Criteria for Reduction of a Nonlinear Parabolic (1+1) Equation by a Point Transformation to the Classical Heat Equation

ABSTRACT This paper investigates the equivalence of parabolic partial differential equations with two independent variables to the classical one‐dimensional heat equation under point transformations. The analysis is carried out within a framework inspired by Lie's approach to the linearization problem of second‐order ordinary differential equations. The study is organized into two main parts. In the first part, necessary constraints are derived to characterize the class of parabolic partial differential equations that may admit equivalence to the heat equation via point transformations. These constraints impose structural restrictions on the form of the equations under consideration. In the second part, the remaining conditions are examined in order to derive sufficient conditions for equivalence and to determine the corresponding differential equations. The admissible cases arising from the analysis are systematically examined, and the results are illustrated through several representative examples. The findings provide an explicit description of parabolic equations that can be reduced to the classical heat equation.