Explicit solutions for linear partial differential equations (Q2717953)

Explicit solutions to linear second-order partial differential equation with constant coefficients of the form NEWLINE\[NEWLINE \sum_{j=0}^{m}a_j\frac{\partial^2u}{\partial x_j^2}+2\sum_{j=0}^{m}b_j\frac{\partial u}{\partial x_j}+4cu=0\quad\text{in } D\subset \mathbb{R}^m, NEWLINE\]NEWLINE where \(a_1=a_2=1\) are considered. In the Theorem 3.1 the representation of finite solutions is obtained and in the Theorems 3.2 and 3.3 the linear independence of those solutions is considered. These results allow to construct a basis of polynomial solutions in \(\mathcal P\) [see \textit{P. S. Pedersen}, Adv. Math. 117, 157-163 (1996; Zbl 0849.35080)]. To illustrate the technique used Laplace equation, heat transfer equation, wave equation, and a particular variable coefficient PDE are solved.