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A heat polynomials method for the two-phase inverse Stefan problem

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Abstract

In this paper, we extend the heat polynomials method (HPM) proposed by the authors for one-dimensional one-phase inverse Stefan problem to the two-phase case. The solution of the problem is presented in the form of linear combination of heat polynomials. The coefficients of this combination can be determined by satisfying the initial and boundary conditions or by the least square method for the boundary of a domain. The inverse problem is ill-posed, therefore, the regularization will be taken into account. Our numerical results are compared with results obtained by another method and show good enough accuracy.

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References

  • Ang DD, Pham Ngoc Dinh A, Tranh D (1996) Regularization of an inverse two-phase Stefan problem. Nonlinear Anal. 34:719–731

    Article  MathSciNet  MATH  Google Scholar 

  • Bragg LR (1965) The radial heat polynomials and the related functions. Trans. Amer. Math. Soc. 119:270–290

    Article  MathSciNet  MATH  Google Scholar 

  • Cannon JR (1984) The One-dimensional Heat Equation. Addison-Wesley, Menlo Park, CA

    Book  MATH  Google Scholar 

  • Cannon JR, Douglas J Jr (1967) The Cauchy problem for the heat equation. SIAM J. Numer. Anal. 4:317–336

    Article  MathSciNet  MATH  Google Scholar 

  • Futakiewicz S, Hozejowski L (1998) Heat polynomials method in solving the direct and inverse heat conduction problems in a cylindrical system of coordinates, Transactions on Engineering Sciences vol. 20, WIT Press, www.witpress.com, ISSN 1743-3533

  • Goldman NL (1997) Inverse Stefan problem. Kluwer, Dordrecht

    Book  Google Scholar 

  • Grysa K (2003) Heat polynomials and their applications. A Archives of Thermodynamics 24(2):107–124

    Google Scholar 

  • Grzymkowski R, Stota D (2002) Multi-phase inverse Stefan problems solved by approximation method, In Parallel Processing and Applied Mathematics, (Edited by R. Wyrzykowski et al.), pp. 679-686, LNCS 2328, Springer, Berlin,

  • Grzymkowski R, Slota D (2006) One-phase inverse Stefan problems solved by Adomain decomposition method. Comput. Math. Appl. 51:33–40

    Article  MathSciNet  MATH  Google Scholar 

  • Gupta SC (2003) The classical Stefan problem. Basic Concepts, Modeling and Analysis, Elsevier, Amsterdam

    MATH  Google Scholar 

  • Haimo DT (1965) \(L^2\) expansions in terms of generalized heat polynomials and treir Appell trasforms. Pacific J. Math. 15:865–875

    Article  MathSciNet  Google Scholar 

  • Hansen PC (1992) Analysis of discrete ill-posed problems by means of the L-curve. SIAM Rev. 34:561–580

    Article  MathSciNet  MATH  Google Scholar 

  • Johansson BT, Lesnic D, Reeve T (2011) A method of fundamental solutions for the one-dimensional inverse Stefan problem. Appl. Math. Model. 35:4367–4378

    Article  MathSciNet  MATH  Google Scholar 

  • Johansson BT, Lesnic D, Reeve T (2013) A meshless method for an inverse two-phase one-dimensional linear Stefan problem. Inverse Prob. Sci. Eng. 21(1):17–33

    Article  MathSciNet  MATH  Google Scholar 

  • Johansson BT, Lesnic D, Reeve T (2014) A meshless method for an inverse two-phase one-dimensional nonlinear Stefan problem. Mathematics and Computers in Simulation 101:61–77

    Article  MathSciNet  MATH  Google Scholar 

  • Kassabek SA, Kharin SN, Suragan D (2021) A heat polynomials method for inverse cylindrical one-phase Stefan problems. Inverse Problems in Science and Engineering 29:3423–3450

    Article  MathSciNet  MATH  Google Scholar 

  • Kassabek SA, Suragan D (2022) Numerical approximation of the one-dimensional inverse Cauchy-Stefan problem using heat polynomials methods, Computational and Applied Mathematics, 41(4),

  • Kassabek S A, Suragan D (2023) Two-phase inverse Stefan problems solved by heat polynomials method, Journal of Computational and Applied Mathematics, 114854,https://doi.org/10.1016/j.cam.2022.114854

