Exact solutions of nonlinear diffusion-reaction equations

Abstract

In the present work, the $(\frac{G^\prime}{G})$-expansion method and its generalized version have been employed for obtaining a variety of exact traveling wave solutions of the nonlinear diffusion reaction equation with quadratic and cubic nonlinearities. We have examined the density independent nonlinear diffusion reaction equation with a convective flux term and successfully have obtained some new and more general solutions in terms of Jacobi elliptic functions. The work highlights the significant features of the employed methods and shows the variety in the obtained solutions.

Appendices

Appendix 1

The general solutions to jacobi elliptic equation and its derivative (see, for example, [16, 31]) are listed below:

e 0	e 1	e 2	G(ξ)	$G^{\prime}(\xi)$
1	−(1 + m 2)	m 2	sn(ξ)	cn(ξ)dn(ξ)
1	−(1 + m 2)	m 2	cd(ξ)	−(1 − m 2)sd(ξ)nd(ξ)
1 − m 2	2m 2 − 1	−m 2	cn(ξ)	−sn(ξ)dn(ξ)
m 2 − 1	2 − m 2	−1	dn(ξ)	−m 2 sn(ξ)cn(ξ)
m 2	−(m 2 + 1)	1	ns(ξ)	−ds(ξ)cs(ξ)
m 2	−(m 2 + 1)	1	dc(ξ)	(1 − m 2)nc(ξ)sc(ξ)
−m 2	2m 2 − 1	1 − m 2	nc(ξ)	sc(ξ) dc(ξ)
−1	2 − m 2	m 2 − 1	nd(ξ)	m 2 sd(ξ)cd(ξ)
1 − m 2	2 − m 2	1	cs(ξ)	−ns(ξ)ds(ξ)
1	2 − m 2	1 − m 2	sc(ξ)	nc(ξ)dc(ξ)
1	2m 2 − 1	m 2(m 2 − 1)	sd(ξ)	nd(ξ)cd(ξ)
m 2(m 2 − 1)	2m 2 − 1	1	ds(ξ)	−cs(ξ)ns(ξ)
$\frac{1}{4}$	$\frac{1}{2}(1-2m^2)$	$\frac{1}{4}$	ns(ξ) ± cs(ξ)	−ds(ξ) cs(ξ) ∓ ns(ξ)ds(ξ)
$\frac{1}{4}(1-m^2)$	$\frac{1}{2}(1+m^2)$	$\frac{1}{4}(1-m^2)$	nc(ξ) ± sc(ξ)	sc(ξ) dc(ξ) ± nc(ξ)dc(ξ)
$\frac{m^2}{4}$	$\frac{1}{2}(m^2-2)$	$\frac{1}{4}$	ns(ξ) ± ds(ξ)	−ds(ξ) cs(ξ) ∓ cs(ξ)ns(ξ)
$\frac{m^2}{4}$	$\frac{1}{2}(m^2-2)$	$\frac{m^2}{4}$	${\text{sn}}(\xi)\pm \upiota {\text{cn}}(\xi)$	${\text{dn}}(\xi) {\text{cn}}(\xi)\mp\upiota {\text{sn}}(\xi){\text{dn}}(\xi)$
0	1	−1	sech(ξ)	−sech(ξ)tanh(ξ)
0	1	1	csch(ξ)	−csch(ξ)coth(ξ)
0	−1	1	sec(ξ)	sec(ξ)tan(ξ)
0	0	1	$\frac{1}{\xi}$	$-\frac{1}{\xi^2}$
0	−(1 + m 2)	m 2	sn(ξ)	cn(ξ)dn(ξ)

where 0 < m < 1 is the modulus of Jacobi elliptic functions and $\upiota=\sqrt{-1}$.

Appendix 2

The Jacobi elliptic functions sn(ξ), cn(ξ), dn(ξ), ns(ξ), cs(ξ), ds(ξ), sc(ξ), sd(ξ) degenerate into hyperbolic functions when m → 1 as follows,

$$ {\text{sn}}(\xi)\rightarrow {\text{tanh}}(\xi),\;{\text{cn}}(\xi)\rightarrow {\text{sech}}(\xi),\;{\text{dn}}(\xi)\rightarrow {\text{sech}}(\xi),\;{\text{ns}}(\xi)\rightarrow {\text{coth}}(\xi),\;{\text{cs}}(\xi)\rightarrow {\text{cosech}}(\xi),\;{\text{ds}}(\xi)\rightarrow {\text{cosech}}(\xi),\;{\text{sc}}(\xi)\rightarrow {\text{sinh}}(\xi),\;{\text{sd}}(\xi)\rightarrow {\text{sinh}}(\xi), $$

and into trigonometric functions when m → 0 as follows,

$$ {\text{sn}}(\xi)\rightarrow {\text{sin}}(\xi),\;{\text{cn}}(\xi)\rightarrow {\text{cos}}(\xi),\;{\text{dn}}(\xi)\rightarrow 1,\;{\text{ns}}(\xi)\rightarrow {\text{cosec}}(\xi),\;{\text{cs}}(\xi)\rightarrow {\text{cot}}(\xi),\;{\text{ds}}(\xi)\rightarrow {\text{cosec}}(\xi),\;{\text{sc}}(\xi)\rightarrow {\text{tan}}(\xi),\;{\text{sd}}(\xi)\rightarrow {\text{sin}}(\xi).$$

Appendix 3

$$ {\text{cd}}(\xi)=\frac{cn(\xi)}{{\text{dn}}(\xi)},\;{\text{dc}}(\xi)=\frac{{\text{dn}}(\xi)}{{\text{cn}}(\xi)},\;{\text{nc}}(\xi)=\frac{1}{{\text{cn}}(\xi)},\;{\text{nd}}(\xi)=\frac{1}{{\text{dn}}(\xi)},\;{\text{cs}}(\xi)=\frac{{\text{cn}}(\xi)}{{\text{sn}}(\xi)},\;{\text{sc}}(\xi)=\frac{{\text{sn}}(\xi)}{{\text{cn}}(\xi)},\;{\text{sd}}(\xi)=\frac{{\text{sn}}(\xi)}{{\text{dn}}(\xi)},\;{\text{ds}}(\xi)=\frac{{\text{dn}}(\xi)}{{\text{sn}}(\xi)}.$$