A novel space–time meshless method for solving the backward heat conduction problem

Abstract

This paper presents a novel space–time meshless method for solving the backward heat conduction problem (BHCP). A numerical approximation is obtained using the Trefftz basis function of the heat equation. The Trefftz method, which differs from conventional collocation methods based on a set of unstructured points in space, is used in this study to collocate boundary points in the space–time coordinate system such that the initial and boundary conditions can both be treated as boundary conditions on the space–time domain boundary. Because the solution in time on the boundary of the domain is unknown, the BHCP can be transformed into an inverse boundary value problem. The numerical solution is obtained by superpositioning the Trefftz base functions that automatically satisfy the governing equation. The validity of the proposed method is established for several test problems, including the one-dimensional BHCP and two-dimensional BHCP. The accuracy of the proposed method is compared with that of a conventional time-marching scheme based on the finite difference method. The results demonstrate that highly accurate numerical solutions can be obtained and errors may not accumulate over the entire time domain. Moreover, the boundary data on the inaccessible boundary can be recovered even when the partial data on the final time boundary are absent.