American Institute of Mathematical Sciences

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

 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

Received  January 2019 Revised  June 2019 Published  December 2019

Fund Project: 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

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.

Citation: Zhijie Cao, Lijun Zhang. Symmetries and conservation laws of a time dependent nonlinear reaction-convection-diffusion equation. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020218
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References:
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= $a=0$ $\tau=c_{1}t+c_{2},$ $c_1(\frac12x\partial_x+t\partial_t-u\partial_u)$ $(c_1t+c_2)k'-$ $b\neq 0$ $\phi=-c_{1}u$ $+c_2\partial_t+c_3\partial_x.$ $\frac{1}{2}c_1k(t)=0$ $\xi=\frac{1}{2}c_{1}x+$ $X=$ $a\neq 0$ $c_4\int{e^{at}kdt}+c_{3},$ $c_1(\frac12x\partial_x+t\partial_t+atu\partial_u)$ $b=0$ $\tau=c_{1}t+c_{2},$ $+c_2\partial_t+ c_3\partial_x+c_4(e^{at}\partial_u$ $(c_1(at+\frac12)+c_5)k$ $\phi=(c_{1}at+c_5)u+c_4e^{at}$ $+\int{e^{at}kdt}\partial_x)+c_5u\partial_u$ $+(c_1t+c_2)k'=0$ ${\xi=c_{3}},$ $X=$ ${c_2k' = 0}$ $a\neq 0$ ${\tau = c_{2}},$ ${c_2\partial_t+c_3\partial_x}.$ $b\neq0$ ${\phi = 0}$
 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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