چکیده (انگلیسی):

The uniqueness of solution of boundary integral equations (BIEs) is studied here
when geometry of boundary and unknown functions is assumed piecewise constant. In fact we will show the BIEs with 3–times monotone radial kernels have unique piecewise constant solution. In this paper non–negative radial function F3 is introduced which has important contribution in proving the uniqueness. It can be found from the paper if 3 is sufficiently small then eigenvalues of the boundary integral operator are bigger than F3/2. Note that there is a smart relation between 3 and boundary discretization which is reported in the paper, clearly. In this article an appropriate constant c0 is found which ensures uniqueness of solution of BIE with logarithmic kernel ln(c0r) as fundamental solution of Laplace equation. Also non–singular BIEs are proposed which can be used in boundary elements method (BEM) instead of singular ones to solve partial differential equations (PDEs). Then singular boundary integrals are vanished from BEM when the non–singular BIEs are used. As a result, an upper bound for condition number of constant Galerkin BEM’s system matrix is obtained when the size of boundary cells decreases. The upper bound found depends on three important issues: geometry of boundary, size of boundary cells and the kernel function. Finally some numerical examples are presented which confirm the analytical results.