Abstract
Abstract In this article, the authors study analytic and numerical solutions of nonlinear diffusion equations of Fisher’s type with the help of classical Lie symmetry method. Lie symmetries are used to reduce the equations into ordinary differential equations (ODEs). Lie group classification with respect to time dependent coefficient and optimal system of one-dimensional sub-algebras is obtained. Then sub-algebras are used to construct symmetry reduction and analytic solutions. Finally, numerical solutions of nonlinear diffusion equations are obtained by using one of the differential quadrature methods.
Highlights
IntroductionWhen g(u) = αu( – u), ( ) stands for the Fisher equation, put forward by Fisher [ ] as a model for the spatial and temporal propagation of a viral gene in an infinite medium
Consider the nonlinear diffusion equation ∂u ∂ u∂t = D ∂x + g(u). ( )When g(u) = αu( – u), ( ) stands for the Fisher equation, put forward by Fisher [ ] as a model for the spatial and temporal propagation of a viral gene in an infinite medium
4 Lie classical analysis for nonlinear diffusion Fisher’s type equations we study the infinitesimal transformations and reductions by onedimensional sub-algebras of ( )-( ) by applying the classical Lie symmetry method [ ] one by one
Summary
When g(u) = αu( – u), ( ) stands for the Fisher equation, put forward by Fisher [ ] as a model for the spatial and temporal propagation of a viral gene in an infinite medium. This equation expresses a one-dimensional reaction-diffusion model for the evolution of the infected population. Applications of traveling wave fronts appear in biology, chemistry, and medicine [ ]. Such wave fronts were studied by Fisher for the first time in s by considering ( )
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