Direct and indirect approximations to positive solution for a nonlinear reaction-diffusion problem. Part I. Direct (variational) approximation

Abstract

Publisher Name
We consider a nonlinear, second-order, two-point boundary value problem that models some reaction-diffusion precesses. When the reaction term has a particular form, f(u) =u3, the problem has a unique positive solution that satisfies a conserved integral condition. We study the bifurcation of this solution with respect to the length of the interval and it turns out that solution bifurcates from infinity. In the first part, we obtain the numerical approximation to the positive solution by direct (variational) methods, while in the second part we consider indirect numerical methods. In order to obtain directly accurate numerical approximations to this positive solution, we characterize it by a variational problem involving a conditional extremum. Then we carry out some numerical experiments by usual finite elements method.

Authors

Calin-Ioan Gheorghiu
Tiberiu Popoviciu Institute of Numerical Analysis

Damian Trif
Babes-Bolyai University, Faculty of Mathematics and Computer Science

Keywords

nonlinear reaction-diffusion; positive solution; conserved integral; bifurcation; variational formulation; Lagrange multiplier; finite elements method.

References

References

[1] Ambrosetti, A. and Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 , pp. 349–381, 1973.
[2] Aronson, D. G. and Weinberger, H. F., Nonlinear diffusion in population genetics, combustion and nerve pulse propagation, Lecture Notes in Math., 446, Springer-Verlag, 1975.
[3] Crandal, M. G. and Rabinowitz, P. H., Nonlinear Sturm-Liouvil le eigenvalue problems and topological degree, J. Math. Mech., 19 , pp. 1083–1102, 1970.
[4] Crandal, M. G. and Rabinowitz, P. H., Bifurcation, perturbation of simple eigenvalues, and linearized stability, Arch. Rat. Mech. Anal., 52, pp. 161–180, 1973.
[5] Elsgolts, L., Differential Equations and the Calculus of Variations , Mir Publishers, Moscow, 1980.
[6] Gheorghiu, C. I., Solution to problem 97-8 by Ph. Korman, SIAM Review, 39, 1997, SIAM Review, 40 , no. 2, pp. 382–385, 1998.
[7] Gheorghiu, C. I.and Trif, D., On the bifurcation and variational approximation of the positive solution of a nonlinear reaction-diffusion problem , Studia UBB, Mathemat- ica, XLX, pp. 29–37, 2000.
[8] Grindrod, P., The Theory and Applications of Reaction-Diffusion Equations; Patern and Waves, Second Edition, Clarendon Press, Oxford, 1996.
[9] Keller, H. B. and Cohen, D. S., Some positone problems suggested by nonlinear heat generation, J. Math. Mech., 16, pp. 1361–1376, 1967.
[10] Korman, Ph., Average temperature in a reaction-diffusion process, Problem 97-8, SIAM Review, 39 , p. 318, 1997.
[11] Laetsch, Th., The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J., 20, pp. 1–13, 1970.
[12] Matkowsky, B. J., A simple nonlinear dynamic stability problem, Bull. Amer. Math.Soc., 76, pp. 620–625, 1970.
[13] Moore, R. A. and Nehari, Z., Nonoscil lation theorems for a class of nonlinear differential equations, Trans. Amer. Math. Soc., 93 , pp. 30–52, 1959.
[14] Sattinger, D. H., Topics in Stability and Bifurcation Theory, Springer-Verlag, 1973.
[15] Sattinger, D. H., Stability of bifurcating solutions by Leray-Schauder degree, Arch. Rat. Mech. Anal., 43, pp. 155–165, 1970.
[16] Sattinger, D. H., Monotone methods in nonlinear el liptic and parabolic boundary value problems, Indiana Univ. Math. J., 21, pp. 979–1000, 1972.
[17] Schwind, W. J., Ji, J. and Koditschek, D. E., A physically motivated further note on the mean value theorem for integrals, Amer. Math. Monthly, 126 , pp. 559–564, 1999.
[18] Simpson, B. R. and Cohen, D. S., Positive solutions of nonlinear el liptic eigenvalue problems, J. Math. Mech., 19, pp. 895–910, 1970.
[19] Turner, R. E. I., Nonlinear eigenvalue problems with nonlocal operators, Comm. Pure Appl. Math., 23, pp. 963–972, 1970.
[20] Wollkind, D. J., Monoranjan, V. S. and Zhang, L., Weakly nonlinear stability analysis of prototype reaction-diffusion model equations , SIAM Review, 36 , no. 2, pp. 176– 214, 1994.

