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On a nonlinear reaction-diffusion boundary-value problem: application of a Lie-Bäcklund symmetry

Published online by Cambridge University Press:  17 February 2009

Philip Broadbridge  and
Colin Rogers
Philip Broadbridge
Affiliation:
Department of Mathematics, University of Wollongong, Wollongong, NSW.
Colin Rogers
Affiliation:
Department of Mathematical Sciences, Loughborough University of Technology, U.K.
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Abstract

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By a systematic search for Lie-Bäetcklund symmetries, a class of linearisable reaction-diffusion equations is obtained that has, as a canonical form, ut = u2uxx + 2u2. One such nonlinear equation is θt = ∂x[a(b - θ)-2 θx] - ma(b -θ)-2 θx - q exp(-mx). This represents an extension of Fokas-Yortsos-Rosen equation (q = 0) to incorporate a reaction term. It is relevant to the modelling of unsaturated flow in a soil with a volumetric extraction mechanism, such as a web of plant roots. Here, a reciprocal transformation is used to solve a nonlinear boundary-value problem for transient flow into a finite layer of a soil subject to a constant flux boundary condition to compensate for such water extraction.

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 34 , Issue 3 , January 1993 , pp. 318 - 332
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Bluman, G. and Kumei, S., “On the remarkable nonlinear diffusion equation (∂/∂x)[a(u+b)-2(∂u/∂x)]-(∂u/∂t) = 0”, J. Math. Phys. 21 (1980) 1019–1023.CrossRefGoogle Scholar
[2]Britton, N. F., Reaction-diffusion equations and their applications to biology (Academic Press, London, 1986).Google Scholar
[3]Broadbridge, P., “Selection of solvable nonlinear evolution equations by systematic searches for Lie-Bäcklund symmetries”, in Nonlinear evolution equations and dynamical system, (eds. Carillo, S. and Ragnisco, O., Lange and Springer, Berlin, 1990).Google Scholar
[4]Broadbridge, P., “Integrable forms of the one-dimensional flow equation for unsaturated heterogeneous porous media”, J. Math. Phys. 29 (1988) 622–627.CrossRefGoogle Scholar
[5]Broadbridge, P. and Godfrey, S. E., “Exact decaying soliton solutions of nonlinear Schrö-dinger equations: Lie-Bäcklund symmetries”, J. Math. Phys. 32 (1991) 8–18.CrossRefGoogle Scholar
[6]Broadbridge, P. and White, I., “Constant rate rainfall infiltration: a versatile nonlinear model. 1. Analytic solution”, Water Resour. Res. 24 (1988) 145–154.CrossRefGoogle Scholar
[7]Carslaw, H. S. and Jaeger, J. C., Conduction of heat in solids, (Clarendon Press, Oxford, 1959).Google Scholar
[8]Cole, J. D., “One a quasilinear parabolic equation occurring in aerodynamics”, Q. Appl. Math. 9 (1951) 225–236.CrossRefGoogle Scholar
[9]Fokas, A. S. and Yortsos, Y. C., “On the exactly solvable equation occurring in two-phase flow in porous media”, SIAM J. Appl. Math 42 (1982) 318–332.CrossRefGoogle Scholar
[10]Freeman, N. C. and Satsuma, J., “Exact solutions describing an interaction of pulses with compact support in a nonlinear diffusive system”, Phys. Lett. 138 (1989) 110–112.CrossRefGoogle Scholar
[11]Galaktionov, V. A., Dorodnitsyn, V. A., Elenin, G. G., Kurdyumov, S. P. and Samarskii, A. A., “A quasilinear heat equation with a source: peaking, localization, symmetry exact solutions, asymptotics, structures”, J. Soviet Math. 41 (1988) 1222–1292.CrossRefGoogle Scholar
[12]Hearn, A. C. (ed.), REDUCE user's manual (Rand Corp, Santa Monica, California, 1987).Google Scholar
