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.
- L. F. Shampine, Concentration-dependent diffusion. II. Singular problems, Quart. Appl. Math. 31 (1973), 287β293. MR 425369, DOI https://doi.org/10.1090/S0033-569X-1973-0425369-3
K. Reichardt, D. R. Nielsen and J. W. Biggar, Scaling of horizontal infiltration into homogeneous soils, Soil Sc. Soc. Am. Proc. 36, 241β245 (1972)
- R. E. Pattle, Diffusion from an instantaneous point source with a concentration-dependent coefficient, Quart. J. Mech. Appl. Math. 12 (1959), 407β409. MR 114505, DOI https://doi.org/10.1093/qjmam/12.4.407
- D. G. Aronson, Regularity propeties of flows through porous media, SIAM J. Appl. Math. 17 (1969), 461β467. MR 247303, DOI https://doi.org/10.1137/0117045
L. A. Peleiter, Asymptotic behaviour of solutions of the porous medium equations. SIAM 21, 542β551 (1973)
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, 193β211 (1974)
- George H. Pimbley Jr., Wave solutions travelling along quadratic paths for the equation $(\partial /\partial )-(k(u)u_{x})_{x}=0$, Quart. Appl. Math. 35 (1977/78), no. 1, 129β138. MR 440178, DOI https://doi.org/10.1090/S0033-569X-1977-0440178-X
J. R. Philip, Theory of infiltration. Adv. in Hydroscience 5, 215β296 (1969)
J. R. Philip, Recent progress in the solution of the nonlinear diffusion equation, Soil Science 117, 257β264 (1974)
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, 522β525 (1962)
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
H. N. Hayhoe, Study of the relative efficiency of finite difference and Galerkin techniques for modelling soil water transfer, Water Resources Res. 14, 97β102 (1978)
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
W. Brutsaert, More on an approximate solution for the nonlinear diffusion equation. Water Resources Res. 10, 1251β1252 (1974)
W. Brutsaert and R. N. Weisman, Comparison of solutions of a nonlinear diffusion equation. Water Resources Res. 6, 642β644 (1970)
- Max A. Heaslet and Alberta Alksne, Diffusion from a fixed surface with a concentration-dependent coefficient, J. Soc. Indust. Appl. Math. 9 (1961), 584β596. MR 137456
J.-Y. Parlange, Theory of water movement in soils. I. One-dimensional absorption. Soil Science 111, 134β137 (1971)
D. K. Babu, Infiltration analysis and perturbation methods. 1. Absorption with exponential diffusivity. Water Resources Res. 12, 89β93 (1976)
D. K. Babu, Infiltration analysis and perturbation methods. 2. Horizontal absorption. Water Resources Res. 12, 1013β1018 (1976)
D. K. Babu, Infiltration analysis and perturbation methods. 3. Vertical infiltration. Water Resources Res. 12, 1019β1024 (1976)
J.-Y. Parlange and D. K. Babu, A comparison of techniques for solving the diffusion equation with an exponential diffusivity, Water Resources Res. 12, 1317β1318 (1976)
J.-Y. Parlange and D. K. Babu, On solving the nonlinear diffusion equationβa comparison of perturbation, iterative and optimal techniques for an arbitrary diffusion equation, Water Resources Res. 13, 213β214 (1977)
Don Kirkham and W. L. Powers, Advanced soil physics, Wiley Interscience, 1972, pp. 87β95
- Ali Hasan Nayfeh, Perturbation methods, John Wiley & Sons, New York-London-Sydney, 1973. Pure and Applied Mathematics. MR 0404788
- Robert E. OβMalley 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
N. Krylov and N. Bogoliuboff, Introduction to nonlinear mechanics, Princeton University Press, 1947
- Milton Van Dyke, Perturbation methods in fluid mechanics, Annotated edition, The Parabolic Press, Stanford, Calif., 1975. MR 0416240
- Robert E. OβMalley Jr. (ed.), Asymptotic methods and singular perturbations, American Mathematical Society, Providence, R.I., 1976. MR 0497408
