High-precision Numerical Solution Of Boundary Integral Equations On Unbounded Domains

Steady problems in engineering can be attributed to the Laplace equation which belongs to elliptic partial differential equations. Such as the torsion of elastic rod,thermal stability,steady flow field,electromagnetic field,etc. Therefore, it is a very significant topic to research the computational method of Laplace equation in half space, and it has been focused on recently.In this paper, we change the Laplace equation of Dirichlet problem on the half plane into the first kind of boundary integral equation and give a detailed analysis of the solvability expressed by the potential of a single layer. Then we mainly use Green function method to get the Poisson integral formulas of Laplace equation's Dirichlet problem on the half plane analytically and give two examples. In another way,we adopt tangent transformation to change the unbounded boundary value problem into a bounded one,then we discretize the first kind of boundary integral equation by collocation method. During this period,with regard to integral kernel of logarithmic singularity,we recur to Maclaurin's formula and multiple-angle cosine formula's expansion to eliminate integral kernel's singularity effectively. By using the mathematical software of Matlab procedure, we give two examples to verify the feasibility and effectiveness of the collocation method towards the same example which has been listed by Green function method.