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Reducing parabolic partial differential equations to canonical form

Published online by Cambridge University Press:  26 September 2008

J. F. Harper
J. F. Harper
Affiliation:
Mathematics Department, Victoria University, Wellington, New Zealand (e-mail: John.Harper@vuw.ac.nz)
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Abstract

A simple method of reducing a parabolic partial differential equation to canonical form if it has only one term involving second derivatives is the following: find the general solution of the first-order equation obtained by ignoring that term and then seek a solution of the original equation which is a function of one more independent variable. Special cases of the method have been given before, but are not well known. Applications occur in fluid mechanics and the theory of finance, where the Black-Scholes equation yields to the method, and where the variable corresponding to time appears to run backwards, but there is an information-theoretic reason why it should.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 5 , Issue 2 , June 1994 , pp. 159 - 164
Copyright
Copyright © Cambridge University Press 1994

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References

Abramowitz, M. & Stegun, I. 1965 Handbook of Mathematical Functions, Dover, New York.Google Scholar
Batishchev, V. A. 1975 Asymptotics of axially symmetric fluid flows with a free boundary in the case of dwindling viscosity. Zh. Prikl. Mekh. Tekh. Fiz. (3), 101–109. (English translation J. Appl. Mech. Tech. Phys. 16 (3), 394–400.)Google Scholar
Batishchev, V. A. & Srubshchik, L. S. 1971 Diffusion of a spherical hill vortex with vanishing viscosity. Dokl. Akad. Nauk SSSR, 197, 1038–1041. (English Translation Soviet Phys. - Dokl., 16, 286–288.)Google Scholar
Black, F. & Scholes, M. 1973 The pricing of options and corporate liabilities. J. Polit. Econ., 81, 637–654.CrossRefGoogle Scholar
Carslaw, H. S. & Jaeger, J. C. 1959 The Conduction of Heat in Solids, 2nd ed.Oxford University Press.Google Scholar
Van Dommelen, L. L. & Shen, S. F. 1985 The flow at a rear stagnation point is eventually determined by exponentially small values of the velocity. J. Fluid Mech., 157, 1–16.CrossRefGoogle Scholar
Goldstein, S. (ed.) 1965 Modern Developments in Fluid Dynamics, Vol. I, Dover, New York.Google Scholar
Gradshteyn, I. S. & Ryzhik, I. M. 1980 Table of Integrals, Series and Products, 4th ed., Academic Press, New York.Google Scholar
Harper, J. F. 1963 Laminar flows at high Reynolds numbers. PhD thesis, University of Cambridge, UK.Google Scholar
Harper, J. F. 1972 The motion of bubbles and drops through liquids. Adv. Appl. Mech. 12, 59–129.CrossRefGoogle Scholar
Hodgson, A., Okunev, J., Tamir, M. & Willett, R. 1992 Some theoretical considerations applying to the modelling of the earnings/returns relationship in empirical research. Discussion Paper 17, Money and Finance Group, Victoria University of Wellington, NZ.Google Scholar
Levich, V. G. 1962 Physicochemical Hydrodynamics, Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Piaggio, H. T. H. 1928 An Elementary Treatise on Differential Equations and their Applications, Bell, London.Google Scholar
Ruckenstein, E. 1971 Transport equations in transfer coordinates. Chem. Eng. Sci., 26, 1795–1802.CrossRefGoogle Scholar
Srubshchik, L. S. & Yudovich, V. I. 1971 Asymptotic behaviour of first-order discontinuities of fluid flows with vanishing viscosity. Dolk. Akad. Nauk SSSR, 199, 563–566. (English translation Soviet Phys. - Dokl., 16, 1972, 530–533.)Google Scholar