Equivalence to the Classical Heat Equation Through Reciprocal Transformations

Abstract

This paper investigates the equivalence of parabolic partial differential equations to the classical one-dimensional heat equation using reciprocal transformations. The equations are assumed to be autonomous, and the methodology applied is similar to S. Lie's approach to solving the linearization problem of second-order ordinary differential equations. The research is structured in two main parts. In the first part, necessary constraints on the class of parabolic partial differential equations with two independent variables, which are equivalent to the classical heat equation under a reciprocal transformation, are identified. In the second part, the remaining conditions are examined, and sufficient conditions are derived. The corresponding differential equations are then obtained. All possible cases that arise are thoroughly analyzed, and the theory is illustrated with several examples.