A similarity solution to a nonlinear diffusion equation of the singular type: a uniformly valid solution by perturbations

Authors:
D. K. Babu and M. Th. van Genuchten

Journal:
Quart. Appl. Math. **37** (1979), 11-21

MSC:
Primary 35C99; Secondary 65P05

DOI:
https://doi.org/10.1090/qam/530666

MathSciNet review:
530666

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The one-dimensional nonlinear diffusion equation is solved by a perturbation technique. It is assumed that the diffusivity varies as a nonnegative power of the concentration, while the concentration at the supply surface varies as another power of time. The resulting similarity solution that has been derived via a perturbation scheme remains valid for all times and all distances. Explicit series formulae are also derived for the location of the concentration front. Since diffusivity vanishes at zero concentration, the study here pertains to a singular problem.

**[1]**L. F. Shampine,*Concentration-dependent diffusion. II. Singular problems*, Quart. Appl. Math.**31**(1973), 287–293. MR**0425369**, https://doi.org/10.1090/S0033-569X-1973-0425369-3**[2]**K. Reichardt, D. R. Nielsen and J. W. Biggar,*Scaling of horizontal infiltration into homogeneous soils*, Soil Sc. Soc. Am. Proc.**36**, 241-245 (1972)**[3]**R. E. Pattle,*Diffusion from an instantaneous point source with a concentration-dependent coefficient*, Quart. J. Mech. Appl. Math.**12**(1959), 407–409. MR**0114505**, https://doi.org/10.1093/qjmam/12.4.407**[4]**D. G. Aronson,*Regularity propeties of flows through porous media*, SIAM J. Appl. Math.**17**(1969), 461–467. MR**0247303**, https://doi.org/10.1137/0117045**[5]**L. A. Peleiter,*Asymptotic behaviour of solutions of the porous medium equations*. SIAM**21**, 542-551 (1973)**[6]**C. Atkinson and C. W. Jones,*Similarity solution in some nonlinear diffusion problems and boundary layer flow of pseudo plastic fluid*. Quart. J. Mech. Appl. Math.**27**, 193-211 (1974)**[7]**George H. Pimbley Jr.,*Wave solutions travelling along quadratic paths for the equation (∂/∂)-(𝑘(𝑢)𝑢ₓ)ₓ=0*, Quart. Appl. Math.**35**(1977/78), no. 1, 129–138. MR**0440178**, https://doi.org/10.1090/S0033-569X-1977-0440178-X**[8]**J. R. Philip,*Theory of infiltration*. Adv. in Hydroscience**5**, 215-296 (1969)**[9]**J. R. Philip,*Recent progress in the solution of the nonlinear diffusion equation*, Soil Science**117**, 257-264 (1974)**[10]**G. Ashcroft, D. R. Marsh, D. D. Evans and L. Boersma,*Numerical methods for solving the diffusion equation. I. Horizontal flow in semi-infinite media*, Soil Sci. Soc. Am. Proc.**26**, 522-525 (1962)**[11]**S. P. Neuman,*Finite element computer programs for flow in saturated-unsaturated media*, second annual report no. A10-SWC-77, Hydr. Eng. Lab. Technion, Haifa, 1972**[12]**H. N. Hayhoe,*Study of the relative efficiency of finite difference and Galerkin techniques for modelling soil water transfer*, Water Resources Res.**14**, 97-102 (1978)**[13]**M. Th. van Genuchten,*Numerical solutions of the one-dimensional saturated-unsaturated flow equation*. Water Resources Program, Research Report 78-WR-9, Dept. of Civil Engineering, Princeton University, 1978**[14a]**W. Brutsaert,*More on an approximate solution for the nonlinear diffusion equation*. Water Resources Res.**10**, 1251-1252 (1974)**[14b]**W. Brutsaert and R. N. Weisman,*Comparison of solutions of a nonlinear diffusion equation*. Water Resources Res.**6**, 642-644 (1970)**[15]**Max A. Heaslet and Alberta Alksne,*Diffusion from a fixed surface with a concentration-dependent coefficient*, J. Soc. Indust. Appl. Math.**9**(1961), 584–596. MR**0137456****[16]**J.-Y. Parlange,*Theory of water movement in soils. I. One-dimensional absorption*. Soil Science**111**, 134-137 (1971)**[17]**D. K. Babu,*Infiltration analysis and perturbation methods. 1. Absorption with exponential diffusivity*. Water Resources Res.**12**, 89-93 (1976)**[18]**D. K. Babu,*Infiltration analysis and perturbation methods. 2. Horizontal absorption*. Water Resources Res.**12**, 1013-1018 (1976)**[19]**D. K. Babu,*Infiltration analysis and perturbation methods. 3. Vertical infiltration*. Water Resources Res.**12**, 1019-1024 (1976)**[20]**J.-Y. Parlange and D. K. Babu,*A comparison of techniques for solving the diffusion equation with an exponential diffusivity*, Water Resources Res.**12**, 1317-1318 (1976)**[21]**J.-Y. Parlange and D. K. Babu,*On solving the nonlinear diffusion equation--a comparison of perturbation, iterative and optimal techniques for an arbitrary diffusion equation*, Water Resources Res.**13**, 213-214 (1977)**[22]**Don Kirkham and W. L. Powers,*Advanced soil physics*, Wiley Interscience, 1972, pp. 87-95**[23]**Ali Hasan Nayfeh,*Perturbation methods*, John Wiley & Sons, New York-London-Sydney, 1973. Pure and Applied Mathematics. MR**0404788****[24]**Robert E. O’Malley Jr.,*Singular perturbation analysis for ordinary differential equations*, Communications of the Mathematical Institute, Rijksuniversiteit Utrecht, vol. 5, Rijksuniversiteit Utrecht, Mathematical Institute, Utrecht, 1977. MR**590621****[25]**N. Krylov and N. Bogoliuboff,*Introduction to nonlinear mechanics*, Princeton University Press, 1947**[26]**Milton Van Dyke,*Perturbation methods in fluid mechanics*, Annotated edition, The Parabolic Press, Stanford, Calif., 1975. MR**0416240****[27]**Robert E. O’Malley Jr. (ed.),*Asymptotic methods and singular perturbations*, American Mathematical Society, Providence, R.I., 1976. MR**0497408**

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
35C99,
65P05

Retrieve articles in all journals with MSC: 35C99, 65P05

Additional Information

DOI:
https://doi.org/10.1090/qam/530666

Article copyright:
© Copyright 1979
American Mathematical Society