# On the integrability and exact solutions
of the nonlinear diffusion equation with a nonlinear source

**K. Vijayakumar
**

Dept of Mathematics, Panjab University,
Chandigarh 160014, India.

### Abstract:

The generalized diffusion equation with a nonlinear source
term which encompasses the Fisher, Newell-Whitehead and Fitzhugh-Nagumo
equations as particular forms and appears in a wide variety
of physical and engineering applications has been analysed for its
generalized symmetries (isovectors) *via* the isovector approach. This
yields a new and exact solution to the generalized diffusion
equation. Further applications of group theoretic techniques on the
travelling wave reductions of the Fisher, Newell-Whitehead and
Fitzhugh-Nagumo equations result in integrability conditions and Lie
vector fields for these equations. The Lie group of transformations
obtained from the exponential vector fields reduces these equations
in generalized form to a standard second-order differential
equation of nonlinear type, which for particular cases become the
Weierstrass and Jacobi elliptic equations. A particular solution to the
generalized case yields the exact solutions that have been
obtained through different techniques. The group-theoretic integrability
relations of the Fisher and Newell-Whitehead equations have been
cross-checked through Painlevé analysis, which yields a new solution
to the Fisher equation in a complex-valued function form.

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© Copyright 1998, Australian Mathematical Society
*TeXAdel Scientific Publishing*

*26/04/2000*