Lie symmetry classification and group invariant solutions of generalized radial heat equation with nonlinear reaction source

Abstract

This work presents a Lie symmetry classification of a generalized nonlinear heat equation with a reaction source term in radial geometry. The model involves three arbitrary constitutive functions that represent thermal capacity, thermal conductivity, and nonlinear heat generation or absorption. Using the classical Lie invariance criterion, the determining equations for point symmetries are derived and simplified through suitable transformations involving the ratios of the constitutive functions. The classification identifies several admissible subclasses for which the principal symmetry algebra is extended, including power-law and logarithmic branches associated with special values of the radial parameter. For these cases, the admitted Lie algebras, commutator structures, and optimal systems of one-dimensional subalgebras are obtained. The corresponding similarity reductions are constructed, reducing the governing partial differential equation to nonlinear ordinary differential equations. Some exact group-invariant solutions are also derived for special parameter choices. The results show that the inclusion of the nonlinear source term significantly enriches the symmetry structure compared with the source-free radial heat equation.