Symmetry analysis of the two-dimensional diffusion equation with a source term

Abstract

A symmetry analysis is performed on a (2 + 1)-dimensional linear diffusion equation with a nonlinear source term involving the dependent variable and its spatial derivatives. In the first part of the paper, we use the classical method to classify source terms where the original equation admits a nontrivial symmetry. In the second part of the paper, we use the nonclassical method and show that we simply recover the classical symmetries.

Similarity Transforming Techniques of Partial Differential Equations and its Applications: A review

“…Thus, an interpretation of the translational parameter m, which is given as u(x,y) = x-my on the x and y components of the velocity u. A symmetry analysis was carried out by (Arrigo et al, 2007) on a (2+1)-dimensional linear diffusion equation that had a source term. The purpose of this research was to identify source words that are capable of admitting nontrivial symmetries.…”

“…The authors uncovered a collection of nonclassical symmetries and demonstrated that determining equations may be built by only enforcing compatibility between the initial equation and a related first-order quasi-linear PDE. Symmetry analysis of the twodimensional diffusion equation with a source term was performed by (Arrigo et al, 2007). The authors made advantage of classical and nonclassical symmetries.…”

First-order compatibility for a (2 + 1)-dimensional diffusion equation

“…was conducted by Arrigo, Suazo and Sule [2], where it was shown that a variety of source terms exist that admit a nontrivial symmetry (those symmetries not obtained by inspection). While both the classical and nonclassical symmetry methods have had tremendous success when applied to a wide variety of physically important nonlinear differential equations, there exist exact solutions to partial differential equations that cannot be explained using symmetry analysis.…”