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On 14 September 2025, at VNUHCM–University of Science, doctoral researcher Nguyễn Vũ Dzũng (Class of 2022) successfully defended his Doctoral thesis in Analytic Mathematics, entitled “Boundary Value Problems for Kirchhoff–Carrier Wave Equations with Discrete Nonlocal Terms”, under the supervision of Assoc. Prof. Lê Thị Phương Ngọc and Dr Nguyễn Thị Thu Vân.

The research focuses on the existence and uniqueness of weak solutions for Robin–Dirichlet boundary value problems in nonlinear Kirchhoff–Carrier wave equations containing viscous-elastic and discrete nonlocal terms. The Kirchhoff–Carrier equation serves as a key mathematical model for describing nonlinear waves in elastic materials or complex dynamical systems, while the discrete nonlocal term makes the solution at a point dependent on values at multiple locations, increasing analytical complexity. The concept of weak solutions allows the study of such problems in broader function spaces, suitable for addressing challenging nonlinear problems. In particular, when the discrete nonlocal terms are expressed as integral sums of Kirchhoff–Carrier terms, the thesis demonstrates convergence of the solution sequence (P_n) to the weak solution of the limiting problem (P_∞), enabling approximate solutions to complex problems and facilitating theoretical validation and numerical implementation.

The thesis achieves key results across five Robin–Dirichlet problems, focusing on two main directions. Firstly, for single nonlinear Kirchhoff–Carrier wave equations with discrete nonlocal terms, it proves the existence and uniqueness of weak solutions and, in many cases, the convergence of solution sequences to the weak solution of the limiting problem. Secondly, for systems of nonlinear Kirchhoff–Carrier wave equations with discrete nonlocal terms, the thesis establishes the existence and uniqueness of weak solutions (uⁿ,vⁿ), confirms convergence to the weak solution of the limiting system, and develops a high-order iterative algorithm for computational simulation of complex physical systems. These results are of substantial theoretical significance in pure mathematics and have potential applications in nonlinear wave modelling, material mechanics, and engineering simulations.

The thesis has been recognised internationally through publications in reputable Scopus- and SCIE-indexed journals, including Results in Nonlinear Analysis (Q3), Nonlinear Functional Analysis and Applications (Q3), Mathematica Bohemica (Q3), Transactions of the National Academy of Sciences of Azerbaijan (Q3), and Mathematical Modelling and Analysis (Q2).

Building on these results, further research directions include investigating dynamic properties of solutions (finite-time blow-up, decay, stability) and developing computational examples to illustrate and verify theoretical findings.

For the full abstract of the thesis, please refer to the official page: