Particular Solutions of Chebyshev Polynomials for Polyharmonic and Poly-Helmholtz Equations

Abstract: 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 ap-proximate 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 par-tial differential equations (PDEs) in which the ho-mogeneous solutions are complementarily solved by the method of fundamental solutions (MFS). Numerical experiments, which include eighth or-der PDEs and three-dimensional cases, are carried out. Due to the exponential convergence of the Chebyshev interpolation and the MFS, our numer-ical results are extremely accurate