The Pseudo Similarity Solutions of Laplace’s Equation with Source Type and Doublet Singularities

The solution of boundary-value problem of Laplace’s equation for a semi-infinite strip domain with Dirichlet boundary conditions was studied by applying the similarity method. For the B.C. $T\left(0,y\right)=f\left(y\right)$ , where $f\left(y\right)=T_0\left(1？m\frac{y}{H}\right)$ with constant $T_0$ , positive integer $m$ , and $H$ in this study. The global similarity solution with $m=0$ was found with similarity variable $\eta \left(x,y\right)$ fitted perfectly all the boundaries. Besides the global similarity solution, the new type of pseudo similarity solution of Laplace’s equation with source type and doublet type singularities in the domain was also studied. For $m=1$ , and $m=2$ , the pseudo similarity solutions were found with similarity variable $\eta \left(x,y\right)$ fitted only part of the boundaries. And the source type singularity can be found analytical near some singular point in the domain. For the boundary condition with Dirac delta function $T\left(0,y\right)=\delta ($