Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 4

heat equation Related Publications

4 Topological Sensitivity Analysis for Reconstruction of the Inverse Source Problem from Boundary Measurement

Authors: Maatoug Hassine, Mourad Hrizi

Abstract:

In this paper, we consider a geometric inverse source problem for the heat equation with Dirichlet and Neumann boundary data. We will reconstruct the exact form of the unknown source term from additional boundary conditions. Our motivation is to detect the location, the size and the shape of source support. We present a one-shot algorithm based on the Kohn-Vogelius formulation and the topological gradient method. The geometric inverse source problem is formulated as a topology optimization one. A topological sensitivity analysis is derived from a source function. Then, we present a non-iterative numerical method for the geometric reconstruction of the source term with unknown support using a level curve of the topological gradient. Finally, we give several examples to show the viability of our presented method.

Keywords: Topological Optimization, heat equation, geometric inverse source problem, topological sensitivity, Kohn-Vogelius formulation

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3 Recovering the Boundary Data in the Two Dimensional Inverse Heat Conduction Problem Using the Ritz-Galerkin Method

Authors: Kamal Rashedi, Saeed Sarabadan

Abstract:

This article presents a numerical method to find the heat flux in an inhomogeneous inverse heat conduction problem with linear boundary conditions and an extra specification at the terminal. The method is based upon applying the satisfier function along with the Ritz-Galerkin technique to reduce the approximate solution of the inverse problem to the solution of a system of algebraic equations. The instability of the problem is resolved by taking advantage of the Landweber’s iterations as an admissible regularization strategy. In computations, we find the stable and low-cost results which demonstrate the efficiency of the technique.

Keywords: Inverse problem, Parabolic Equations, heat equation, Ritz-Galerkin method, Landweber iterations

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2 Lagrange and Multilevel Wavelet-Galerkin with Polynomial Time Basis for Heat Equation

Authors: Montri Maleewong, Watcharakorn Thongchuay, Puntip Toghaw

Abstract:

The Wavelet-Galerkin finite element method for solving the one-dimensional heat equation is presented in this work. Two types of basis functions which are the Lagrange and multi-level wavelet bases are employed to derive the full form of matrix system. We consider both linear and quadratic bases in the Galerkin method. Time derivative is approximated by polynomial time basis that provides easily extend the order of approximation in time space. Our numerical results show that the rate of convergences for the linear Lagrange and the linear wavelet bases are the same and in order 2 while the rate of convergences for the quadratic Lagrange and the quadratic wavelet bases are approximately in order 4. It also reveals that the wavelet basis provides an easy treatment to improve numerical resolutions that can be done by increasing just its desired levels in the multilevel construction process.

Keywords: heat equation, Galerkin finite element method, Lagrange basis function, Wavelet basis function

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1 A Comparison of Some Splines-Based Methods for the One-dimensional Heat Equation

Authors: Ahmad Izani Md. Ismail, Joan Goh, Ahmad Abd. Majid

Abstract:

In this paper, collocation based cubic B-spline and extended cubic uniform B-spline method are considered for solving one-dimensional heat equation with a nonlocal initial condition. Finite difference and θ-weighted scheme is used for time and space discretization respectively. The stability of the method is analyzed by the Von Neumann method. Accuracy of the methods is illustrated with an example. The numerical results are obtained and compared with the analytical solutions.

Keywords: heat equation, Collocation based, Cubic Bspline, Extended cubic uniform B-spline

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