Similarity solutions of the nonlinear diffusion equation
Author:
R. E. Grundy
Journal:
Quart. Appl. Math. 37 (1979), 259-280
MSC:
Primary 35K60
DOI:
https://doi.org/10.1090/qam/548987
MathSciNet review:
548987
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Abstract: The paper considers similarity solutions of the nonlinear diffusion equation of the form ${t^\alpha }f\left ( \eta \right )$ where $\eta = r{t^{ - \delta }}$ or $\exp \left ( {\alpha t} \right )f\left ( \eta \right )$ where $\eta = r\exp \left ( { - \delta t} \right )$. The novel feature of the paper is that the second-order differential equation for $f$ is reduced to a system of first-order equations and a phase plane analysis of one member of the system can be made. In this way we may discuss the existence and uniqueness of all the solutions for $f\left ( n \right )$. Restricting the discussion to plane geometry, we list all the continuous solutions to the basic problem on $0 \le \eta \le \infty$ with $f\left ( 0 \right ) = U \ge 0$ and $f\left ( \infty \right ) = 0$. Solutions of previous authors are identified as special cases.
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W. F. Ames, Similarity for the nonlinear diffusion equation, I & EC Fundamentals 4, 72–76 (1965)
G. I. Barenblatt, On some unsteady motions of a liquid and a gas in porous medium, Prikl. Mat. Meh. 16, 67–78 (1952)
G. I. Barenblatt, On a class of exact solutions of the plane one-dimensional problem of unsteady filtration into a porous medium, Prikl. Mat. Meh. 17, 739–742 (1953)
G. I. Barenblatt and Ya. B. Zel’dovich, On the dipole-type solution in problems of unsteady gas filtration in the polytropic regime, Prikl. Mat. Meh. 21, 718–720 (1957)
R. H. Boyer, On some solutions of a nonlinear diffusion equation, J. Math. Phys. 40, 41–45 (1961)
B. H. Gilding and L. A. Peletier, On a class of similarity solutions of the porous media equation, J. Math. Anal. Appl. 55, 351–364 (1976)
M. A. Heaslet and A. Alksne, Diffusion from a fixed surface with a concentration-dependent coefficient, J. Soc. Ind. Appl. Math. 9, 584–596 (1961)
C. W. Jones, On reducible nonlinear differential equations occurring in mechanics, Proc. Roy. Soc. A217, 327–343 (1953)
R. E. Marshak, Effect of radiation on shock wave behavior, Phys. Fluids 1, 24–29 (1958)
R. E. Pattle, Diffusion from an instantaneous point source with concentration-dependent coefficient, Quart. J. Mech. Appl. Math. 12, 407–409 (1959)
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© Copyright 1979
American Mathematical Society