A Numerical Method for the Solution of the Two-Phase Fractional Lamé–Clapeyron–Stefan Problem
In this paper, we present a numerical solution of a two-phase fractional Stefan problem with time derivative described in the Caputo sense. In the proposed algorithm, we use a special case of front-fixing method supplemented by the iterative procedure, which allows us to determine the position of th...
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| Format: | Article |
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MDPI AG
2020-12-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/12/2157 |
| _version_ | 1797545812961525760 |
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| author | Marek Błasik |
| author_facet | Marek Błasik |
| author_sort | Marek Błasik |
| collection | DOAJ |
| description | In this paper, we present a numerical solution of a two-phase fractional Stefan problem with time derivative described in the Caputo sense. In the proposed algorithm, we use a special case of front-fixing method supplemented by the iterative procedure, which allows us to determine the position of the moving boundary. The presented method is an extension of a front-fixing method for the one-phase problem to the two-phase case. The novelty of the method is a new discretization of the partial differential equation dedicated to the second phase, which is carried out by introducing a new spatial variable immobilizing the moving boundary. Then, the partial differential equation is transformed to an equivalent integro-differential equation, which is discretized on a homogeneous mesh of nodes with a constant spatial and time step. A new convergence criterion is also proposed in the iterative algorithm determining the location of the moving boundary. The motivation for the development of the method is that the analytical solution of the considered problem is impossible to calculate in some cases, as can be seen in the figures in the paper. Moreover, the change of the boundary conditions makes obtaining a closed analytical solution very problematic. Therefore, creating new numerical methods is very valuable. In the final part, we also present some examples illustrating the comparison of the analytical solution with the results received by the proposed numerical method. |
| first_indexed | 2024-03-10T14:20:24Z |
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| id | doaj.art-5785b571e88c4645b443b3c687004aa6 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T14:20:24Z |
| publishDate | 2020-12-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj.art-5785b571e88c4645b443b3c687004aa62023-11-20T23:24:16ZengMDPI AGMathematics2227-73902020-12-01812215710.3390/math8122157A Numerical Method for the Solution of the Two-Phase Fractional Lamé–Clapeyron–Stefan ProblemMarek Błasik0Institute of Mathematics, Czestochowa University of Technology, Armii Krajowej 21, 42-201 Czestochowa, PolandIn this paper, we present a numerical solution of a two-phase fractional Stefan problem with time derivative described in the Caputo sense. In the proposed algorithm, we use a special case of front-fixing method supplemented by the iterative procedure, which allows us to determine the position of the moving boundary. The presented method is an extension of a front-fixing method for the one-phase problem to the two-phase case. The novelty of the method is a new discretization of the partial differential equation dedicated to the second phase, which is carried out by introducing a new spatial variable immobilizing the moving boundary. Then, the partial differential equation is transformed to an equivalent integro-differential equation, which is discretized on a homogeneous mesh of nodes with a constant spatial and time step. A new convergence criterion is also proposed in the iterative algorithm determining the location of the moving boundary. The motivation for the development of the method is that the analytical solution of the considered problem is impossible to calculate in some cases, as can be seen in the figures in the paper. Moreover, the change of the boundary conditions makes obtaining a closed analytical solution very problematic. Therefore, creating new numerical methods is very valuable. In the final part, we also present some examples illustrating the comparison of the analytical solution with the results received by the proposed numerical method.https://www.mdpi.com/2227-7390/8/12/2157moving boundary problemsfractional derivatives and integralsstefan problemsphase changesnumerical method |
| spellingShingle | Marek Błasik A Numerical Method for the Solution of the Two-Phase Fractional Lamé–Clapeyron–Stefan Problem Mathematics moving boundary problems fractional derivatives and integrals stefan problems phase changes numerical method |
| title | A Numerical Method for the Solution of the Two-Phase Fractional Lamé–Clapeyron–Stefan Problem |
| title_full | A Numerical Method for the Solution of the Two-Phase Fractional Lamé–Clapeyron–Stefan Problem |
| title_fullStr | A Numerical Method for the Solution of the Two-Phase Fractional Lamé–Clapeyron–Stefan Problem |
| title_full_unstemmed | A Numerical Method for the Solution of the Two-Phase Fractional Lamé–Clapeyron–Stefan Problem |
| title_short | A Numerical Method for the Solution of the Two-Phase Fractional Lamé–Clapeyron–Stefan Problem |
| title_sort | numerical method for the solution of the two phase fractional lame clapeyron stefan problem |
| topic | moving boundary problems fractional derivatives and integrals stefan problems phase changes numerical method |
| url | https://www.mdpi.com/2227-7390/8/12/2157 |
| work_keys_str_mv | AT marekbłasik anumericalmethodforthesolutionofthetwophasefractionallameclapeyronstefanproblem AT marekbłasik numericalmethodforthesolutionofthetwophasefractionallameclapeyronstefanproblem |