L. F. Shampine, Concentration-dependent diffusion. 2. Singular problems. Quart. Appl. Math 31, 287β293 (1973)
K. Reichardt, D. R. Nielsen and J. W. Biggar, Scaling of horizontal infiltration into homogeneous soils, Soil Sc. Soc. Am. Proc. 36, 241β245 (1972)
R. E. Pattle, βDiffusion from an instantaneous point source with concentration-dependent coefficient", Quart. J. Mech. Appl. Math. 12, 407β409 (1959)
D. G. Aronson, Regularity properties of flows through porous media. SIAM 17, 461β467 (1969)
L. A. Peleiter, Asymptotic behaviour of solutions of the porous medium equations. SIAM 21, 542β551 (1973)
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, 193β211 (1974)
G. H. Pimbeley, Jr., Wave solutions travelling along quadratic paths for equation ${U_t} - {\left ( {k\left ( u \right ){u_x}} \right )_x} = 0$, Quart. Appl. Math. 35, 129β138 (1977)
J. R. Philip, Theory of infiltration. Adv. in Hydroscience 5, 215β296 (1969)
J. R. Philip, Recent progress in the solution of the nonlinear diffusion equation, Soil Science 117, 257β264 (1974)
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, 522β525 (1962)
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
H. N. Hayhoe, Study of the relative efficiency of finite difference and Galerkin techniques for modelling soil water transfer, Water Resources Res. 14, 97β102 (1978)
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
W. Brutsaert, More on an approximate solution for the nonlinear diffusion equation. Water Resources Res. 10, 1251β1252 (1974)
W. Brutsaert and R. N. Weisman, Comparison of solutions of a nonlinear diffusion equation. Water Resources Res. 6, 642β644 (1970)
M. A. Heaslet and A. Alksne, Diffusion from a fixed surface with a concentration-dependent coefficient, J. SIAM 9, 584β596 (1961)
J.-Y. Parlange, Theory of water movement in soils. I. One-dimensional absorption. Soil Science 111, 134β137 (1971)
D. K. Babu, Infiltration analysis and perturbation methods. 1. Absorption with exponential diffusivity. Water Resources Res. 12, 89β93 (1976)
D. K. Babu, Infiltration analysis and perturbation methods. 2. Horizontal absorption. Water Resources Res. 12, 1013β1018 (1976)
D. K. Babu, Infiltration analysis and perturbation methods. 3. Vertical infiltration. Water Resources Res. 12, 1019β1024 (1976)
J.-Y. Parlange and D. K. Babu, A comparison of techniques for solving the diffusion equation with an exponential diffusivity, Water Resources Res. 12, 1317β1318 (1976)
J.-Y. Parlange and D. K. Babu, On solving the nonlinear diffusion equationβa comparison of perturbation, iterative and optimal techniques for an arbitrary diffusion equation, Water Resources Res. 13, 213β214 (1977)
Don Kirkham and W. L. Powers, Advanced soil physics, Wiley Interscience, 1972, pp. 87β95
A. H. Nayfeh, Perturbation methods, Wiley Interscience, 1973
R. E. OβMalley, Jr., Singular perturbation analysis for ordinary differential equations (lecture notes), Communications of the Mathematical Institute, Rijksuniversiteit Utrecht-5-1977 (The Netherlands)
N. Krylov and N. Bogoliuboff, Introduction to nonlinear mechanics, Princeton University Press, 1947
M. Van Dyke, Perturbation methods in fluid mechanics, The Parabolic Press, Stanford, California, 1975
Asymptotic methods and singular perturbations, SIAM-AMS Proceedings X, American Math. Society, Providence, Rhode Island, 1976
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
35C99,
65P05
Retrieve articles in all journals
with MSC:
35C99,
65P05
Additional Information
Article copyright:
© Copyright 1979
American Mathematical Society