Abstract
The nonlinear diffusion equation arises in many important areas of science and technology and most of the known exact solutions turn out to be similarity solutions. For a general similarity solution involving an arbitrary parameter λ, a new integration procedure is proposed which enables first integrals to be obtained for special values of λ. The best known exact solutions arise from this analysis when the integration constant is taken to be zero and the procedure provides a natural way of deducing other special exact solutions. A new exact solution is obtained for the power law diffusivity of index —4/3 and new first integrals are deduced for a general equation which includes nonlinear cylindrical and spherical symmetrical diffusion and one-dimensional nonlinear diffusion with an inhomogeneous diffusivity. The procedure has given rise to an extensive number of first-order ordinary differential equations which include a wide variety of differing physical situations and which warrant further study either analytically to determine exact integrals or numerically for particular boundary value problems.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
W.F. Ames, Nonlinear partial differential equations in engineering, Academic Press, New York (1965).
D.K. Babu and M.Th. van Genuchten, A similarity solution to a nonlinear diffusion equation of the singular type: a uniformly valid solution by perturbations, Quart. Appl. Math. 37 (1979) 11–21.
G.I. Barenblatt, On some unsteady motions of a liquid and a gas in a porous medium, Prikl. Math. Mech. 16 (1952) 67–78.
G.I. Barenblatt, On a class of exact solutions for the plane one dimensional problem of unsteady filtration into a porous medium, Prikl. Math. Mech. 17 (1953) 739–742.
G.I. Barenblatt and Ya.B. Zel'dovich, On the dipole type solution in the problem of unsteady gas filtration in the polytropic regime, Prikl. Math. Mech. 21 (1957) 718–720.
G.W. Bluman and J.D. Cole, Similarity methods for differential equations, Springer-Verlag, New York (1974).
R.H. Boyer, On some solutions of a nonlinear diffusion equation, J. Math and Phys. (now Studies in Appl. Math.) 40 (1961) 41–45.
W. Brutsaert, Some exact solutions for nonlinear desorptive diffusion, Z. angew. Math. Phys. 33 (1982) 540–546.
J. Crank, The mathematics of diffusion, 2nd edn, Clarendon Press, Oxford (1983).
H. Fujita, The exact pattern of a concentration-dependent diffusion in a semi-infinite medium, Part I, Text. Res. J. 22 (1952) 757–760.
H. Fujita, The exact pattern of a concentration-dependent diffusion in a semi-infinite medium, Part II, Text. Res. J. 22 (1952) 823–827.
H. Fujita, The exact pattern of a concentration-dependent diffusion in a semi-infinite medium, Part III, Text. Res. J. 24 (1954) 234–240.
B.H. Gilding and L.A. Peletier, On a class of similarity solutions of the porous media equation, J. Math. Anal. Appl. 55 (1976) 351–364.
B.H. Gilding and L.A. Peletier, On a class of similarity solutions of the porous media equation II, J. Math. Anal. Appl. 57 (1977) 522–538.
R.E. Grundy, Similarity solutions of the nonlinear diffusion equation, Quart. Appl. Math. 37 (1979) 259–280.
J.M. Hill, Solution of differential equations by means of one-parameter groups, Research Notes in Mathematics 63, Pitman, London (1982).
J.M. Hill and V.G. Hart, The Stefan problem in nonlinear heat conduction, Z. angew. Math. Phys. 37 (1986) 206–229.
J.H. Knight and J.R. Philip, Exact solutions in nonlinear diffusion, J. Engng. Math. 8 (1974) 219–227.
A.A. Lacey, J.R. Ockendon and A.B. Tayler, “Waiting-time” solutions of a nonlinear diffusion equation, SIAM J. Appl. Math. 42 (1982) 1252–1264.
K.E. Lonngren, W.F. Ames, A. Hirose and J. Thomas, Field penetration into a plasma with nonlinear conductivity, Phys. Fluids 17 (1974) 1919–1920.
O.A. Oleinik, A.S. Kalashnikov and Shzou Yui-Lin, The Cauchy problem and boundary problems for equations of the type of unsteady filtration, Izv. Akad. Nauk. SSSR. Ser. Mat. 22 (1958) 667–704. (In Russian.)
J.-Y. Parlange and R.D. Braddock, A note on some similarity solutions of the diffusion equation, Z. angew. Math. Phys. 31 (1980) 653–656.
J.-Y. Parlange, R.D. Braddock and B.T. Chu, First integrals of the diffusion equation; an extension of the Fujita solutions, Soil Sci. Soc. Am. J. 44 (1980) 809–911.
R.E. Prattle, Diffusion from an instantaneous point source with a concentration-dependent coefficient, Quart. J. Mech. Appl. Math. 12 (1959) 407–409.
J.R. Philip, General method of exact solution of the concentration-dependent diffusion equation, Aust. J. Phys. 13 (1960) 1–12.
L.F. Shampire, Concentration-dependent diffusion, Quart. Appl. Math. 30 (1973) 441–452.
L.F. Shampire, Concentration-dependent diffusion II. Singular problems, Quart. Appl. Math. 31 (1973) 287–293.
N.F. Smyth and J.M. Hill, High order nonlinear diffusion, IMA J. Appl. Math. 35 (1988) 73–86.
M.L. Storm, Heat conduction in simple metals, J. Appl. Phys. 22 (1951) 940–951.
B. Tuck, Some explicit solutions to the nonlinear diffusion equation, J. Phys. D, 9 (1976) 1559–1569.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hill, J.M. Similarity solutions for nonlinear diffusion — a new integration procedure. J Eng Math 23, 141–155 (1989). https://doi.org/10.1007/BF00128865
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00128865