[13]Hopf, E., “The partial differential equation u1+uux=uuxx”, Comm. Pure Appl. Math. 33 (1950) 201–230.CrossRefGoogle Scholar
[14]Ibragimov, N. H., Transformation groups applied to mathematical physics (D. Reidel, Boston, 1985).CrossRefGoogle Scholar
[15]Knight, J. H. and Philip, J. R., “Exact solutions in nonlinear diffusion”, J. Eng. Math. 8 (1974) 219–227.CrossRefGoogle Scholar
[16]Konopelchenko, B. and Rogers, C., in Nonlinear equations in applied science (Academic Press, New York, to appear, 1991).Google Scholar
[17]Lomen, D. O. and Warrick, A. W., “Solution of the one-dimensional linear moisture flow equation with implicit water extraction functions”, Soil Sci. Soc. Am. J. 40 (1976) 342–344.CrossRefGoogle Scholar
[18]Ludford, G. S., Combustion and chemical reactions (Amer. Math. Soc., Providence, Rhode Island, 1986).Google Scholar
[19]Molz, F. J. and Remson, I., “Extraction term models of soil moisture use by transpiring plants”, Water Resour. Res. 6 (1970) 1346–1356.CrossRefGoogle Scholar
[20]Olver, P. J., Application of Lie groups of differential equations (Springer, New York, 1988).Google Scholar
[21]Philip, J. R., “Theory of infiltration”, Adv. Hydrosciences 5 (1969) 215–296.CrossRefGoogle Scholar
[22]Rogers, C., “Application of a reciprocal transformation to a two-phase Stefan problem”, J. Phys. A: Mathematical and General 18 (1985) 105–109.CrossRefGoogle Scholar
[23]Rogers, C., “On a class of moving boundary problems in non-linear heat conduction: application of a Bäckhand transformation”, Int. J. Nonlinear Mech 21 (1986) 249–256.CrossRefGoogle Scholar
[24]Rogers, C. and Ames, W. F., Nonlinear boundary value problems in science and engineering (Academic Press, New York, 1989).Google Scholar
[25]Rogers, C. and Broadbridge, P., “On a nonlinear moving boundary problem with hetero-geneity: application of a Bäcklund transformation”, Zeit. Angew. Math. Phys. 39 (1988) 122–128.CrossRefGoogle Scholar
[26]Rogers, C., Stallybrass, M. P. and Clements, D. L., “On two phase filtration under gravity and with boundary infiltration: application of a Bäcklund transformation”, J. Nonlinear Analysis, Theory, Methods and Applications 7 (1983) 785–799.Google Scholar
[27]Rogers, C. and Wong, P., “On reciprocal Bäcklund transformations of inverse scattering schemes” Physica Scripta 30 (1984) 10–14.CrossRefGoogle Scholar
[28]Rose, C. W. and Stern, W. R., “Determination of withdrawal of water from the soil by crop roots as a function of depth and time”, Austral. J. Soil Res. 5 (1967) 11–19.CrossRefGoogle Scholar
[29]Rosen, G., “Method for the exact solution of a nonlinear diffusion-convection equation”, Phys. Lett. 49 (1982) 1844–1846.CrossRefGoogle Scholar
[30]Sander, G. C., Parlange, J.-Y., Kühnel, V., Hogarth, W. L., Lockington, D. and O'Kane, J. P. J., “Exact nonlinear solution for constant flux infiltration”, J. Hydrol. 97 (1988) 341–346.CrossRefGoogle Scholar
[31]Svinolupov, S. I., “Second order evolution equations with symmetries”, Usp. Mat. Nauka 40 (1985) 241–242.Google Scholar
[32]Storm, M. L., “Heat conduction in simple metals”, J. Appl. Phys. 22 (1951) 940–951.CrossRefGoogle Scholar
[33]Talsma, T. and Gardner, E. A., “Soil water extraction by a mixed eucalypt forest during a drought period”, Austr. J. Soil Res. 24 (1986) 25–32.CrossRefGoogle Scholar
[34]White, I. and Broadbridge, P., “Constant-rate rainfall infiltration: a versatile nonlinear model 2. Applications of solutions”, Water Resour. Res. 24 (1988) 155–162.CrossRefGoogle Scholar
[35]White, I. and Sully, M. J., “Macroscopic and microscopic capillary length and time scales from field infiltration”, Water Resour. Res. 23 (1987) 1514–1522.CrossRefGoogle Scholar