Abstract
We consider a numerical enclosure method for solutions of an inverse Dirichlet eigenvalue problem. When the finite number of prescribed eigenvalues are given, we reconstruct a potential function, with guaranteed error bounds, for which the corresponding elliptic operator exactly has those eigenvalues including the ordering property. All computations are executed with numerical verifications based upon the finite and infinite fixed point theorems using interval arithmetic. Therefore, the results obtained are mathematically correct. We present numerical examples which confirm us the enclosure algorithm works on real problems.
Similar content being viewed by others
References
G. Alefeld, On the convergence of some interval-arithmetic modifications of Newton’s method. SIAM J. Numer. Anal.,21 (1984), 363–372.
A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems. Springer, New York, 1996.
R. Lohner, Enclosing the solutions of ordinary initial and boundary value problems. Computerarithmetic (eds. E. Kaucher, et al.), B.G. Teubner Stuttgart, 1987, 255–286.
K. Nagatou, N. Yamamoto and M.T. Nakao, An approach to the numerical verification of solutions for nonlinear elliptic problems with local uniqueness. Numer. Funct. Anal. Optim.,20 (1999), 543–565.
K. Nagatou, A numerical method to verify the elliptic eigenvalue problems including a uniqueness property. Computing,63 (1999), 109–130.
M.T. Nakao, A numerical verification method for the existence of weak solutions for nonlinear boundary value problems. J. Math. Anal. Appl.,164 (1992), 489–507.
M.T. Nakao, Solving nonlinear elliptic problems with result verification using an H−1 residual iteration. Comput. Suppl.,9 (1993), 161–173.
M.T. Nakao and N. Yamamoto, Numerical verification of solutions for nonlinear elliptic problems using L∞ residual method. J. Math. Anal. Appl.,217 (1998), 246–262.
M.T. Nakao, N. Yamamoto and K. Nagatou, Numerical verifications of eigenvalues of secondorder elliptic operators. Japan J. Indust. Appl. Math.,16 (1999), 307–320.
M. Neher, Ein Einschließungsverfahren für das inverse Dirichletproblem. Dissertation, University of Karlsruhe, 1993.
M. Neher, Enclosing solutions of an inverse Sturm-Liouville problem with finite data. Computing,53 (1994), 379–395.
S.M. Rump, Solving algebraic problems with high accuracy. A New Approach to Scientific Computation (eds. U. Kulisch and W.L. Miranker), Academic Press, New York, 1983.
Y. Watanabe and M.T. Nakao, Numerical verifications of solutions for nonlinear elliptic equations. Japan J. Indust. Appl. Math.,10 (1993), 165–178.
N. Yamamoto and M.T. Nakao, Numerical verifications for solutions to elliptic equations using residual iterations with higher order finite element. J. Comput. Appl. Math.,60 (1995), 271–279.
N. Yamamoto, A numerical verification method for solutions of boundary value problems with local uniqueness by Banach’s fixed point theorem. SIAM J. Numer. Anal.,35 (1998), 2004–2013.
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Nakao, M.T., Watanabe, Y. & Yamamoto, N. Verified numerical computations for an inverse elliptic eigenvalue problem with finite data. Japan J. Indust. Appl. Math. 18, 587–602 (2001). https://doi.org/10.1007/BF03168592
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF03168592