FUNCTIONS AND INTEGRALS CONNECTED WITH SOLUTIONS OF THE DIFFUSION OR HEAT FLOW EQUATION.
MARYLAND UNIV COLLEGE PARK INST FOR FLUID DYNAMICS AND APPLIED MATHEMATICS
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The paper contains an investigation of the properties of certain functions connected with solutions of the ordinary diffusion or heat flow equation. In the first part of the paper the functions of interest are defined, and various properties of these functions are derived from a study of the partial differential equation and the behavior of its solutions when the boundary curve is subjected to variations. An integro-differential equation is obtained for one of the main functions, from which a series development is derived. - In the second part the boundary curve is always a parabola. A transformation of the field makes it possible to change this boundary into a straight line, the differential equation obtaining a slightly more complicated form. The new equation is solved by an application of Bessel functions of the order 13. New types of series development are obtained for the functions defined in the first part of the paper. - In the third part certain multiple integrals are evaluated, depending upon these functions defined. In an Appendix the relation of these investigations is indicated with previous work on statistical problems connected with the solution of a nonlinear diffusion equation. References are given to the main earlier publications, and a comparison is made of the notations used. Author
- Physical Chemistry
- Numerical Mathematics
- Fluid Mechanics