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Solvability of a non-linear Cauchy problem for an elliptic equation
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0002-2681-8965
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Makerere Univ, Uganda.
2019 (English)In: International Journal of Computer Mathematics, ISSN 0020-7160, E-ISSN 1029-0265, Vol. 96, no 12, p. 2317-2333Article in journal (Refereed) Published
Abstract [en]

We study a non-linear operator equation originating from a Cauchy problem for an elliptic equation. The problem appears in applications where surface measurements are used to calculate the temperature below the earth surface. The Cauchy problem is ill-posed and small perturbations to the used data can result in large changes in the solution. Since the problem is non-linear certain assumptions on the coefficients are needed. We reformulate the problem as an non-linear operator equation and show that under suitable assumptions the operator is well-defined. The proof is based on making a change of variables and removing the non-linearity from the leading term of the equation. As a part of the proof we obtain an iterative procedure that is convergent and can be used for evaluating the operator. Numerical results show that the proposed procedure converges faster than a simple fixed point iteration for the equation in the the original variables.

Place, publisher, year, edition, pages
TAYLOR & FRANCIS LTD , 2019. Vol. 96, no 12, p. 2317-2333
Keywords [en]
Cauchy problem; iterative method; elliptic equation; non-linear
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-161369DOI: 10.1080/00207160.2019.1569640ISI: 000490180800002OAI: oai:DiVA.org:liu-161369DiVA, id: diva2:1367570
Available from: 2019-11-04 Created: 2019-11-04 Last updated: 2019-11-04

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