The Nonlinear Diffusion Equation: Asymptotic Solutions and Statistical Problems

✍ Scribed by J. M. Burgers (auth.)

• Publisher: Springer Netherlands
• Year: 1974
• Tongue: English
• Leaves: 183
• Edition: 1
• Category: Library

✦ Synopsis

Since the 'Introduction' to the main text gives an account of the way in which the problems treated in the following pages originated, this 'Preface' may be limited to an acknowledgement of the support the work has received. It started during the pe­ riod when I was professor of aero- and hydrodynamics at the Technical University in Delft, Netherlands, and many discussions with colleagues ha ve in:fluenced its devel­ opment. Oftheir names I mention here only that ofH. A. Kramers. Papers No. 1-13 ofthe list given at the end ofthe text were written during that period. Severa! ofthese were attempts to explore ideas which later had to be abandoned, but gradually a line of thought emerged which promised more definite results. This line began to come to the foreground in pa per No. 3 (1939}, while a preliminary formulation ofthe results was given in paper No. 12 (1954}. At that time, however, there still was missing a practica! method for manipulating a certain distribution function of central interest. A six months stay at the Hydrodynamics Laboratories ofthe California Institute of Technology, Pasadena, California (1950-1951}, was supported by a Contract with the Department of the Air F orce, N o. AF 33(038}-17207. A course of lectures was given during this period, which were published in typescript under the title 'On Turbulent Fluid Motion', as Report No. E-34. 1, July 1951, of the Hydrodynamics Laboratory.

✦ Table of Contents

Front Matter....Pages I-X
Introduction....Pages 1-8
The Hopf-Cole Solution of the Nonlinear Diffusion Equation and Its Geometrical Interpretation for the Case of Small Diffusivity....Pages 9-20
Digression on Generalizations of the Geometric Method of Solution. — Solutions of Equation (1.1) for the Domain x > 0 with a Boundary Condition at x = 0....Pages 21-34
Statistical Problems Connected with the Solutions of Chapter I, for v →+0 and t →∞....Pages 35-45
Solutions of the Linear Diffusion Equation with a Boundary Condition Referring to a Parabola....Pages 46-71
Development of the Functions Ψ, E, F in Terms of Exponentials Multiplied By Bessel Functions....Pages 72-83
Evaluation of Integrals and Sums Depending on the Functions ѱ, E, F ....Pages 84-123
Mean Values Connected with the Sawtooth Curve of Figure 5....Pages 124-131
Distribution Functions Referring to Sets of Two Consecutive Arcs....Pages 132-151
Correlation Functions and Distribution Functions Referring to Sets of More Than Two Consecutive Arcs....Pages 152-173
Back Matter....Pages 174-174

✦ Subjects

Analysis