Localized Boundary Knot Method for Solving Two-Dimensional Inverse Cauchy Problems

Yang Wu, Junli Zhang, Shuang Ding and Yan-Cheng Liu
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Yang Wu: School of Civil Engineering and Architecture, Nanchang University, Nanchang 330031, China
Junli Zhang: School of Civil Engineering and Architecture, Nanchang University, Nanchang 330031, China
Shuang Ding: Shanghai Engineering Research Center of Underground Infrastructure Detection and Maintenance Equipment, Shanghai 200092, China
Yan-Cheng Liu: School of Civil Engineering and Architecture, Nanchang University, Nanchang 330031, China

Mathematics, 2022, vol. 10, issue 8, 1-17

Abstract: 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.

Keywords: inverse Cauchy problem; Laplace equation; localized boundary knot method; noise; multiply domain (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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