ANALYSIS OF THE TIME FRACTIONAL NONLINEAR DIFFUSION EQUATION FROM DIFFUSION PROCESS

Jian-Gen Liu; Xiao-Jun Yang; Yi-Ying Feng; Hong-Yi Zhang

doi:10.11948/20190186

2020 Volume 10 Issue 3

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Article navigation > Journal of Applied Analysis & Computation > 2020 > 10(3): 1060-1072

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Jian-Gen Liu, Xiao-Jun Yang, Yi-Ying Feng, Hong-Yi Zhang. ANALYSIS OF THE TIME FRACTIONAL NONLINEAR DIFFUSION EQUATION FROM DIFFUSION PROCESS[J]. Journal of Applied Analysis & Computation, 2020, 10(3): 1060-1072. doi: 10.11948/20190186

Citation:

Jian-Gen Liu, Xiao-Jun Yang, Yi-Ying Feng, Hong-Yi Zhang. ANALYSIS OF THE TIME FRACTIONAL NONLINEAR DIFFUSION EQUATION FROM DIFFUSION PROCESS[J]. Journal of Applied Analysis & Computation, 2020, 10(3): 1060-1072. doi: 10.11948/20190186

ANALYSIS OF THE TIME FRACTIONAL NONLINEAR DIFFUSION EQUATION FROM DIFFUSION PROCESS

1.
School of Mathematics, China University of Mining and Technology, Xuzhou, Jiangsu, 221116, China
2.
State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou, Jiangsu, 221116, China
3.
School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou, Jiangsu, 221116, China

Corresponding author: Email address:xjyang@cumt.edu.cn(X.J. Yang)

Abstract

Abstract

Under investigation in this paper is a time fractional nonlinear diffusion equation which can be utilized to express various diffusion processes. The symmetry of this considered equation has been obtained via fractional Lie group approach with the sense of Riemann-Liouville (R-L) fractional derivative. Based on the symmetry, this equation can be changed into an ordinary differential equation of fractional order. Moreover, some new invariant solutions of this considered equation are found. Lastly, utilising the Noether theorem and the general form of Noether type theorem, the conservation laws are yielded to the time fractional nonlinear diffusion equation, respectively. Our discovery that there are no conservation laws under the general form of Noether type theorem case. This result tells us the symmetry of this equation is not variational symmetry of the considered functional. These rich results can give us more information to interpret this equation.
- fractional Lie group approach /
- time fractional nonlinear diffusion equation /
- invariant solutions /
- conservation laws
MSC: 22E70, 35D99, 35K05, 35L65, 35Q51

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