THE NONLINEAR DIFFUSION EQUATION

THE NONLINEAR DIFFUSION EQUATION

THE NONLINEAR DIFFUSION EQUATION Asymptotic Solutions and Statistica! Problems by J. M. BURGERS Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, Md., U.S.A. SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

First published under the title Statistica/ Problems Connected with Asymptotic Solutions of the One-Dimensional Non/inear Diffusion Equation in 1973 by the Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, Lecture Series No. 52 Library ofcongress Catalog Card Number 74-81936 ISBN 978-94-010-1747-3 ISBN 978-94-010-1745-9 (ebook) DOI 10.1007/978-94-010-1745-9 Ali Rights Reserved Copyright 1974 by Springer Science+Business Media Dordrecht Originally published by D. Reidel Publishing Company, Dordrecht, Holland in 1974. Softcover reprint of the hardcover 1 st Edition 1974 No part of this book may be reproduced in any form, by print, photoprint, microfilm, or any other means, without written permission from the publisher

TABLE OF CONTENTS PREFACE IX INTRODUCTION CHAPTER I 1 THE HOPF-COLE SOLUTION OF THE NONLINEAR DIFFUSION EQUATION AND ITS GEOMETRICAL INTERPRETA TION FOR THE CASE OF SMALL DIFFUSIVITY 9 1. Basic Solution 9 2. Geometric Interpretation of the Solution. - Possibility of Multiple Roots 11 3. Steep Fronts or 'Shock Waves'. General Form of the Solution for Large Values of t 16 4. Alternative Derivation of the Approximate Solution of Equation (1.1) in the Neighborhood of a Steep Front. Additional Observations 18 CHAPTER II 1 DIGRESSION ON GENERALIZATIONS OF THE GEOMETRIC METHOD OF SOLUTION. - SOLUTIONS OF EQUATION (1.1) FOR THE DOMAIN x>o WITH A BOUNDARY CONDITIONAT x=o 21 5. Application ofthe Geometric Method ofsolution to Slightly More General Equations 21 6. Solutions of the Nonlinear Diffusion Equation with a Boundary Conditionat x=o. Two Preliminary Cases 24 7. Generalization of Solution (6.6) 26 8. Extension to a Continuous Function 31 CHAPTER III 1 STA TISTICAL PROBLEMS CONNECTED WITH THE SOL U- TIONS OF CHAPTER I, FOR V-> +0 AND t--+00 35 9. Ensemble of Initial Data-Curves. The Parameter J 35 10. Statistics of a Chain of Parabolic Arcs 38 11. Transformation of the Statistica! Problem 40 12. Additional Observations 44 CHAPTER IV 1 SOLUTIONS OF THE LINEAR DIFFUSION EQUATION WITH A BOUNDARY CONDITION REFERRING TO A PARABOLA 13. Basic Functions 46 46

VI TABLE OF CONTENTS 14. Application of Green's Theorem 50 15. Solution of Equation (13.1) Assuming Prescribed Values on the Boundary 51 16. Change of the Form of the Boundary 52 17. Transformation to a Coordinate System with Inclined x-axis 53 18. Extension ofequation (16.2} 54 19. Application of the Results of Sections 16 and 17 to a Parabolic Boundary 55 20. Integro-Differential Equation for 8(z) 57 21. Series Development for 8(z) 58 22. The Function E(x) 60 23. Application to a Parabolic Boundary 62 24. Effect of a Change of the Boundary upon E(x) 63 25. The Integral P 0 65 26. The Normalization Condition (10.10) 69 27. A Theorem Connected with the Integral P 1 for a Parabolic Boundary 70 CHAPTER V 1 DEVELOPMENT OF THE FUNCTIONS '1', E, F IN TERMS OF EXPONENTIALS MUL TIPLIED BY BESSEL FUNCTIONS 72 28. The Transformation of the Differential Equation and its Solution 72 29. Properties ofthe Functions q 0 (v), qn(v) 74 30. The Green's Function Connected with (28.6} 75 31. Summation Formulas 77 32. Series for the Functions 1/J, '1', e, cp 80 33. Expression for E(x) 82 CHAPTER VI 1 EVAL UATION OF INTEGRALS AND SUMS DEPENDING ON THE FUNCTIONS '1', E, F 84 34. Basic Definitions 84 35. Initial Integrations and Summations 86 36. Series for the Functions Rm and /m 90 37. More Summation Formulas 94 38. Continuation from Section 35 97 39. Results Obtained for the Integrals P 0, P 1,... P* 102 40. Results Obtained for Some Additional Integrals 108 List oflntegrals 110 APPENDIX TO CHAPTER VI 112 41. Additional Summation F ormulas 112

