Recovery of an unknown flux in parabolic problems with nonstandard boundary conditions: Error estimates

Marián Slodička

Applications of Mathematics (2003)

  • Volume: 48, Issue: 1, page 49-66
  • ISSN: 0862-7940

Abstract

top
In this paper, we consider a 2nd order semilinear parabolic initial boundary value problem (IBVP) on a bounded domain Ω ⊂ ℝ N , with nonstandard boundary conditions (BCs). More precisely, at some part of the boundary we impose a Neumann BC containing an unknown additive space-constant α ( t ) , accompanied with a nonlocal (integral) Dirichlet side condition. We design a numerical scheme for the approximation of a weak solution to the IBVP and derive error estimates for the approximation of the solution u and also of the unknown function α .

How to cite

top

Slodička, Marián. "Recovery of an unknown flux in parabolic problems with nonstandard boundary conditions: Error estimates." Applications of Mathematics 48.1 (2003): 49-66. <http://eudml.org/doc/33133>.

@article{Slodička2003,
abstract = {In this paper, we consider a 2nd order semilinear parabolic initial boundary value problem (IBVP) on a bounded domain $\Omega \subset \mathbb \{R\}^N$, with nonstandard boundary conditions (BCs). More precisely, at some part of the boundary we impose a Neumann BC containing an unknown additive space-constant $\alpha (t)$, accompanied with a nonlocal (integral) Dirichlet side condition. We design a numerical scheme for the approximation of a weak solution to the IBVP and derive error estimates for the approximation of the solution $u$ and also of the unknown function $\alpha $.},
author = {Slodička, Marián},
journal = {Applications of Mathematics},
keywords = {nonlocal boundary condition; parameter identification; parabolic IBVP; parabolic initial boundary value problem; nonlocal boundary condition},
language = {eng},
number = {1},
pages = {49-66},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Recovery of an unknown flux in parabolic problems with nonstandard boundary conditions: Error estimates},
url = {http://eudml.org/doc/33133},
volume = {48},
year = {2003},
}

TY - JOUR
AU - Slodička, Marián
TI - Recovery of an unknown flux in parabolic problems with nonstandard boundary conditions: Error estimates
JO - Applications of Mathematics
PY - 2003
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 48
IS - 1
SP - 49
EP - 66
AB - In this paper, we consider a 2nd order semilinear parabolic initial boundary value problem (IBVP) on a bounded domain $\Omega \subset \mathbb {R}^N$, with nonstandard boundary conditions (BCs). More precisely, at some part of the boundary we impose a Neumann BC containing an unknown additive space-constant $\alpha (t)$, accompanied with a nonlocal (integral) Dirichlet side condition. We design a numerical scheme for the approximation of a weak solution to the IBVP and derive error estimates for the approximation of the solution $u$ and also of the unknown function $\alpha $.
LA - eng
KW - nonlocal boundary condition; parameter identification; parabolic IBVP; parabolic initial boundary value problem; nonlocal boundary condition
UR - http://eudml.org/doc/33133
ER -

References

top
  1. Global existence and blow up in a parabolic problem with nonlocal dynamical boundary conditions, Adv. Differential Equations 1 (1996), 729–752. (1996) MR1392003
  2. 10.1051/m2an/1985190100071, RAIRO Modél. Math. Anal. Numér. 19 (1985), 7–32. (1985) MR0813687DOI10.1051/m2an/1985190100071
  3. 10.1051/m2an/1994280709031, RAIRO Modél. Math. Anal. Numér. 28 (1994), 903–919. (1994) MR1309419DOI10.1051/m2an/1994280709031
  4. 10.21914/anziamj.v42i0.611, ANZIAM Journal (C) 42 (2000), 518–535. (2000) MR1810647DOI10.21914/anziamj.v42i0.611
  5. Partial Differential Equations, Robert E.  Krieger Publishing Company, Hungtinton, New York, 1976. (1976) MR0454266
  6. Variational Principles and Free-Boundary Problems, Wiley, New York, 1982. (1982) Zbl0564.49002MR0679313
  7. Method of Rothe in Evolution Equations. Teubner Texte zur Mathematik Vol. 80, Teubner, Leipzig, 1985. (1985) MR0834176
  8. Les méthodes directes en théorie des équations elliptiques, Academia, Prague, 1967. (1967) MR0227584
  9. Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. (1992) Zbl0777.35001MR1212084
  10. 10.1002/(SICI)1097-0363(19960315)22:5<325::AID-FLD307>3.0.CO;2-Y, Internat. J. Numer. Methods Fluids 22 (1996), 325–352. (1996) MR1380844DOI10.1002/(SICI)1097-0363(19960315)22:5<325::AID-FLD307>3.0.CO;2-Y
  11. The Method of Discretization in Time and Partial Differential Equations, Reidel Publishing Company, Dordrecht-Boston-London, 1982. (1982) Zbl0522.65059MR0689712
  12. A monotone linear approximation of a nonlinear elliptic problem with a non-standard boundary condition, In: Algoritmy 2000, A. Handlovičová, M. Komorníková, K. Mikula and D. Ševčovič (eds.), Slovak University of Technology, Faculty of Civil Engineering, Department of Mathematics and Descriptive Geometry, Bratislava, 2000, pp. 47–57. (2000) 
  13. 10.1051/m2an:2001132, RAIRO Modél. Math. Anal. Numér. 35 (2001), 691–711. (2001) Zbl0997.65124MR1862875DOI10.1051/m2an:2001132
  14. 10.1016/S0096-3003(01)00057-1, Appl. Math. Comput. 129 (2002), 469–480. (2002) MR1905411DOI10.1016/S0096-3003(01)00057-1
  15. 10.1007/BF02905911, Arch. Comput. Methods Engrg. 5 (1999), 385–443. (1999) MR1675223DOI10.1007/BF02905911
  16. Numerical modelling for the recovery of an unknown flux in semilinear parabolic problems with nonstandard boundary conditions, In: Proceedings European Congress on Computational Methods in Applied Sciences and Engineering, Barcelona, E. Onate, G. Bugeda and B. Suárez (eds.), Barcelona, 2000. (2000) 
  17. Numerical techniques for the recovery of an unknown Dirichlet data function in semilinear parabolic problems with nonstandard boundary conditions, In: Numerical Analysis and Its Applications, L. Vulkov, J. Wasniewski and P. Yalamov (eds.), Springer, 2001, pp. 467–474. (2001) MR1938440

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.