An Idea to Compute DtN Maps of PDE with Constant Coefficients
Abstract
Consider the following problem related to an ordinary differential equation: `For given constans u_{a},u_{b},M,N, function f(x) in [a,b], find ∂u/∂n at x = a,b which satisfies u″+Mu'+Nu = f in (a,b), u(a) = u_{a},u(b) = u_{b}.' This kind of problem is called the "DirichletNeumann map problem". This problem is usually solved in two steps. The solution is obtained in the first steps. In the second step, u'(a),u'(b) is computed by differentiating the solution. However, this twostep procedure is inefficient because the solution in (a,b) obtained in the first step is not essentially required. In this paper, the author presents a new strategy for obtaining Neumann data directly via a boundary integral equation formulation. Using this strategy, an explicit analytical expression of the DirichletNeumann map of this problem can be directly obtained by solving 2×2 matrices. Furthermore, an extension of the strategy to partial differential equations in twodimensional space and numerical algorithms is also presented.
 Publication:

Numerical Analysis and Applied Mathematics ICNAAM 2011: International Conference on Numerical Analysis and Applied Mathematics
 Pub Date:
 September 2011
 DOI:
 10.1063/1.3636768
 Bibcode:
 2011AIPC.1389..481A
 Keywords:

 partial differential equations;
 finite difference methods;
 finite element analysis;
 integration;
 02.30.Jr;
 02.70.Bf;
 02.70.Dh;
 02.60.Jh;
 Partial differential equations;
 Finitedifference methods;
 Finiteelement and Galerkin methods;
 Numerical differentiation and integration