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Confluent hypergeometric solutions of heat conduction equation. (English) Zbl 0707.76087

Summary: The unsteady one-dimensional heat conduction equation is transformed into an ordinary differential equation, called Kummer’s equation, unifiedly in the linear, cylindrical and spherical coordinate systems. Kummer’s equation is solved in terms of the confluent hypergeometric functions and thus the similarity solutions are obtained. These solutions exist on the conditions that boundaries lie at the origin and infinity, or otherwise move with their positions proportional to the square root of time, and that the strength of heat source is a power function of time. For the already known similarity solutions expressed in terms of other functions, the corresponding confluent hypergeometric expressions are shown. If the conduction similarity solutions are applied to solve moving boundary problems with phase change, only one solution exists in each coordinate system.

MSC:

76R99 Diffusion and convection
80A20 Heat and mass transfer, heat flow (MSC2010)
35Q35 PDEs in connection with fluid mechanics
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