Multiplier method and exact solutions for a density dependent reaction-diffusion equation

i	j	w_j	z_j	u_j
1	1	v₃	$\begin{matrix} t^{\frac{n - m + 1}{2 m - 2}} x \end{matrix}$ $\begin{array}{} \displaystyle {t}^{\frac{n-m+1}{2\,m-2}}\,x \end{array}$	$\begin{matrix} h x^{\frac{2}{n - m + 1}} \end{matrix}$ $\begin{array}{} \displaystyle h\,{x}^{\frac{2}{n-m+1}} \end{array}$
2	2	v₄	$\begin{matrix} t^{\frac{n - m}{2 m}} x \end{matrix}$ $\begin{array}{} \displaystyle {t}^{\frac{n-m}{2\,m}}\,x \end{array}$	$\begin{matrix} \frac{2 \log (x)}{n - m} + h \end{matrix}$ $\begin{array}{} \displaystyle \frac{2\,\log\left( x\right) }{n-m}+h \end{array}$
3	3	v₅	x	$\begin{matrix} h e^{- \frac{c_{1} t}{n}} \end{matrix}$ $\begin{array}{} \displaystyle h\,e^ {- {{{\it c_1}\,t}\over{n}} } \end{array}$
4	4	bv₁ + v₃	$\begin{matrix} \frac{8 x - 3 b \log t}{8} \end{matrix}$ $\begin{array}{} \displaystyle {{8\,x-3\,b\,\log t}\over{8}} \end{array}$	$\begin{matrix} h t^{\frac{3}{4}} \end{matrix}$ $\begin{array}{} \displaystyle h\,t^{{{3}\over{4}}} \end{array}$
4	5	cv₃ + v₆	$\begin{matrix} - \frac{4 c e^{- \frac{2 x}{\sqrt{3}}}}{\sqrt{3}} \end{matrix}$ $\begin{array}{} \displaystyle -{{4\,c\,e^ {- {{2\,x}\over{\sqrt{3}}} }}\over{\sqrt{3}}} \end{array}$–log t	$\begin{matrix} h e^{- \sqrt{3} c e^{- \frac{2 x}{\sqrt{3}}} - \sqrt{3} x} \end{matrix}$ $\begin{array}{} \displaystyle h\,e^{-\sqrt{3}\,c\,e^ {- {{2\,x}\over{\sqrt{3}}} }-\sqrt{3}\,x} \end{array}$
4	6	dv₃ + v₇	$\begin{matrix} \frac{4 d e^{\frac{2 x}{\sqrt{3}}}}{\sqrt{3}} \end{matrix}$ $\begin{array}{} \displaystyle {{4\,{\it d}\,e^{{{2\,x}\over{\sqrt{3}}}}}\over{\sqrt{3}}} \end{array}$–log t	$\begin{matrix} h e^{\sqrt{3} d e^{\frac{2 x}{\sqrt{3}}} + \sqrt{3} x} \end{matrix}$ $\begin{array}{} \displaystyle h\,e^{\sqrt{3}\,{\it d}\,e^{{{2\,x}\over{\sqrt{3}}}}+\sqrt{3 }\,x} \end{array}$

Reduced ODEs:

i	j	ODEs_j
1	1	hⁿ(n – m + 1)²( $\begin{matrix} {h_{z}}^{2} n \end{matrix}$ $\begin{array}{} \displaystyle { {h}_{z} }^{2} \,n \end{array}$+hh_zz) z²+4hⁿ⁺¹h_z(n + 1) (n – m + 1)z+2hⁿ⁺²(n + m + 1)+h^m+1(n – m + 1)² = 0
2	2	$\begin{matrix} h_{z z} {(n - m)}^{2} e^{\frac{h n^{2} + h m^{2}}{n - m}} z^{\frac{2 n + m}{n - m} + 2} + {h_{z}}^{2} n {(n - m)}^{2} e^{\frac{h n^{2} + h m^{2}}{n - m}} z^{\frac{4 n - m}{n - m}} + 4 h_{z} n (n - m) e^{\frac{h n^{2} + h m^{2}}{n - m}} z^{\frac{3 n}{n - m}} \\ + ((2 n + 2 m) e^{\frac{h n^{2} + h m^{2}}{n - m}} + (n^{2} - 2 m n + m^{2}) e^{\frac{2 h m n}{n - m}}) z^{\frac{2 n + m}{n - m}} = 0. \end{matrix}$ $\begin{array}{} \displaystyle {h}_{zz}\,{\left( n-m\right) }^{2}\,{e}^{\frac{h\,{n}^{2}+h\,{m}^{2}}{n-m}}\,{z}^{\frac{2\,n+m}{n-m}+2}+{ {h}_{z} }^{2}\,n\,{\left( n-m\right) }^{2}\,{e}^{\frac{h\,{n}^{2}+h\,{m}^{2}}{n-m}}\,{z}^{\frac{4\,n-m}{n-m}} +4\, {h}_{z} \,n\,\left( n-m\right) \,{e}^{\frac{h\,{n}^{2}+h\,{m}^{2}}{n-m}}\,{z}^{\frac{3\,n}{n-m}} \\ +\left( \left( 2\,n+2\,m\right) \,{e}^{\frac{h\,{n}^{2}+h\,{m}^{2}}{n-m}}+\left( {n}^{2}-2\,m\,n+{m}^{2}\right) \,{e}^{\frac{2\,h\,m\,n}{n-m}}\right) \,{z}^{\frac{2\,n+m}{n-m}} =0. \end{array}$
3	3	hⁿ(h_z)²n+c₂hⁿ⁺²+hⁿ⁺¹h_zz = 0
4	4	–12hh_zz+16(h_z)²– $\begin{matrix} \frac{9 b h^{\frac{7}{3}} h_{z}}{2} + 9 h^{\frac{10}{3}} \end{matrix}$ $\begin{array}{} \displaystyle {{9\,b\,h^{{{7}\over{3}}} \,h_{z}}\over{2}}+9\,h^{{{10}\over{3}}} \end{array}$–12h² = 0
4	5	27 $\begin{matrix} h^{\frac{7}{3}} \end{matrix}$ $\begin{array}{} \displaystyle h^{{{7}\over{3}}} \end{array}$h_ze^z+192c²hh_zz–256c²(h_z)²–96c²hh_z–36c²h² = 0
4	6	27 $\begin{matrix} h^{\frac{7}{3}} \end{matrix}$ $\begin{array}{} \displaystyle h^{{{7}\over{3}}} \end{array}$h_ze^z+192d²hh_zz–256d²(h_z)²–96d²hh_z–36d²h² = 0

