Solutions of boundary detection problem for modified Helmholtz equation by Trefftz method

Abstract

The boundary detection problem, which is governed by modified Helmholtz equation, is analysed by the Trefftz method and the exponentially convergent scalar homotopy algorithm (ECSHA). The spatial position for part of boundary with given boundary condition is unknown; therefore, it is very difficult to analyse the boundary detection problem by any numerical scheme. In this study, we used the Trefftz method to analyse the boundary detection problem, governed by the modified Helmholtz equation, and the numerical solution of Trefftz method is expressed as a linear combination of the T-complete functions. By using the Trefftz method to analyse the problem, a system of non-linear algebraic equations will be formed and then solved by the ECSHA which will converge exponentially. The unknown coefficients in Trefftz method and the spatial position of the unknown boundary positions can be simultaneously found via evolutionary process of the ECSHA. Some numerical examples will be provided to validate the accuracy and efficacy of the proposed scheme.