The Modified Collocation Trefftz Method and Exponentially Convergent Scalar Homotopy Algorithm for the Inverse Boundary Determination Problem for the Biharmonic Equation

Abstract

AbstractIn this paper, the modified collocation Trefftz method (MCTM) and the exponentially convergent scalar homotopy algorithm (ECSHA) are adopted to analyze the inverse boundary determination problem governed by the biharmonic equation. The position for part of the boundary with given boundary condition is unknown and the position for the rest of the boundary with overspecified Cauchy boundary conditions is given a priori. Since the spatial position for portion of boundary is not given a priori, it is extremely difficult to solve such a boundary determination problem by any numerical scheme. In order to stably solve the boundary determination problem, the MCTM will be adopted in this study owing to that it can avoid the generation of mesh grid and numerical integration. When this problem is modeled by MCTM, a system of nonlinear algebraic equations will be formed and then be solved by ECSHA. Some numerical examples will be provided to demonstrate the ability and accuracy of the proposed scheme. In addition, the stability of the proposed meshless method will be tested by adding some noise into the prescribed boundary conditions and then to see how does that affect the numerical results.