Research ArticleNo Access

Symmetry analysis, optimal system, conservation laws and exact solutions of time-fractional diffusion-type equation

Jicheng Yu

https://orcid.org/0000-0001-6125-263X

School of Science, Wuhan University of Science and Technology, Wuhan 430081, Hubei, P. R. China

E-mail Address: yjicheng@126.com

Corresponding author.

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and

Yuqiang Feng

https://orcid.org/0000-0002-1208-0509

School of Science, Wuhan University of Science and Technology, Wuhan 430081, Hubei, P. R. China

Hubei Province Key Laboratory of Systems Science in Metallurgical Process, Wuhan 430081, Hubei, P. R. China

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https://doi.org/10.1142/S0219887824502864Cited by:2 (Source: Crossref)

Abstract

In Liu et al. [On the generalized time fractional diffusion equation: Symmetry analysis, conservation laws, optimal system and exact solutions, Int. J. Geom. Methods Mod. Phys.17(01) (2020) 2050013], the generalized time-fractional diffusion equation $D_{t}^{α} u = a u^{p} u_{x x} + b u^{q} u_{x}^{2}$ $D_{t}^{α} u = a u^{p} u_{x x} + b u^{q} u_{x}^{2}$ is studied by the symmetry analysis method. Liu et al. obtained two group generators $X_{1} = \frac{\partial}{\partial x},$ $X_{1} = \frac{\partial}{\partial x},$ $X_{2} = \frac{2 t}{α} \frac{\partial}{\partial t} + x \frac{\partial}{\partial x}$ $X_{2} = \frac{2 t}{α} \frac{\partial}{\partial t} + x \frac{\partial}{\partial x}$ and only one trivial solution $u (t, x) = f (t) = \frac{C_{0}}{Γ (α)} t^{α - 1}$ $u (t, x) = f (t) = \frac{C_{0}}{Γ (α)} t^{α - 1}$ for the equation. In this paper, we classify the Lie symmetry group admitted by the equation, and for the case $p = q + 1$ $p = q + 1$ , we obtain a richer set of group generators and some nontrivial solutions, including unprecedented exact solutions and power series solutions. In addition, we construct the one-dimensional optimal system of the Lie symmetry group admitted by the time-fractional diffusion-type equation by Olver’s method, and obtain the conservation laws for all the obtained Lie symmetries using the general method developed by Ibragimov. For the novel exact solutions and power series solutions, we analyze their dynamic behavior graphically.

Keywords:

AMSC: 35B06, 35R11, 76M60

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