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Exact and analytic-numerical solutions of strongly coupled mixed diffusion problems

Published online by Cambridge University Press:  20 January 2009

L. Jódar
Affiliation:
Departamento de Matemática Aplicada, Universidad Politécnica de Valencia, P.O. Box 22012, Valencia, Spain (ljodar@mat.upv.es)
E. Navarro
Affiliation:
Departamento de Matemática Aplicada, Universidad Politécnica de Valencia, P.O. Box 22012, Valencia, Spain (ljodar@mat.upv.es)
J. A. Martin
Affiliation:
Departamento de Análisis Matemático y Matemática Aplicada, Universidad de Alicante, Ap. Correos 99, E-03080 Alicante, Spain (jose.martin@ua.es)
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Abstract

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This paper deals with the construction of exact and analytical-numerical solutions with a priori error bounds for systems of the type ut = Auxx, A1u(0, t) + B1ux (0, t) = 0, A2u (1, t) + B2ux (1, t) = 0, 0 < x < 1, t > 0, u(x, 0) = f(x), where A1, A2, B1 and B2 are matrices for which no simultaneous diagonalizable hypothesis is assumed, and A is a positive stable matrix. Given an admissible error ε and a bounded subdomain D, an approximate solution whose error with respect to an exact series solution is less than ε uniformly in D is constructed.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2000

References

1. Alexander, M. H. and Manolopoulos, D. E., A stable linear reference potencial algorithm for solution of the quantum close-coupled equations in molecular scattering theory, J. Chem. Phys. 86 (1987), 20442050.Google Scholar
2. Apostol, T. M., Mathematical analysis (Addison-Wesley, Reading, MA, 1977).Google Scholar
3. Atkinson, F. V., Discrete and continuous boundary value problems (Academic, New York, 1964).Google Scholar
4. Atkinson, F. V., Krall, A. M., Leaf, G. K. and Zettel, A., On the numerical computation of eigenvalues of Sturm–Liouville problems with matrix coefficients, Technical Report, Argonne National Laboratory, 1987.Google Scholar
5. Axelsson, O., Iterative solution methods (Cambridge University Press, 1994).Google Scholar
6. Campbell, S. L. and Meyer, C. D. Jr, Generalized inverses of linear transformations (Pitman, London, 1979).Google Scholar
7. Coddington, E. A. and Levinson, N., Theory of ordinary differential equations (McGraw-Hill, New York, 1967).Google Scholar
8. Crank, J., The mathematics of diffusion, 2nd edn (Oxford University Press, 1995).Google Scholar
9. Dunford, N. and Schwartz, J., Linear operators, part I (Interscience, New York, 1957).Google Scholar
10. Folland, G. B., Fourier analysis and its applications (Wadworth & Brooks, Pacific Grove, CA, 1992).Google Scholar
11. Golub, G. H. and Van Loan, C. F., Matrix computation (The Johns Hopkins University Press, Baltimore, MD. 1989).Google Scholar
12. Greenberg, L., A Prüfer method for calculating eigenvalues of self-adjoint systems of ordinary differential equations, parts 1 and 2, University of Maryland, Technical Report TR91–24.Google Scholar
13. Hueckel, T., Borsetto, M. and Peano, A., Modelling of coupled thermo-elastoplastic hydraulic response of clays subjected to nuclear waste heat, in Numerical methods in transient and coupled problems (ed. Lewis, R. W., Hinton, E., Bettes, P. and Scherefler, B. A.), pp. 213235 (Wiley, New York, 1987).Google Scholar
14. Inge, E. L., Ordinary differential equations (Dover, New York, 1927).Google Scholar
15. Jódar, L. and Ponsoda, E., Continuous numerical solutionand error bounds for time dependent systems of partial differential equations: mixed problems, Comp. Math. Appl. 29 (1995), 6371.Google Scholar
16. Levine, R. D., Shapiro, M. and Johnson, B., Transition probabilities in molecular collisions: computational studies of rotational excitation, J. Chem. Phys. 53 (1970), 17551766.Google Scholar
17. Lill, J. V., Schmalz, T. G. and Light, J. C., Imbedded matrix Green's functions in atomic and molecular scattering theory, J. Chem. Phys. 78 (1983), 44564463.Google Scholar
18. Marletta, M., Theory and implementation of algorithms for Sturm–Liouville systems, PhD thesis, Royal Military College of Science, Cranfield, 1991.Google Scholar
19. Melezhik, V. S., Puzynin, I. V., Puzynina, T. P. and Somov, L. N., Numerical solution of a system of integrodifferential equations arising from the quantum-mechanical three-body problem with Coulomb interaction, J. Comput. Phys. 54 (1984), 221236.Google Scholar
20. Mikhailov, M. D. and Ösizik, M. N., Unifield analysis and solutions of heat and mass diffusion (Wiley, New York, 1984).Google Scholar
21. Moler, C. B. and Van Loan, C. F., Nineteen dubious ways to compute the exponential of a matrix, SIAM Rev. 20 (1978), 801836.Google Scholar
22. Mrugala, F. and Secrest, D., The generalized log-derivate method for inelastic and reactive collisions, J. Chem. Phys. 78 (1983), 59545961.Google Scholar
23. Navarro, E., Ponsoda, E. and Jódar, L., A matrix approach to the analytic-numerical solution of mixed partial differential systems, Comp. Math. Appl. 30 (1995), 99109.Google Scholar
24. Pryce, J. D., Numerical solution of Sturm–Liouville problems (Clarendon, Oxford, 1993).Google Scholar
25. Pryce, J. D. and Marletta, M., Automatic solution of Sturm–Liouville problems using Pruess method, J. Comput. Appl. Math. 39 (1992), 5778.Google Scholar
26. Rao, C. R. and Mitra, S. K., Generalized inverse of matrices and its applications (Wiley, New York, 1971).Google Scholar
27. Reid, W. T., Ordinary differential equations (Wiley, New York, 1971).Google Scholar
28. Shapiro, M. and Balint-Kurti, G. G., A new method for the exact calculation of vibrational-rotational energy levels of triatomic molecules, J. Chem. Phys. 71 (1979), 14611469.Google Scholar
29. Stakgold, I., Green's functions and boundary value problems (Wiley, New York, 1979).Google Scholar