Normalized systems of polyharmonic polynomials and their application to the solution of Riquier's problem (Q2743014)

The authors consider the new class of normalized polyharmonic polynomials [see \textit{E. P. Miles} and \textit{E. Williams}, Duke Math. J. 26, 35-40(1959; Zbl 0084.06602)] of the form NEWLINE\[NEWLINE\begin{multlined} H_{a,b,c}^{n+ 2(p-1),m,p}(x,y,z) =\\ \sum_{i=p-1}^{l+ p-m-1} \sum_{k=0}^m(-1)^{i+k-p+1}\binom{i+k}{k}\binom{i}{p-1}x^{2i+2k+a,!} y^{2m-2k+b,!} z^{n+2(p-1)-2m-2i-a-b,!}, \end{multlined}NEWLINE\]NEWLINE where \(a,b=0, 1\), \(t^{k,!}=t^k/k!\), \(m=0,1,\dotsc, l\), \(c=2\{\frac 12(n-a-b)\}\), \(l=[\frac 12(n-a-b)]\), and \([\cdot]\) and \(\{\cdot\}\) denote the integer and fractional parts of a number, respectively. Some properties of the above polyharmonic polynomials are investigated. On the base of these properties an approximate solution to the boundary value problem: \(\Delta^\nu u(x,y,z)=0\) in \(\Omega \subset \mathbb{R}^3\); \(\Delta^p u(x,y,z)_{\mid\partial\Omega}=f_p(s)\) for \(p=0, 1,\dotsc,\nu-1\), is obtained.