Multiple solutions of diffusion equations and hydrodynamics

Background. The main goal of the paper is to construct a new class of solutions of the two-dimensional diffusion equation (heat conductivity), which are multivalued functions. New solutions are associated with quasilinear first-order equations that have a hydrodynamic analogy in the class of flows of an ideal fluid. We compare the classical hydrodynamic analogy of the diffusion process with the flow of a viscous fluid and a new analogy with the flow of an ideal fluid. The general role of branch points in the identification of uniquely determined solutions is considered. New solutions of diffusion equations are constructed. Materials and methods. The method of investigation is the analysis of solutions of the diffusion equations written in coordinates on the complex plane. Results. We found general formulas for calculating the exact multivalued solutions of the two-dimensional diffusion equation based on their connection with quasilinear first-order equations. A new hydrodynamic analogy of these solutions is established with the flows of an ideal liquid in the plane. Specific examples of solutions for several important practical problems are given. Conclusions. Developed in this paper, it is shown that the diffusion (thermal conductivity) equations have many-valued functions as solutions, the number of sheets of which is determined by the initial conditions. The developed method gives a new approach to the constancy of the solutions of the diffusion equations both classical and in the class of multivalued functions.