  • Kharin SN (2014) The analytical solution of the two-phase Stefan problem with boundary flux condition. Mathematical Journal 14(1 (51)):55–76

    MATH  Google Scholar 

  • Kharin SN, Sarsengeldin MM, Nouri H (2016) Analytical solution of two-phase spherical Stefan problem by heat polynomials and integral error functions. AIP Conference Proceedings 1759:020031

    Article  Google Scholar 

  • Kharin SN (2017) Mathematical models of phenomena in electrical contacts. A.P. Ershov Institute of Informatics System The Russian Academy of Sciences, Siberian Branch, Novosibirsk

    Google Scholar 

  • Nasim C (1988) On generalized heat polynomials, International. J. Math. & Math. Sci. Vol. ii No. 2, pp. 393-400,

  • Reemtsen R, Kirsch A (1984) A method for the numerical solution of the one-dimensional inverse Stefan problem. Numerische Mathematik 45:253–273

    Article  MathSciNet  MATH  Google Scholar 

  • Rosenbloom PC, Widder DV (1959) Expansions in terms of heat polynomials and associated functions. Trans. Amer. Math. Soc. 92:220–336

    Article  MathSciNet  MATH  Google Scholar 

  • Rubinstain LI (1971) The Stefan problem. American Mathemagical Society, Providence, RI

    Google Scholar 

  • Sadoun N, Si-Ahmed EK, Colinet P (2006) On the Refined heat balance integral method for the one-Phase Stefan problem with time-dependent boundary conditions. Appl. Math. Modelling 30(6):531–544

    Article  MATH  Google Scholar 

  • Sarsengeldin M, Kharin SN (2017) Method of the integral error functions for the solution of the one- and two-phase Stefan problems and its application. Filomat journal 31(4):1017–1029

    Article  MathSciNet  MATH  Google Scholar 

  • Słiota D (2011), Homotopy perturbation method for solving the two-phase inverse Stefan problem, Numerical Heat Transfer, Part A 59, pp. 775-768, (2011)

  • Scott W (2021) McCue, Maud El-Hachem, and Matthew. J. Simpson, Exact sharp-fronted travelling wave solutions of the Fisher-KPP equation, Applied Mathematics Letters 114:106918

    Google Scholar 

  • Trefftz E (1926) Ein Gegensruek zum Ritz’schen Verfahren, in: Proccedings 2nd International Congress of Apllied Mechanics (Zurich), pp. 131-137, (1926)

  • Wood AS (2001) A new look at the heat balance integral method. Appl. Math. Modelling 25(10):815–824

    Article  MATH  Google Scholar 

  • Yano H, Fukutani S, Kieda A (1983) A boundary residual metod with heat polynomilas for solving usteady heat conduction problems. J. of the Franklin Inst. 316(4):219–298

    Article  MATH  Google Scholar 

  • Du Y, Lin Z (2010) Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary. SIAM Journal on Mathematical Analysis 42:377–405

    Article  MathSciNet  MATH  Google Scholar 

  • Yu Y, Luo XC, Cui HJ (2015) The solution of two-phase inverse stefan problem based on a hybrid method with optimization. Math Probl Eng. https://doi.org/10.1155/2015/319054.

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Acknowledgements

This research was funded by Nazarbayev University under Collaborative Research Program Grant 20122022CRP1601. This publication is also supported by the target program AP09258948 from the Ministry of Science and Education of the Republic of Kazakhstan.

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Authors and Affiliations

  1. Department of Mathematics, Nazarbayev University, Astana, Kazakhstan

    Samat A. Kassabek & Durvudkhan Suragan

  2. Department of Computational and Data Science, Astana IT University, Nur-Sultan, Kazakhstan

    Samat A. Kassabek

  3. Department of Mathematics and Natural Sciences, Suleyman Demirel University, Kaskelen, Kazakhstan

    Samat A. Kassabek

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  1. Samat A. Kassabek
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  2. Durvudkhan Suragan
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Correspondence to Samat A. Kassabek.

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Communicated by Wei Gong.

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This research was funded by Nazarbayev University under Collaborative Research Program Grant 20122022CRP1601

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Kassabek, S.A., Suragan, D. A heat polynomials method for the two-phase inverse Stefan problem. Comp. Appl. Math. 42, 129 (2023). https://doi.org/10.1007/s40314-023-02259-0

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