Polynomial solutions of partial differential equations (Q1296538)

A method of finding polynomial solutions for second-order partial differential equations with constant coefficients in two variables with polynomial data is introduced. The polynomial solutions to the equation \(u_{xx}-\varepsilon u_{yy}=0, (\varepsilon=\pm 1)\), satisfying the condition \(u(x,y)=P_n(x,y)\) on \(S=\{(x,y)\in\mathbb R^2\mid x^2-\eta y^2=1\), \(\eta=\pm 1\}\), where \(P_n\) is a polynomial of degree \(n\) in \(x,y\) is discussed. Some examples when \(\varepsilon=-1\), \(S=\{(x,y)\in\mathbb R^2\mid \frac{x^2}{a^2}+\eta\frac{y^2}{b^2}=1, \eta=\pm 1\}\) and \(\varepsilon=-1\), \(S=\{(x,y)\in\mathbb R^2\mid x^4+y^4=r^4\}\) are considered. Existence of polynomial solutions to the equation \(\alpha u_{xx}+2\gamma u_{xy}+\beta u_{yy}=0\), satisfying the condition \(u(x,y)=Q_n(x,y)\) on the second degree curve \(S\), is proved. Polynomial solutions to the heat equation \(u_{xx}-u_{y}=0\) are discussed too. At the end of the paper one example for the biharmonic equation is given.