Applications of the Method of Fundamental Solutions to the Helmholtz, Diffusion and Burgers’ Equations
Date Issued
2005
Date
2005
Author(s)
Hu, Shu-Ping
DOI
en-US
Abstract
The method of fundamental solutions (MFS) is one of the popular meshless methods, gaining attention in the recent past. Since this method is free from the integration of the singular functions, this method has been applied for the solution of partial differential equations representing many engineering problems. The present thesis focuses on the application of the MFS to simulate problems of elliptical waveguides, Stokes’ first and second problems and Burgers’ equation.
Initially the MFS was utilized to solve elliptical waveguide problems by solving the Helmholtz equation using the singular value decomposition (SVD) method. The method could predict the results for the cutoff wavelengths in close agreement with analytical results. Later the MFS was applied to solve unsteady Stokes’ first and second problems. The time derivatives are handled by a time-space domain concept, which completely avoids the requirement of Laplace transformation or the finite difference scheme to discretize the time derivatives. Results obtained for the unsteady Stokes’ first and second problems indicate that the MFS could predict results closer to the analytical solutions. An error analysis carried out also demonstrates that the proposed numerical scheme based on the MFS can produce stable numerical results for unsteady problems solved on semi-infinite domain.
Finally, the MFS procedure was extended to solve non-linear Burgers’ equation in combination with the Eulerian-Lagrangian method and the Cole-Hopf transformation independently. The numerical experiments demonstrate that the MFS performs very well in combination with the above schemes to solve non-linear partial differential equations as well. Results obtained for many test cases of the non-linear Burgers’ equations in 1-D and 2-D domains indicate the present scheme could produce results closer to the analytical results. The results discussed in the thesis show that the MFS is a powerful meshless numerical scheme to solve non-linear partial differential equations.
Subjects
無網格法
基本解法
赫姆霍茲方程式
特徵值
奇異值分解法
波導管
擴散方程式
史托克斯第一問題
史托克斯第二問題
尤拉-拉格朗日法
柏格斯方程式
柯爾霍普夫轉換
Meshless
Method of fundamental solutions
Helmholtz equation
Eigenvalue
Singular value decomposition
Waveguides
Stokes’ first problem
Stokes’ second problem
Burgers’ equation
Eulerian-Lagrangian method
Cole-Hopf transformation.
Type
thesis
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