Optimal systems.

i
1	av₁ + v₂, v₃
2	av₁ + v₂, v₄
3	av₁ + v₂, v₅
4	av₁ + v₂, bv₁ + bv₃, cv₃ + v₆, dv₁ + v₇

Functions and generators.

i	f_i	g_i	v_k
1	u^m	uⁿ	v₁, v₂, v₃ = (n – m + 1)x∂x + 2(1 – m)t∂t + 2u∂u (m ≠ n + 1)
2	e^nu	e^mu	v₁, v₂, v₄ = (m – n)x∂x – 2nt∂t + 2∂u (n ≠ m)
3	$\begin{matrix} c_{2} u^{n + 1} - \frac{c_{1} u}{n} \end{matrix}$ $\begin{array}{} \displaystyle {\it c_2}\,u^{n+1}-{{{\it c_1}\,u}\over{n}} \end{array}$	uⁿ	v₁, v₂, v₅ = e^c₁t∂t – $\begin{matrix} \frac{c_{1} e^{c_{1} t} u}{n} \partial u \end{matrix}$ $\begin{array}{} \displaystyle \frac{c_1e^{c_1t}u}{n}{\partial u} \end{array}$ (n ≠ 0)
4	$\begin{matrix} u^{- \frac{1}{3}} \end{matrix}$ $\begin{array}{} \displaystyle u^{-\frac{1}{3}} \end{array}$	$\begin{matrix} u^{\frac{- 4}{3}} \end{matrix}$ $\begin{array}{} \displaystyle u^{\frac{-4}{3}} \end{array}$	v₁, v₂, v₃, v₆ = $\begin{matrix} e^{\frac{2 x}{\sqrt{3}}} \partial x - \sqrt{3} e^{\frac{2 x}{\sqrt{3}}} u \partial u, v_{7} = e^{- \frac{2 x}{\sqrt{3}}} \partial x + \sqrt{3} e^{- \frac{2 x}{\sqrt{3}}} u \partial u \end{matrix}$ $\begin{array}{} \displaystyle e^{\frac{2x}{\sqrt{3}}} {\partial x}- \sqrt{3} e^{\frac{2x}{\sqrt{3}}}u {\partial u}, \, {\bf v}_7 =\displaystyle e^{-\frac{2x}{\sqrt{3}}} {\partial x}+ \sqrt{3} e^{-\frac{2x}{\sqrt{3}}}u {\partial u} \end{array}$

eISSN:: 2444-8656
Lingua:: Inglese

Frequenza di pubblicazione:: 2 volte all'anno
Argomenti della rivista:: Life Sciences, other, Mathematics, Applied Mathematics, General Mathematics, Physics

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Multiplier method and exact solutions for a density dependent reaction-diffusion equation

Pubblicato online: 01 lug 2016

Pagine: 311 - 320

Ricevuto: 05 mar 2016

Accettato: 01 lug 2016

DOI: https://doi.org/10.21042/AMNS.2016.2.00026

Parole chiaveLie symmetries, Partial differential equations, Conservation laws, Fisher equation

© 2016 M. Rosa, M. L Gandarias, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Fig. 1

Similarity solutions.

Reduced ODEs:

Optimal systems.

Functions and generators.

Parole chiave
Lie symmetries, Partial differential equations, Conservation laws, Fisher equation