Chebyshev tau meshless method based on the integration-differentiation for Biharmonic-type equations on irregular domain

This paper reports a new method, Chebyshev tau meshless method based on the integration–differentiation (CTMMID) for numerically solving Biharmonic-type equations on irregularly shaped domains with complex boundary conditions. In general, the direct application of Chebyshev spectral method based on differentiation process to the fourth order equations leads to the corresponding differentiation matrix with large condition numbers. From another aspect, the strategy based on the integral formula of a Chebyshev polynomial could not only create sparsity, but also improve the accuracy, however it requires a lot of computational cost for directly solving high order two-dimensional problems. In this paper, the construction of the Chebyshev approximations is to start from the mix partial derivative uxxyy(x,y) rather than the unknown function u(x,y). The irregular domain is embedded in a domain of rectangle shape and the curve boundary can be efficiently treated by CTMMID. The numerical results show that compared with the existing results, our method yields spectral accuracy, and the main distinguishing feature is reducing the condition number of fourth order equations on rectangle domain from O(N8) to O(N4). It also appears that CTMMID is effective for the problems on irregular domains.