functional-analysis-and-pdes

Solving PDEs using weak formulations and functional analysis (Lax-Milgram, drift-diffusion)

https://github.com/georgessakr/functional-analysis-and-pdes

Science Score: 44.0%

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Repository

Solving PDEs using weak formulations and functional analysis (Lax-Milgram, drift-diffusion)

Basic Info

Host: GitHub
Owner: GeorgesSakr
License: mit
Default Branch: main
Size: 594 KB

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Created 8 months ago · Last pushed 8 months ago

Metadata Files

Readme License Citation

README.md

Weak-Form Solutions of Drift–Diffusion PDEs via Functional Analysis

This project presents a rigorous functional-analytic treatment of a family of drift–diffusion partial differential equations (PDEs).

Prove the Lax–Milgram theorem in the symmetric case and use it to establish existence and uniqueness of weak solutions for elliptic problems.
Formulate and solve a steady-state drift–diffusion equation on the upper-right quarter unit disk (Ω ⊂ ℝ²) with mixed Dirichlet–Neumann conditions.
Analyse the solution operator T : L²(Ω) → H₀ proving boundedness, compactness, and describing its spectrum via the Fredholm alternative.
Solve an associated Sturm–Liouville eigenvalue problem, obtaining an infinite countable spectrum {λₙ} with λₙ → ∞.
Extend the framework to the time-dependent drift–diffusion PDE, cast it in semi-variational form, and apply Lions’ theorem to prove well-posedness with an energy estimate E(t).
Discuss applicability of Picard’s theorem for an abstract second-order evolution equation in a Hilbert space setting.

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Project Structure

```text ├── Final_Report.pdf # Complete write‐up (proofs, derivations, results) ├── README.md # You are here ├── LICENSE # MIT license text └── CITATION.cff # Machine-readable citation metadata

Owner

Login: GeorgesSakr
Kind: user

Repositories: 1
Profile: https://github.com/GeorgesSakr

Citation (CITATION.cff)

cff-version: 1.2.0
title: "Weak-Form Solutions of Drift–Diffusion PDEs via Functional Analysis"
authors:
  - family-names: Sakr
    given-names: Georges
  - family-names: Tayyar
    given-names: Ali
  - family-names: Joe
    given-names: Naimeh
date-released: "2025-05-17"
version: "1.0"
doi: ""
url: "https://github.com/GeorgesSakr/pde-functional-analysis"
license: "MIT"
message: "If you use this work, please cite it as shown below."

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