Abstract
Although the nonlinear diffusion equation has been extensively studied and there exists substantial literature in many diverse areas of science and technology, the number of exact concentration profiles is nevertheless limited. In a recent article in this journal (Hill [1]) a brief review of known exact results is given, as well as an elementary integration procedure which appears to be a general device for obtaining integrals associated with similarity solutions. This paper extends the results given in [1] and for particular power law diffusivitiesc m (such asm = −/12, −1, −/32 and −2) presents a number of new exact solutions obtained by fully integrating the ordinary differential equations derived in [1]. In addition new results are found for a general nonlinear diffusion equation which includes one-dimensional diffusion with an inhomogenouus and nonlinear diffusivityc mxmas well as symmetric nonlinear diffusion in cylinders and spheres. Moreover by a separate and ad-hoc procedure a new solution is obtained of the travelling wave type but with a variable wave speed. Some of the new exact solutions obtained for one-dimensional nonlinear diffusion with power law diffusivitiesc mare illustrated graphically.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.M. Hill, Similarity solutions for nonlinear diffusion — a new integration procedure,J. Engng. Math. 23 (1989) 141–155.
C.W. Jones, On reducible nonlinear differential equations occurring in mechanics,Proc. Roy. Soc. London Ser. A 217 (1953) 327–343.
A.A. Lacey, J.R. Ockendon and A.B. Tayler, “Waiting-time” solutions of n nonlinear diffusion equation,SIAM J. Appl. Math. 42 (1982) 1252–1264.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hill, D.L., Hill, J.M. Similarity solutions for nonlinear diffusion — further exact solutions. J Eng Math 24, 109–124 (1990). https://doi.org/10.1007/BF00129869
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00129869