scholarly journals INverse problem of identifying a time-dependent coefficient and free boundary in heat conduction equation by using the meshless local Petrov-Galerkin (MLPG) method via moving least squares approximation

Filomat ◽  
2020 ◽  
Vol 34 (10) ◽  
pp. 3319-3337
Author(s):  
Akbar Karami ◽  
Saeid Abbasbandy ◽  
Elyas Shivanian
Keyword(s):  
Inverse Problem ◽  
Heat Conduction ◽  
Free Boundary ◽  
Least Squares ◽  
Boundary Problem ◽  
Heat Conduction Equation ◽  
Time Dependent ◽  
Moving Least Squares ◽  
Fixed Boundary ◽  
Mlpg Method

In this paper we investigated the inverse problem of identifying an unknown time-dependent coefficient and free boundary in heat conduction equation. By using the change of variable we reduced the free boundary problem into a fixed boundary problem. In direct solver problem we employed the meshless local Petrov-Galerkin (MLPG) method based on the moving least squares (MLS) approximation. Inverse reduced problem with fixed boundary is nonlinear and we formulated it as a nonlinear least-squares minimization of a scalar objective function. Minimization is performed by using of f mincon routine from MATLoptimization toolbox accomplished with the Interior - point algorithm. In order to deal with the time derivatives, a two-step time discretization method is used. It is shown that the proposed method is accurate and stable even under a large measurement noise through several numerical experiments.

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

2019 ◽  
Vol 24 (4) ◽  
pp. 101
Author(s):  
A. Karami ◽  
Saeid Abbasbandy ◽  
E. Shivanian
Keyword(s):  
Free Boundary ◽  
Least Squares ◽  
Boundary Problem ◽  
Moving Boundary ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Moving Least Squares ◽  
Heaviside Step Function ◽  
Mlpg Method ◽  
Mls Approximation

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.

Stabilised MLS in MLPG method for heat conduction problem

2019 ◽  
Vol 36 (4) ◽  
pp. 1323-1345
Author(s):  
Rituraj Singh ◽  
Krishna Mohan Singh
Keyword(s):  
Heat Conduction ◽  
Design Methodology ◽  
Heat Conduction Problem ◽  
Moving Least Squares ◽  
Moment Matrix ◽  
Content Type ◽  
Patch Tests ◽  
Mlpg Method ◽  
The One ◽  
Method Analysis

Purpose The purpose of this paper is to assess the performance of the stabilised moving least squares (MLS) scheme in the meshless local Petrov–Galerkin (MLPG) method for heat conduction method. Design/methodology/approach In the current work, the authors extend the stabilised MLS approach to the MLPG method for heat conduction problem. Its performance has been compared with the MLPG method based on the standard MLS and local coordinate MLS. The patch tests of MLS and modified MLS schemes have been presented along with the one- and two-dimensional examples for MLPG method of the heat conduction problem. Findings In the stabilised MLS, the condition number of moment matrix is independent of the nodal spacing and it is nearly constant in the global domain for all grid sizes. The shifted polynomials based MLS and stabilised MLS approaches are more robust than the standard MLS scheme in the MLPG method analysis of heat conduction problems. Originality/value The MLPG method based on the stabilised MLS scheme.

Long-time existence of classical solutions to a one-dimensional swelling gel

2014 ◽  
Vol 25 (01) ◽  
pp. 165-194 ◽  
Author(s):  
M. Carme Calderer ◽  
Robin Ming Chen
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Lagrangian Formulation ◽  
Classical Solutions ◽  
Governing Equations ◽  
Fixed Boundary ◽  
One Dimensional ◽  
Long Time Existence ◽  
Long Time ◽  
Time Existence

In this paper, we derived a model which describes the swelling dynamics of a gel and study the system in one-dimensional geometry with a free boundary. The governing equations are hyperbolic with a weakly dissipative source. Using a mass-Lagrangian formulation, the free boundary is transformed into a fixed boundary. We prove the existence of long-time C1-solutions to the transformed fixed boundary problem.

scholarly journals Analytical Solution of the Hyperbolic Heat Conduction Equation for Moving Semi-Infinite Medium under the Effect of Time-Dependent Laser Heat Source

2009 ◽  
Vol 2009 ◽  
pp. 1-18 ◽  
Author(s):  
R. T. Al-Khairy ◽  
Z. M. AL-Ofey
Keyword(s):  
Heat Conduction ◽  
Analytical Solution ◽  
Heat Source ◽  
Heat Conduction Equation ◽  
Closed Form Solution ◽  
Infinite Medium ◽  
Form Solution ◽  
Time Dependent ◽  
Hyperbolic Heat Conduction ◽  
Laser Heat Source

This paper presents an analytical solution of the hyperbolic heat conduction equation for moving semi-infinite medium under the effect of time dependent laser heat source. Laser heating is modeled as an internal heat source, whose capacity is given by while the semi-infinite body has insulated boundary. The solution is obtained by Laplace transforms method, and the discussion of solutions for different time characteristics of heat sources capacity (constant, instantaneous, and exponential) is presented. The effect of absorption coefficients on the temperature profiles is examined in detail. It is found that the closed form solution derived from the present study reduces to the previously obtained analytical solution when the medium velocity is set to zero in the closed form solution.

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