Chebyshev polynomial solutions of second-order linear partial differential equations
Approximate polynomial solutions to second-order linear PDE with variable coefficients of the following form \[ A(x,y)\frac{\partial^2 u}{\partial x^2}+ B(x,y)\frac{\partial^2 u}{\partial x\partial y}+ C(x,y)\frac{\partial^2 u}{\partial y^2}+ D(x,y)\frac{\partial u}{\partial x}+ E(x,y)\frac{\partial u}{\partial y}+F(x,y)= G(x,y)\tag{*} \] are considered. The proposed method is based on taking the truncated Chebyshev expansions of the unknown function \(u(x,y)=\sum_{r=0}^n \sum_{s=0}^na_{r,s}T_{r,s}(x,y)\), where \(T_{r,s}(x,y)\) are the Chebyshev polynomials defined in [\textit{N. K. Basu}, SIAM J. Numer. Anal. 10, 496-505 (1973; Zbl 0254.41011)] and also of all coefficients of equation (*) and all functions included in the boundary value conditions to equation (*). The original boundary value problem (or initial value problem) is reduced to the system of algebraic equations for determining coefficients \(a_{r,s}\). Three examples are given.
scientific article; zbMATH DE number 1861288
scientific article