On Numerical Solution of Integral Equation of the First Kind

Many problems of computational electromagnetics [1,2], stationary heat transfer [3,4] and elasticity theory [5] can be naturally reduced to integral equations of the first kind (IEFK) with a weak singularity. So we shall name the equations with a logarithmic peculiarity in one-dimensional case and equation with a peculiarity of an aspect 1 0 rMM in two-dimensional case, where rMM 0 is distance between points M and M0 over surface of integration. In particular, Dirichlet problem for the one-dimensional equation of Laplace or Helmholtz [2] can be reduced to the solution of IEFK. Many work (see [1,2,5]) are devoted to properties of IEFKs with a logarithmic percularity in non-periodic and periodic cases. These equations are incorrect in the uniform metrics. If we applying quadrature formulas for their numerical solution, the system of the linear algebraic equations (SLAE) will be received. All elements of this matrix uniformly tend to zero at magnification of number of the equations to infinity. Therefore in the given situation round-off errors begin to influence on numerical solution. If we differentiate this equation, we shall receive (together with some additional conditions) a singular integral equation (SIE) of the first kind, equivalent to the initial equation. This SIE is also incorrect in the uniform metrics, but for it the method of a numerical solution such as method of discrete percularities (MDP) has been developed [3]. MDP is self-regularizating method in the uniform metrics. The determinant of a MDP matrix does not tend to zero if we increase a number of equations. In [3] was considered the influence of number of equations in this SLAE and magnitude of round-off errors on accuracy of numerical solution. In present work the similar method is offered for a IEFK with a percularity 1 0 rMM on a rectangle.