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.

In this paper, a localized boundary knot method is proposed, based on the local concept in the localized method of fundamental solutions. The localized boundary knot method is formed by combining the classical boundary knot method and the localization approach. The localized boundary knot method is truly free from mesh and numerical quadrature, so it has great potential for solving complicated engineering applications, such as multiply connected problems. In the proposed localized boundary knot method, both of the boundary nodes and interior nodes are required, and the algebraic equations at each node represent the satisfaction of the boundary condition or governing equation, which can be derived by using the boundary knot method at every subdomain. A sparse system of linear algebraic equations can be yielded using the proposed localized boundary knot method, which can greatly reduce the computer time and memory required in computer calculations. In this paper, several cases of simply connected domains and multi-connected domains of the Laplace equation and bi-harmonic equation are demonstrated to evidently verify the accuracy, convergence and stability of this proposed meshless method.

In this study, an efficient localized method of fundamental solution (LMFS) is applied to nonlinear heat conduction with mixed boundary conditions. Since the thermal conductivity is temperature-dependent, the Kirchhoff transformation is used to transform the nonlinear partial differential equations (PDEs) into Laplace equations with nonlinear boundary conditions. Then the LMFS is applied to the governing equation, and the nonlinear equations are treated by the fictitious time integration method (FTIM). Both 2D and 3D numerical examples are proposed to verify the effectiveness of the LMFS.

In this paper, a localized boundary knot method is adopted to solve two-dimensional inverse Cauchy problems, which are controlled by a second-order linear differential equation. The localized boundary knot method is a numerical method based on the local concept of the localization method of the fundamental solution. The approach is formed by combining the classical boundary knot method with the localization method. It has the potential to solve many complex engineering problems. Generally, in an inverse Cauchy problem, there are no boundary conditions in specific boundaries. Additionally, in order to be close to the actual engineering situation, a certain level of noise is added to the known boundary conditions to simulate the measurement error. The localized boundary knot method can be used to solve two-dimensional Cauchy problems more stably and is truly free from mesh and numerical quadrature. In this paper, the stability of the method is verified by using multi-connected domain and simply connected domain examples in Laplace equations.