Note on a boundary value problem of Poisson's equation (Q1891677)

The boundary value problem of Poisson's equation in the half-plane \(- \infty< x<\infty\), \(y>0\), \(\Delta \varphi= Q(x, y)\) and \(\varphi (x, 0)= f(x)\) is solved in an integral form using Green's function. For some source functions \(Q(x, y)\) these integrals become difficult to evaluate. In such cases, the similarity method of differential equations to evaluate analytically these integrals can be tried. By considering a special class of functions \(Q(x, y)\), the authors show in this note that the similarity analysis is reliable in this context.