Numerical analysis, Partial differential equations
On non-denseness for a method of fundamental solutions with source points fixed in time for parabolic equations
Comptes Rendus. Mathématique, Volume 359 (2021) no. 6, pp. 733-738.

Linear combinations of fundamental solutions to the parabolic heat equation with source points fixed in time is investigated. The open problem whether these linear combinations generate a dense set in the space of square integrable functions on the lateral boundary of a space-time cylinder, is settled in the negative. Linear independence of the set of fundamental solutions is shown to hold. It is outlined at the end, for a particular example, that such linear combinations constitute a linearly independent and dense set in the space of square integrable functions on the upper top part (where time is fixed) of the boundary of this space-time cylinder.

Des combinaisons linéaires de solutions fondamentales avec des points sources indépendants du temps pour la résolution de l’équation de la chaleur sont étudiées. On étudie la question ouverte de savoir si ces combinaisons linéaires génèrent un ensemble dense dans l’espace des fonctions de carrés intégrables sur la limite latérale d’un cylindre espace-temps et on montre que la réponse à cette question est négative. L’indépendance linéaire de l’ensemble des solutions fondamentales est démontrée. Il est souligné à la fin pour un cas particulier que de telles combinaisons linéaires sont linéairement indépendantes et denses dans l’espace des fonctions de carrés intégrables définies sur la partie supérieure (où le temps est fixe) de la limite du cylindre espace-temps.

Received:
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Accepted:
Published online:
DOI: 10.5802/crmath.204
Classification: 35K05, 35A08, 65N80
Johansson, B. Tomas 1

1 Mathematics, ITN, Campus Norrköping, Linköping University, 601 74, Norrköping, Sweden.
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Johansson, B. Tomas. On non-denseness for a method of fundamental solutions with source points fixed in time for parabolic equations. Comptes Rendus. Mathématique, Volume 359 (2021) no. 6, pp. 733-738. doi : 10.5802/crmath.204. http://www.numdam.org/articles/10.5802/crmath.204/

[1] Alves, Carlos J. S. On the choice of source points in the method of fundamental solutions, Eng. Anal. Bound. Elem., Volume 33 (2009) no. 12, pp. 1348-1361 | DOI | MR | Zbl

[2] Borachok, I.; Chapko, R.; Johansson, B. Tomas A method of fundamental solutions for heat and wave propagation from lateral Cauchy data, Numer. Algorithms (2021) | DOI

[3] Cannon, John R. The One-Dimensional Heat Equation, Encyclopedia of Mathematics and Its Applications, 23, Addison-Wesley Publishing Group; Cambridge University Press, 1984 | MR | Zbl

[4] Chantasiriwan, Somchart; Johansson, B. Tomas; Lesnic, Daniel The method of fundamental solutions for free surface Stefan problems, Eng. Anal. Bound. Elem., Volume 33 (2009) no. 4, pp. 529-538 | DOI | MR | Zbl

[5] Cohen, Alan M. Numerical Methods for Laplace Transform Inversion, Numerical Methods and Algorithms, 5, Springer, 2007 | MR | Zbl

[6] Costabel, Martin Boundary integral operators for the heat equation, Integral Equations Oper. Theory, Volume 13 (1990) no. 4, pp. 498-552 | DOI | MR | Zbl

[7] Fairweather, Graeme; Karageorghis, Andreas The method of fundamental solutions for elliptic boundary value problems, Adv. Comput. Math., Volume 9 (1998) no. 1-2, pp. 69-95 | DOI | MR | Zbl

[8] Fattorini, Hector O. Boundary control of temperature distributions in a parallelepipedon, SIAM J. Control, Volume 13 (1975), pp. 1-13 | DOI | MR | Zbl

[9] Fattorini, Hector O.; Russell, David L. Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Rational. Mech. Anal., Volume 43 (1971), pp. 272-292 | DOI | MR | Zbl

[10] Friedman, Avner Partial Differential Equations of Parabolic Type, Prentice Hall, 1964 | Zbl

[11] Fursikov, Andreĭ V.; Imanuvilov, Oleg Yu. Controllability of evolution equations, Lecture Notes Series, 34, Seoul National University, 1996 | MR | Zbl

[12] Golberg, M. A.; Chen, C. S. The method of fundamental solutions for potential, Helmholtz and diffusion problems, Boundary Integral Methods: Numerical and Mathematical Aspects (Golberg, M. A., ed.) (Computational Engineering), Volume 1, WIT Press/ Computational Mechanics Publications, 1999, pp. 103-176 | MR | Zbl

[13] Guenther, Ronald Some elementary properties of the fundamental solution of parabolic equations, Math. Mag., Volume 39 (1966), pp. 294-298 | DOI | MR | Zbl

[14] Hon, Yiu-Ching; Wei, Ting A fundamental solution method for inverse heat conduction problem, Eng. Anal. Bound. Elem., Volume 28 (2004) no. 5, pp. 489-495 | Zbl

[15] Johansson, B. Tomas Properties of a method of fundamental solutions for the parabolic heat equation, Appl. Math. Lett., Volume 65 (2017), pp. 83-89 | DOI | MR | Zbl

[16] Johansson, B. Tomas; Lesnic, Daniel A method of fundamental solutions for transient heat conduction, Eng. Anal. Bound. Elem., Volume 32 (2008) no. 9, pp. 697-703 | DOI | Zbl

[17] Johansson, B. Tomas; Lesnic, Daniel; Reeve, Thomas A meshless regularization method for a two-dimensional two-phase linear inverse Stefan problem, Adv. Appl. Math. Mech., Volume 5 (2013) no. 6, pp. 825-845 | DOI | MR

[18] John, Fritz Partial Differential Equations, Applied Mathematical Sciences, 1, Springer, 1982 | Zbl

[19] Karageorghis, Andreas; Lesnic, Daniel; Marin, Liviu A survey of applications of the MFS to inverse problems, Inverse Probl. Sci. Eng., Volume 19 (2011) no. 3, pp. 309-336 | DOI | MR | Zbl

[20] Kupradze, Viktor D. A method for the approximate solution of limiting problems in mathematical physics, U.S.S.R. Comput. Math. Math. Phys., Volume 4 (1967) no. 6, pp. 199-205 translation from Zh. Vychisl. Mat. Mat. Fiz. 4, 1118–1121 (1964) | DOI | Zbl

[21] Lebeau, Gilles; Robbiano, Luc Contrôle exact de l’équation de la chaleur, Comm. Partial Differential Equations, Volume 20 (1995) no. 1-2, pp. 335-356 | DOI | Zbl

[22] Martin, Philippe; Rosier, Lionel; Rouchon, Pierre Null controllability of the heat equation using flatness, Automatica, Volume 50 (2014) no. 12, pp. 3067-3076 | DOI | MR | Zbl

[23] Mera, Nicolae S. The method of fundamental solutions for the backward heat conduction problem, Inverse Probl. Sci. Eng., Volume 13 (2005) no. 1, pp. 65-78 | DOI | MR | Zbl

[24] Russell, David L. Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Rev., Volume 20 (1978), pp. 639-739 | DOI | MR | Zbl

[25] Shidfar, Abdollah; Darooghehgimofrad, Z. Numerical solution of two backward parabolic problems using method of fundamental solutions, Inverse Probl. Sci. Eng, Volume 25 (2017) no. 2, pp. 155-168 | DOI | MR | Zbl

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