Xiaodan Zhao

Xiaodan Zhao, Li-Lian Wang, Ziqing Xie

This paper provides a rigorous and delicate analysis for exponential decay of Jacobi polynomial expansions of analytic functions associated with the Bernstein ellipse. Using an argument that can recover the best estimate for the Chebyshev expansion, we derive various new and sharp bounds of the expansion coefficients, which are featured with explicit dependence of all related parameters and valid for degree $n\ge 1$. We demonstrate the sharpness of the estimates by comparing with existing ones, in particular, the very recent results in [38, SIAM J. Numer. Anal., 2012]. We also extend this argument to estimate the Gegenbauer-Gauss quadrature remainder of analytic functions, which leads to some new tight bounds for quadrature errors.

Li-Lian Wang, Bo Wang, Xiaodan Zhao

This paper is concerned with fast and accurate computation of exterior wave equations truncated via exact circular or spherical nonreflecting boundary conditions (NRBCs, which are known to be nonlocal in both time and space). We first derive analytic expressions for the underlying convolution kernels, which allow for a rapid and accurate evaluation of the convolution with $O(N_t)$ operations over $N_t$ successive time steps. To handle the onlocality in space, we introduce the notion of boundary perturbation, which enables us to handle general bounded scatters by solving a sequence of wave equations in a regular domain. We propose an efficient spectral-Galerkin solver with Newmark's time integration for the truncated wave equation in the regular domain. We also provide ample numerical results to show high-order accuracy of NRBCs and efficiency of the proposed scheme.

Ziqing Xie, Li-Lian Wang, Xiaodan Zhao

This paper is devoted to a rigorous analysis of exponential convergence of polynomial interpolation and spectral differentiation based on the Gegenbauer-Gauss and Gegenbauer-Gauss-Lobatto points, when the underlying function is analytic on and within an ellipse. Sharp error estimates in the maximum norm are derived.

Xiaodan Zhao, Li-Lian Wang

We report in this paper a fast and accurate algorithm for computing the exact spherical nonreflecting boundary condition (NRBC) for time-dependent Maxwell's equations. It is essentially based on a new formulation of the NRBC, which allows for the use of an analytic method for computing the involved inverse Laplace transform. This tool can be generically integrated with the interior solvers for challenging simulations of electromagnetic scattering problems. We provide some numerical examples to show that the algorithm leads to very accurate results.

Sanjay Krishna Anbalagan, Xinrui Nie, Umesh Mohan et al.

Domain specific chatbot applications often involve multi step interactions, such as refining search filters, selecting multiple items, or performing comparisons. Traditional graphical user interfaces (GUIs) handle these workflows by providing explicit "Submit" (commit data) and "Reset" (discard data) actions, allowing back-end systems to track user intent unambiguously. In contrast, conversational agents rely on subtle language cues, which can lead to confusion and incomplete context management. This paper proposes modeling these GUI inspired metaphors acknowledgment (submit like) and context switching (reset-like) as explicit tasks within large language model (LLM) prompts. By capturing user acknowledgment, reset actions, and chain of thought (CoT) reasoning as structured session data, we preserve clarity, reduce user confusion, and align domain-specific chatbot interactions with back-end logic. We demonstrate our approach in hotel booking and customer management scenarios, highlighting improvements in multi-turn task coherence, user satisfaction, and efficiency.