Title: Advanced Spectral Approximations for Quadratic Pencils and Schrödinger Operators: A \nu-Convergence and Integral Framework

Abstract: This plenary lecture surveys recent breakthroughs in the theoretical analysis and high-performance numerical computation of non-linear spectral problems and integral equations. Focusing on quadratic operator pencils of the form L(\lambda) = \lambda^2 A + \lambda B + C, we demonstrate their critical role in modeling complex Schrödinger operators. Building upon joint work with S. Kamouche and A. Khellaf, we introduce an extended Kantorovich projection strategy embedded within a generalized \nu-convergence framework. This combined approach systematically overcomes the long-standing challenge of spectral pollution, suppressing spurious modes and delivering robust convergence guarantees for exact eigenvalues. Expanding on these operator-theoretic techniques, we also highlight advanced solvers for non-linear Volterra integral equations with weakly singular kernels, emphasizing product integration formulas and Picard-based iteration protocols.

Title: High order nonlocal abstract Boussinesq equations and applications

Abstract: In this talk, the Cauchy problem for linear and nonlinear nonlocal Boussinesq equations is studied. The equation involves convolution terms including abstract operator functions in Fourier type Banach space E. Here, assuming enough smoothness on the initial data and the growth assumptions on operator functions, the local, global existence, uniqueness and regularity properties of solutions are established in terms of fractional powers of given sectorial operator. We can obtain a different classes of nonlocal Boussinesq equations by choosing the space E and linear operator, which occur in a wide variety of physical systems.