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Polynomial solutions of linear partial differential equations

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  • 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.

    Mathematics Subject Classification: Primary: 35B99, 35G05; Secondary: 35A25, 35C99.


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