The High Precision Numerical Methods Of The Second Kind Boundary Integral Equations

Laplace equation, Helmholtz equation, the biharmonic equation, the Stocksequation, the elastic mechanics equations and the fluid mechanics equation and so onare commonly used in engineering. These equations with boundary value condition canbe converted into the second boundary integral equations by the theory of the doublelayer potential. These boundary integral equations possess extremely importanttheoretical significance and practical background. The second boundary integralequations satisfy the Fredholm alternating theorem. However, it is impossible to get theanalytical solution of these equations in practical problem. We can only get thenumerical solution. Thus, the numerical solutions to these equations have receivedextensive attention in recent years.Because integral kernels of the equations are the normal derivative of the basicsolution at the boundary, it is hard for computer to accomplish automatically. Besides,the numerical differential calculation is unstable. When the border is a smooth curve orsmooth surface, the kernel of the integral equation and solution are continuous; andwhen there is angular point on the boundary curve or corner boundary surface is notsmooth, the kernel of the integral equation and solution have singularity, this leadsthat there are some difficulties to get the numerical solution of the second kind ofboundary integral equation. Now although there are many methods for solving this kindof integral equations, a large number of studies are projection methods based on theprojection operator theory framework. The numerical solutions have low accuracy bythe methods for solving the integral equation. The singularity are difficult to deal with.In this paper, firstly, we study the second kind of boundary integral equation. Thesingular reduction method is used to dealing with angular point, and get rid ofsingularity of integral equations of th integral kernals and solutions. The integraloperator converts into a compact operator.Secondly, we use the Galerkin method, collocation method and the Nystom methodfor solving the second kind of boundary integral equation. Numerical examples are given. By comparing with the numerical results of various methods, we illustrates theadvantages of each method.Finally, we present the iterative Galerkin method, the discrete iterative Galerkinmethod, iterative method, and discrete collocation method for solving the second kindof boundary integral equation. Numerical examples are given. The numerical results ofvarious methods are compared.