TABLE OF CONTENTS VII CHAPTER VII 1 MEAN VALUES CONNECTED WITH THE SA WTOOTH CURVE OF FIGURE 5 124 42. Recapitulation ofprevious Results 124 43. Mean Values Referring to a Single Shock 125 44. Other Mean Va1ues Which can be Expressed in a Simple Way 128 45. The Distribution Function for the Arc Lengths ~k 130 CHAPTER VIII 1 DISTRIBUTION FUNCTIONS REFERRING TO SETS OF TWO CONSECUTIVE ARCS 132 46. Integrals with Two Successive 'l'-functions 132 47. Weighting Function for the Wave1engths Ak 135 48. Alternative Expressions for the Functions K* and L* 138 49. Relations Invo1ving the Quantity clj 0 139 50. The Value of clj 0 141 51. Evaluation of< ~k)* for A-+ O 143 52. Calcu1ation of< ~k~k+ 1 )* for A-+ O 146 53. Calculation of <a~k+ 1)* for A-+ O 148 54. Application of Expressions (48.5) and (48.6) for K* and L* 150 CHAPTER IX 1 CORRELA TION FUNCTIONS AND DISTRIBUTION FUNC- TIONS REFERRING TO SETS OF MORE THAN TWO CONSECUTIVE ARCS 55. Calculation of u 1 u 2 152 56. Calculation of uiu 2 157 57. Behavior for Small Values of z 159 58. Behavior of the Sums Occurring in Equation (55.18) for Indefinitely Increasing Values of z 161 59. Behavior of the Sums Occurring in Equation (56.10) or (56.12) for Indefinitely Increasing Va1ues of z 165 60. Integrals Referring to a Series of Consecutive Arcs 167 61. Continuation 169 62. The Integra1s for the Functions W 0 ;(z),..., W 3 ;(z), lntroduced in Sections 55 and 56 172 LIST OF PAPERS BY THE AUTHOR, CONNECTED WITH THE PRESENT MONOGRAPH 174

PREFACE 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 period 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 development. 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. Main progress carne after my return to Delft, when the Hopf-Cole transformation of the nonlinear equation into a linear one had opened a new way of attack. Since the end of 1955 until1971 the work was carried on at the Institute for Fluid Dynamics and Applied Mathematics ofthe University ofmaryland.lt was partially supported by the Air Force Oftice of Scientific Research, as a subject coming under Contracts AF-18(600}993, AF-49(638}401, F-44620-67-C-0023, and Grants AFOSR- 141-63, AFOSR-141-64, AFOSR-141-65, and AFOSR-141-66. For ali support and for the general interest of many colleagues, deep gratitude is expressed. At one stage ofthe work it was surmised that the integrals considered in Chapter VI below might be related to Wiener integrals. This was discussed in the beginning of 1963 with Dr F. Pollaczek, at that time a visitor at the Institute for Fluid Dynamics and Applied Mathematics; and also with Dr M. Kac of the Rockefeller University in New York. Dr Kac pointed out that an important property of Wiener integrals is that they lead to a partial differential equation, a subject which is treated in his book, Probability and Related Topics in Physical Sciences, Interscience Publishers, New York, 1959, Chapter IV, pp. 161-182 (in particular p. 171}.1t appeared that the corresponding differential equation for my case [Equation (28.3} below] could be obtained

X PREFACE by a direct transformation ofthe original equation, so that there was no need to give further attention to Wiener integrals. This is developed in Chapters V and VI of the monograph. It furnished the hasis for the evaluation of ali integrals connected with the central distribution function. The results of the calculations ha ve led to definite numbers, without adjustable constants. Numerica! calculations sometimes were desirable in order to obtain certain transcendental numbers. Mr Y.-K. Hsu and Mr S.-K. Oh gave help with the calcu Iation ofthe coefficients ofthe series (21.3) for the function <f; (z) and for the evaluation of E(O) according to Equation (23.7).- Mr David Wilson made calculations for the numbers m 1 (Equation (38.17a)) and m 3 (Equation (38.17b)). Special gratitude is due to the secretaries who have typed out, often repeatedly, the extensive manuscript notes with their long and complicated equations. I mention in particular Miss Helga Lincke for her work in 1964-1965; and Mrs Irene Huvos for the still more extensive!ater work and her never ending patience, during the years 1970-1974. To Professor Bruce Kellogg of the Institute I am indebted for much he1p and care given when the Institute decided to distribute the typewritten text as No. 52 of the Institute's 'Lecture Series', in November 1973. I consider it a great honor that D. Reidel Publishing Company in Dordrecht, The Netherlands, has decided to publish this monograph in the form of a book. I am well aware that the method followed here in treating asymptotic solutions of the non1inear diffusion equation and their statistics is a specialized one. Many approaches have been explored by a variety of authors starting from other points of view. It is impossib1e to extend the present monograph in such a way that an adequate digest of the existing literature can be given. This has therefore been left aside. A survey of published so1utions until 1972 has been presented by E. R. Benton and G. W. Platzmann; see the reference in footnote* at the beginning of Chapter 1. New papers are continually forthcoming. A final remark: whi1e great care has been taken to check the numerous equations by comparing them with the original material, a full guarantee against misprints cannot be given and the clemency of the reader is invoked. J.M.B.