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# Meshless Local Petrov–Galerkin Formulation of Inverse Stefan Problem via Moving Least Squares Approximation

Department of Applied Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin 34149-16818, Iran
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Math. Comput. Appl. 2019, 24(4), 101; https://doi.org/10.3390/mca24040101
Received: 6 November 2019 / Revised: 5 December 2019 / Accepted: 9 December 2019 / Published: 10 December 2019
In this paper, we study the meshless local Petrov–Galerkin (MLPG) method based on the moving least squares (MLS) approximation for finding a numerical solution to the Stefan free boundary problem. Approximation of this problem, due to the moving boundary, is difficult. To overcome this difficulty, the problem is converted to a fixed boundary problem in which it consists of an inverse and nonlinear problem. In other words, the aim is to determine the temperature distribution and free boundary. The MLPG method using the MLS approximation is formulated to produce the shape functions. The MLS approximation plays an important role in the convergence and stability of the method. Heaviside step function is used as the test function in each local quadrature. For the interior nodes, a meshless Galerkin weak form is used while the meshless collocation method is applied to the the boundary nodes. Since MLPG is a truly meshless method, it does not require any background integration cells. In fact, all integrations are performed locally over small sub-domains (local quadrature domains) of regular shapes, such as intervals in one dimension, circles or squares in two dimensions and spheres or cubes in three dimensions. A two-step time discretization method is used to deal with the time derivatives. It is shown that the proposed method is accurate and stable even under a large measurement noise through several numerical experiments. View Full-Text
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MDPI and ACS Style

Karami, A.; Abbasbandy, S.; Shivanian, E. Meshless Local Petrov–Galerkin Formulation of Inverse Stefan Problem via Moving Least Squares Approximation. Math. Comput. Appl. 2019, 24, 101. https://doi.org/10.3390/mca24040101

AMA Style

Karami A, Abbasbandy S, Shivanian E. Meshless Local Petrov–Galerkin Formulation of Inverse Stefan Problem via Moving Least Squares Approximation. Mathematical and Computational Applications. 2019; 24(4):101. https://doi.org/10.3390/mca24040101

Chicago/Turabian Style

Karami, A., Saeid Abbasbandy, and E. Shivanian 2019. "Meshless Local Petrov–Galerkin Formulation of Inverse Stefan Problem via Moving Least Squares Approximation" Mathematical and Computational Applications 24, no. 4: 101. https://doi.org/10.3390/mca24040101

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