-fractional diffusion/Harry Dym equation. 2, ð1Þ and further study nonlinear model of space-fractional diffusion/Harry Dym equation. For n − 1 < β < n, we have dβ f dtβ 1 dn ðt Γðn − βÞ dtn 0 ðt f ðτÞ − τÞβ+1−n dτ: ð5Þ

Travelling-Wave Similarity Solution of TimeFractional Harry Dym Equation

Numerical Solutions of Time-Fractional Diffusion Equation

Conclusions