Cite this paper as

C.I. Gheorghiu, D. Trif, Direct and indirect approximations to positive solution for a nonlinear reaction-diffusion problem. Part I. Direct (variational) approximation, Rev. Anal. Numér. Théor. Approx. 31 (2002) 61-70.

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Journal
Revue d’Analyse Numerique et de Theorie de l’Approximation
Publisher
Editions de l’Academie Roumaine
Print ISSN

1222-9024

Online ISSN

2457-8126

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[1] Ambrosetti, A. and Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. Funct. Anal., 14, pp. 349-381, 1973.

[2] Aronson, D. G. and Weinberger, H. F., Nonlinear diffusion in population genetics, combustion and nerve pulse propagation, Lecture Notes in Math., 446, Springer-Verlag, 1975.

[3] Crandal, M. G. and Rabinowitz, P. H., Nonlinear Sturm-Liouville eigenvalue problems and topological degree, J. Math. Mech., 19, pp. 1083-1102, 1970.

[4] Crandal, M. G. and Rabinowitz, P. H., Bifurcation, perturbation of simple eigenvalues, and linearized stability, Arch. Rat. Mech. Anal., 52, pp. 161-180, 1973.

[5] Elsgolts, L., Differential Equations and the Calculus of Variations, Mir Publishers, Moscow, 1980.

[6] Gheorghiu, C. I., Solution to problem 97-8 by Ph. Korman, SIAM Review, 39, 1997, SIAM Review, 40, no. 2, pp. 382-385, 1998.

[7] Gheorghiu, C. I. and Trif, D., On the bifurcation and variational approximation of the positive solution of a nonlinear reaction-diffusion problem, Studia UBB, Mathematica, XLV, pp. 29-37, 2000.

[8] Grindrod, P., The Theory and Applications of Reaction-Diffusion Equations; Patern and Waves, Second Edition, Clarendon Press, Oxford, 1996.

[9] Keller, H. B. and Cohen, D. S., Some positone problems suggested by nonlinear heat generation, J. Math. Mech., 16, pp. 1361-1376, 1967.

[10] Korman, Ph., Average temperature in a reaction-diffusion process, Problem 97-8, SIAM Review, 39, pp.318, 1997.

[11] Laetsch, Th., The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J., 20, pp. 1-13, 1970.

[12] Matkowsky, B. J., A simple nonlinear dynamic stability problem, Bull. Amer. Math. Soc., 76, pp. 620-625, 1970.

[13] Moore, R. A. and Nehari, Z., Nonoscillation theorems for a class of nonlinear differential equations, Trans. Amer. Math. Soc., 93, pp. 30-52, 1959.

[14] Sattinger, D. H., Topics in Stability and Bifurcation Theory, Springer-Verlag, 1973.

[15] Sattinger, D. H., Stability of bifurcating solutions by Leray-Schauder degree, Arch. Rat. Mech. Anal., 43, pp. 155-165, 1970.

[16] Sattinger, D. H., Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21, pp. 979-1000, 1972.

[17] Schwind, W. J., Ji, J. and Koditschek, D. E., A physically motivated further note on the mean value theorem for integrals, Amer. Math. Monthly, 126, pp. 559-564, 1999.

[18] Simpson, B. R. and Cohen, D. S., Positive solutions of nonlinear elliptic eigenvalue problems, J. Math. Mech., 19, pp. 895-910, 1970.

[19] Turner, R. E. I., Nonlinear eigenvalue problems with nonlocal operators, Comm. Pure Appl. Math., 23, pp. 963-972, 1970.

[20] Wollkind, D. J., Monoranjan, V. S. and Zhang, L., Weakly nonlinear stability analysis of prototype reaction-diffusion model equations, SIAM Review, 36, no. 2, pp. 176-214, 1994.

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