Daniele Funaro Libri

Focusing on spectral methods, this book explores approximate solution techniques for differential equations using classical orthogonal polynomials. It highlights their growing popularity as a competitive alternative to traditional methods, particularly in analyzing periodic solutions through trigonometric and algebraic polynomials. Aimed at beginners, it provides foundational mathematical concepts, essential formulas, and theorems, all presented in an accessible manner without requiring prior knowledge of functional analysis. The book encourages practical experimentation and skill enhancement in numerical applications.

The book explores a revised model of electromagnetism that incorporates solitary waves, leading to a fresh perspective on atomic structures. By treating photons deterministically, it challenges traditional quantum physics foundations, depicting atoms and molecules as structured aggregates of nuclei and electrons connected by resonating photon layers. This innovative approach not only addresses technical issues in physical chemistry but also prompts deeper epistemological inquiries, potentially reshaping established scientific viewpoints.

In recent years, there has been increasing interest in developing numerical techniques for approximating differential model problems with multiscale solutions. These problems often feature functions that behave smoothly except in specific regions with sudden, sharp variations, such as internal or boundary layers. When the discretization process lacks sufficient degrees of freedom to finely resolve these layers, stabilization procedures are necessary to prevent oscillatory effects without introducing excessive artificial viscosity. In finite element analysis, methods like streamline diffusion, Galerkin least-squares, and bubble function approaches effectively address transport equations of elliptic type with small diffusive terms, known in fluid dynamics as advection-diffusion equations. This work aims to guide readers in constructing a computational code based on the spectral collocation method using algebraic polynomials. It focuses on approximating elliptic type boundary-value partial differential equations in 2-D, particularly transport-diffusion equations where second-order diffusive terms are significantly overshadowed by first-order advective terms. The applications discussed will highlight cases where nonlinear systems of partial differential equations can be simplified to a sequence of transport-diffusion equations.