# Article

 Title: A matrix constructive method for the analytic-numerical solution of coupled partial differential systems (English) Author: Jódar, Lucas Author: Navarro, Enrique Author: Ferrer, M. V. Language: English Journal: Applications of Mathematics ISSN: 0862-7940 (print) ISSN: 1572-9109 (online) Volume: 40 Issue: 5 Year: 1995 Pages: 391-400 Summary lang: English . Category: math . Summary: In this paper we construct analytic-numerical solutions for initial-boundary value systems related to the equation $u_t-Au_{xx}-Bu=0$, where $B$ is an arbitrary square complex matrix and $A$ ia s matrix such that the real part of the eigenvalues of the matrix $\frac{1}{2}(A+A^H)$ is positive. Given an admissible error $\varepsilon$ and a finite domain $G$, and analytic-numerical solution whose error is uniformly upper bounded by $\varepsilon$ in $G$, is constructed. (English) Keyword: Schur decomposition Keyword: partial differential system Keyword: eigenvalues bound Keyword: matrix norms Keyword: analytic-numerical solution Keyword: error bounds MSC: 15A24 MSC: 15A45 MSC: 15A60 MSC: 35K15 MSC: 65F15 MSC: 65M70 MSC: 65N15 idZBL: Zbl 0841.65087 idMR: MR1342368 DOI: 10.21136/AM.1995.134302 . Date available: 2009-09-22T17:49:07Z Last updated: 2020-07-28 Stable URL: http://hdl.handle.net/10338.dmlcz/134302 . Reference: [1] T.M. Apostol: Mathematical Analysis.Addison-Wesley, Reading M.A., 1977. Reference: [2] F.L. Bauer, A.S. Householder: Absolute norms and characteristic roots.Numerische Math. 3 (1961), 241–246. MR 0130105, 10.1007/BF01386023 Reference: [3] I.E. Butsenieks, E.V. Shcherbinin: Magnetohydrodynamic flow in electrodynamically coupled rectangular ducts I.Magn. Girodinam 2 (1976), 35–40. Reference: [4] J.R. Cannon, R.E. Klein: On the observability and stability of the temperature distribution in a composite head conductor.SIAM J. Appl. Math. 24 (1973), 596–602. MR 0370318, 10.1137/0124061 Reference: [5] H.H. Chiu: Theory of irreducible linear systems.Quart. Appl. Maths. XXVII (1969), 87–104. MR 0244596, 10.1090/qam/244596 Reference: [6] R.V. Churchill, J.W. Brown: Fourier Series and Boundary Value Problems.McGraw-Hill, New York, 1978. MR 0513825 Reference: [7] R. Courant, D. Hilbert: Methods of Mathematical Physics, Vol. II.Interscience, New York, 1962. MR 0065391 Reference: [8] G.H. Golub, C.F. van Loan: Matrix Computations.Johns Hopkins Univ. Press, Baltimore M.A., 1985. Reference: [9] K. Gopalsamy: An arms race with deteriorating armaments.Math. Biosci. 37 (1977), 191–203. Zbl 0368.90101, MR 0682615, 10.1016/0025-5564(77)90092-X Reference: [10] J.M. Hill, J.R. Willis: Analytic aspects of heat flow in random two-component laminates.Quart. Appl. Maths. 46 (1988), no. 2, 353–364. MR 0950607, 10.1090/qam/950607 Reference: [11] L. Jódar: Computing accurate solutions for coupled systems of second order partial differential equations.Int. J. Computer Math. 37 (1990), 201–212. 10.1080/00207169008803948 Reference: [12] L. Jódar: Computing accurate solutions for coupled systems of second order partial differential equations II.Int. J. Computer Math. 46 (1992), 63–75. 10.1080/00207169208804139 Reference: [13] A.I. Lee, J.M. Hill: On the general linear coupled system for diffusion in media with two diffusivities.J. Math. Anal. Appl. 89 (1982), 530–557. MR 0677744, 10.1016/0022-247X(82)90116-0 Reference: [14] M. Mascagni: An initial-boundary value problem of physiological significance for equations of nerve conduction.Com. Pure Appl. Math. XLII (1989), 213–227. Zbl 0664.92007, MR 0978705, 10.1002/cpa.3160420206 Reference: [15] M. Mascagni: The backward Euler method for numerical solution of the Hodgkin-Huxley equations of nerve condition.SIAM J. Numer. Anal. 27 (1990), no. 4, 941–962. MR 1051115, 10.1137/0727054 Reference: [16] M. Mascagni: A parallelizing algorithm for compution solutions to arbitrarily branched cable neutron models.J. Neurosc. Methods 36 (1991), 105–114. 10.1016/0165-0270(91)90143-N Reference: [17] C.B. Moler, C.F. van Loan: Nineteen dubious ways to compute the exponential of a matrix.SIAM Review 20 (1978), 801–836. MR 0508383, 10.1137/1020098 Reference: [18] M. Sezgin: Magnetohydrodynamic flow in an infinite channel.Int. J. Numer. Methods Fluids 6 (1986), 593–609. Zbl 0597.76116, 10.1002/fld.1650060903 Reference: [19] J. Stoer, R. Bulirsch: Introduction to Numerical Analysis.Springer-Verlag, New York, 1980. MR 0557543 Reference: [20] K.S. Yee: Explicit solutions for the initial boundary value problem of a system of lossless transmission lines.SIAM J. Appl. Math. 24 (1975), 62–80. MR 0318657, 10.1137/0124008 Reference: [21] E.C. Zachmanoglou, D.W. Thoe: Introduction to Partial Differential Equations with Applications.William and Wilkins, Baltimore, 1976. MR 0463623 .

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