TY  - EJOU
AU  - Tsai, Chia-Cheng 

TI  - Particular Solutions of Chebyshev Polynomials for Polyharmonic and Poly-Helmholtz Equations
T2  - Computer Modeling in Engineering \& Sciences

PY  - 2008
VL  - 27
IS  - 3
SN  - 1526-1506

AB  - In this paper we develop analytical particular solutions for the polyharmonic and the products of Helmholtz-type partial differential operators with Chebyshev polynomials at right-hand side. Our solutions can be written explicitly in terms of either monomial or Chebyshev bases. By using these formulas, we can obtain the approximate particular solution when the right-hand side has been represented by a truncated series of Chebyshev polynomials. These formulas are further implemented to solve inhomogeneous partial differential equations (PDEs) in which the homogeneous solutions are complementarily solved by the method of fundamental solutions (MFS). Numerical experiments, which include eighth order PDEs and three-dimensional cases, are carried out. Due to the exponential convergence of the Chebyshev interpolation and the MFS, our numerical results are extremely accurate.
KW  - Particular solution
KW  -  Chebyshev polynomials
KW  -  polyharmonic equation
KW  -  product of Helmholtz equation
KW  -  boundary element method
KW  -  method of fundamental solutions
KW  -  Trefftz method
KW  -  radial basis function

DO  - 10.3970/cmes.2008.027.151