May  2009, 8(3): 1053-1065. doi: 10.3934/cpaa.2009.8.1053

Polynomial solutions of linear partial differential equations

1. 

Department of Engineering Sciences, Division of Applied Mathematics and Mechanics, University of Patras, 26500 Patras, Greece

2. 

Department of Mathematics, University of Patras, 26500 Patras, Greece

Received  March 2007 Revised  October 2007 Published  February 2009

In this paper it is proved that the condition

$\lambda=a_1 (n-2)(n-1)+\gamma_1 (m-2)(m-1)+\beta_1 (n-1)(m-1)+\delta_1 (n-1)+\epsilon_1 (m-1),$

where $n=1,2,...,N$, $m=1,2,...,M$ is a necessary and sufficient condition for the linear partial differential equation

$(a_1x^2+a_2x+a_3)u_{x x}+(\beta_1xy+\beta_2x+\beta_3y+\beta_4)u_{x y} $

$+(\gamma_1y^2+\gamma_2y+\gamma_3)u_{y y}+(\delta_1x+\delta_2)u_x+(\epsilon_1y+\epsilon_2)u_y=\lambda u, $

where $a_i$, $\beta_j$, $\gamma_i$, $\delta_s$, $\epsilon_s$, $i=1,2,3$, $j=1,2,3,4$, $s=1,2$ are real or complex constants, to have polynomial solutions of the form

$u(x,y)=\sum_{n=1}^N\sum_{m=1}^Mu_{n m}x^{n-1}y^{m-1}.$

The proof of this result is obtained using a functional analytic method which reduces the problem of polynomial solutions of such partial differential equations to an eigenvalue problem of a specific linear operator in an abstract Hilbert space. The main result of this paper generalizes previously obtained results by other researchers.

Citation: Eugenia N. Petropoulou, Panayiotis D. Siafarikas. Polynomial solutions of linear partial differential equations. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1053-1065. doi: 10.3934/cpaa.2009.8.1053
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