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An adaptive boundary algorithm for the reconstruction of boundary and initial data using the method of fundamental solutions for the inverse Cauchy–Stefan problem

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Abstract

In this paper, a recent algorithm, based around the method of fundamental solutions (MFS), for reconstructing boundary data in inverse Stefan problems is extended and applied to inverse Cauchy–Stefan problems, wherein initial data must also be reconstructed. A key feature of the algorithm is that it is adaptive and iterates to find the optimal locations of the source points that are required by the method. Tikhonov regularization is used to take care of the ill-conditioned matrix that the MFS generates, with the algorithm being able to determine the optimal regularization parameter automatically. The effects of accuracy and random noise on the optimal location and number of source points are also evaluated. In addition, we consider a nonlinear variant of the inverse problem where one has to identify the moving boundary along with the missing initial data. Numerical experiments, carried out on five different benchmark examples, show promising results.

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Acknowledgements

G. M. M. Reddy would like to thank FAPESP (Fundação de Amparo a Pesquisa do Estado de São Paulo) for the financial support received [Grant Number 2016/19648-9]. G. M. M. Reddy and P. Nanda would like to thank the Department of Science and Technology for support through the grant SRG/2019/001973. M. Vynnycky acknowledges the award of visiting researcher grants from the University of São Paulo, CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) [Grant Number 401945/2012-0] and FAPESP [Grant Number 2018/07643-8].

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Authors and Affiliations

  1. Department of Mathematics, Birla Institute of Technology and Science, Pilani, Hyderabad Campus, Jawahar Nagar, Kapra Mandal, Medchal, Telangana, 500078, India

    G. M. M. Reddy & P. Nanda

  2. Department of Mathematics and Statistics, University of Limerick, Limerick, V94 T9PX, Ireland

    M. Vynnycky

  3. Department of Applied Mathematics and Statistics, Institute of Mathematical and Computer Sciences, University of São Paulo at São Carlos, PO Box 668, São Carlos, São Paulo, 13560-970, Brazil

    J. A. Cuminato

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  1. G. M. M. Reddy
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  2. P. Nanda
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  3. M. Vynnycky
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  4. J. A. Cuminato
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Correspondence to M. Vynnycky.

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Communicated by Antonio José Silva Neto.

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Reddy, G.M.M., Nanda, P., Vynnycky, M. et al. An adaptive boundary algorithm for the reconstruction of boundary and initial data using the method of fundamental solutions for the inverse Cauchy–Stefan problem. Comp. Appl. Math. 40, 99 (2021). https://doi.org/10.1007/s40314-021-01454-1

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  • DOI: https://doi.org/10.1007/s40314-021-01454-1

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