Symmetries and conservation laws of a time dependent nonlinear reaction-convection-diffusion equation

Zhijie Cao; Lijun Zhang

doi:10.3934/dcdss.2020218

Article Contents

2020, Volume 13, Issue 10: 2703-2717. Doi: 10.3934/dcdss.2020218

This issue Previous Article Lie symmetries, conservation laws and exact solutions of a generalized quasilinear KdV equation with degenerate dispersion Next Article Quasilinearization applied to boundary value problems at resonance for Riemann-Liouville fractional differential equations

Symmetries and conservation laws of a time dependent nonlinear reaction-convection-diffusion equation

Zhijie Cao^1, and
Lijun Zhang^2,3, ,

1.
Department of Mathematics, College of Science, China Three Gorges University, Yichang, Hubei 443002, China
2.
College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, Shandong 266590, China
3.
African Institute for Mathematical Sciences, Muizenberg, Cape Town, South Africa

^* Corresponding author: Lijun Zhang
^* Corresponding author: Lijun Zhang

Received: January 2019

Revised: June 2019

Early access: December 2019

Published: July 2020

The first author is funded by the China Scholarship Fund (No.201908420198). The second author is supported by NSF grant No. 11672270. The revision was done during L Zhang's research visit to African Institute for Mathematical Sciences South Africa.

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Abstract

In this paper, we study a nonlinear diffusion-convection-reaction equation with a variable coefficient which has applications in many fields. The Lie point symmetries of this equation are derived, according to which this equation is classified into four different kinds. Conservation laws for this equation are constructed by using the conservation theorem of Ibragimov.

Keywords:

Lie point symmetries,
a diffusion-convection-reaction equation with a variable coefficient,
conservation laws.

Mathematics Subject Classification: 76M60, 37K05, 35K10.

Citation:

Full Text(HTML)

Table 1. The results for symmetries of equation (1.1)

Cases	Sol. of determining eq.	Infinitesimal generator	Restriction on $ k $
	$ \xi=\frac{1}{2}c_{1}x+ $	$ X= $
$ a=0 $	$ c_4\int{k(t)dt}+c_{3}, $	$ c_1(\frac12 x\partial_x+t\partial_t) $
$ b=0 $	$ \tau=c_{1}t+c_{2}, $	$ +c_2\partial_t+c_3\partial_x+c_5u\partial_u $	$ (c_5+\frac{1}{2}c_1)k+ $
	$ \phi=c_{5}u+c_{4}. $	$ +c_4(\int{k(t)dt}\partial_x+\partial_u) $	$ (c_1t+c_2)k'=0 $
	$ \xi=\frac{1}{2}c_{1}x+c_{3}, $	X=
$ a=0 $	$ \tau=c_{1}t+c_{2}, $	$ c_1(\frac12x\partial_x+t\partial_t-u\partial_u) $	$ (c_1t+c_2)k'- $
$ b\neq 0 $	$ \phi=-c_{1}u $	$ +c_2\partial_t+c_3\partial_x. $	$ \frac{1}{2}c_1k(t)=0 $
	$ \xi=\frac{1}{2}c_{1}x+ $	$ X= $
$ a\neq 0 $	$ c_4\int{e^{at}kdt}+c_{3}, $	$ c_1(\frac12x\partial_x+t\partial_t+atu\partial_u) $
$ b=0 $	$ \tau=c_{1}t+c_{2}, $	$ +c_2\partial_t+ c_3\partial_x+c_4(e^{at}\partial_u $	$ (c_1(at+\frac12)+c_5)k $
	$ \phi=(c_{1}at+c_5)u+c_4e^{at} $	$ +\int{e^{at}kdt}\partial_x)+c_5u\partial_u $	$ +(c_1t+c_2)k'=0 $
	$ {\xi=c_{3}}, $	$ X= $	${c_2k' = 0}$
$a\neq 0$	${\tau = c_{2}}, $	${c_2\partial_t+c_3\partial_x}.$
$b\neq0$	${\phi = 0}$

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