Search

Showing total 2 results
2 results

1. Numerical solutions of elliptic partial differential equations using Chebyshev polynomials.

Author

Khatri Ghimire, B., Tian, H.Y., and Lamichhane, A.R.

Subjects
*NUMERICAL analysis, *CHEBYSHEV polynomials, *ELLIPTIC differential equations, *APPROXIMATION theory, *INTERPOLATION, *STOCHASTIC convergence, *BOUNDARY value problems
Abstract

We present a simple and effective Chebyshev polynomial scheme (CPS) combined with the method of fundamental solutions (MFS) and the equilibrated collocation Trefftz method for the numerical solutions of inhomogeneous elliptic partial differential equations (PDEs). In this paper, CPS is applied in a two-step approach. First, Chebyshev polynomials are used to approximate a particular solution of a PDE. Chebyshev nodes which are the roots of Chebyshev polynomials are used in the polynomial interpolation due to its spectral convergence. Then the resulting homogeneous equation is solved by boundary type methods including the MFS and the equilibrated collocation Trefftz method. Numerical results for problems on various irregular domains show that our proposed scheme is highly accurate and efficient. [ABSTRACT FROM AUTHOR]

Published
2016
Full Text
View/download PDF

2. Shape variable radial basis function and its application in dual reciprocity boundary face method

Author

Zhou, Fenglin, Zhang, Jianming, Sheng, Xiaomin, and Li, Guangyao

Subjects
*RADIAL basis functions, *MULTIVARIATE analysis, *APPROXIMATION theory, *BOUNDARY value problems, *INTERPOLATION, *COMPARATIVE studies, *NUMERICAL analysis, *DIFFERENTIAL equations
Abstract

Abstract: The radial basis functions (RBFs) is an efficient tool in multivariate approximation, but it usually suffers from an ill-conditioned interpolation matrix when interpolation points are very dense or irregularly spaced. The RBFs with variable shape parameters can usually improve the interpolation matrix condition number. In this paper a new shape parameter variation scheme is implemented. Comparison studies with the constant shaped RBF on convergence and stability are made. Results show that under the same accuracy level, the interpolation matrix condition number by our scheme grows much slower than that of the constant shaped RBF interpolation matrix with increase in the number of interpolation points. As an application example, the dual reciprocity method equipped with the new RBF is combined with the boundary face method to solve boundary value problems governed by Poisson equations. Numerical results further demonstrate the robustness and better stability of the new RBF. [ABSTRACT FROM AUTHOR]

Published
2011
Full Text
View/download PDF