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

  • * Corresponding author: Lijun Zhang

    * Corresponding author: Lijun Zhang

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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  • 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.

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

    Citation:

    \begin{equation} \\ \end{equation}
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  • 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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