☁️Functional Analysis and Variational Methods for PDEs

The rigorous study of Partial Differential Equations (PDEs) heavily relies on functional analysis, particularly Hilbert and Sobolev spaces, which provide the essential framework for defining weak solutions, establishing their existence, uniqueness, and regularity, and underpinning variational methods crucial for both theoretical understanding and numerical approaches.

This "Cloud Computing" project section, "Functional Analysis and Variational Methods for PDEs," explores advanced mathematical concepts like Sobolev Space and weak derivatives, the Dirac Delta as a Distribution, the Cauchy–Schwarz Inequality, the Poincaré Inequality (discrete form), and the variational formulation of PDEs using the Poisson equation as an example.

🎬Animated result

Hilbert Space

Dirac Delta as a Distribution

Cauchy–Schwarz inequality for two vectors in 2D

Poincaré Inequality