Symmetry analysis and group-invariant solutions to inhomogeneous nonlinear diffusion equation

2015 ◽  
Vol 28 (1-3) ◽  
pp. 50-65 ◽  
Author(s):  
Wei Feng ◽  
Lina Ji
Keyword(s):  
Diffusion Equation ◽  
Nonlinear Diffusion ◽  
Nonlinear Diffusion Equation ◽  
Symmetry Analysis ◽  
Invariant Solutions ◽  
Group Invariant ◽  
Group Invariant Solutions

Time-fractional inhomogeneous nonlinear diffusion equation: Symmetries, conservation laws, invariant subspaces, and exact solutions

2018 ◽  
Vol 32 (32) ◽  
pp. 1850401 ◽  
Author(s):  
Wei Feng ◽  
Songlin Zhao
Keyword(s):  
Diffusion Equation ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Nonlinear Diffusion ◽  
Fractional Derivatives ◽  
Nonlinear Diffusion Equation ◽  
Group Invariant ◽  
Liouville Fractional Derivative ◽  
Group Invariant Solutions ◽  
Dimensional Dynamical System

In this paper, a class of time-fractional inhomogeneous nonlinear diffusion equation (tFINDE) with Riemann–Liouville fractional derivative is studied. All point symmetries admitted by this equation are derived. The optimal system of one-dimensional subalgebras is classified to perform the symmetry reductions. It is shown that the tFINDE can be reduced to fractional ordinary differential equations (FODEs), including Erdélyi–Kober fractional derivatives. As the results, some explicit group-invariant solutions are obtained. Through nonlinear self-adjointness, all conservation laws admitted by tFINDE arising from these point symmetry groups are listed. The method of invariant subspace is also applied to reduce the tFINDE to a two-dimensional dynamical system (DS). The admitted point symmetries of DS are used to derive the exact solutions of DS, which determine the exact solutions of the original tFINDE.

scholarly journals Lie symmetry analysis and group invariant solutions of the nonlinear Helmholtz equation

2018 ◽  
Vol 331 ◽  
pp. 457-472 ◽  
Author(s):  
K. Sakkaravarthi ◽  
A.G. Johnpillai ◽  
A. Durga Devi ◽  
T. Kanna ◽  
M. Lakshmanan
Keyword(s):  
Helmholtz Equation ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Invariant Solutions ◽  
Lie Symmetry Analysis ◽  
Group Invariant ◽  
Group Invariant Solutions
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