Polynomial solutions of linear partial differential equations
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.
Keywords:
- Elliptic partial differential equation
- Mathematical analysis
- Hessian equation
- FTCS scheme
- Polynomial
- Separable partial differential equation
- d'Alembert's formula
- Mathematics
- Symbol of a differential operator
- Linear differential equation
- Combinatorics
- Homogeneous differential equation
- First-order partial differential equation
- Hilbert space
- Eigenvalues and eigenvectors
- Correction
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