This thesis focuses on the study of three mathematical models consisting of degenerate diffusion equations and fractional diffusion equations arising from different study needs. The first one is taken from the study of self-organized criticality phenomena arising from recent papers by Barbu. About the second one, that is connected to the obstacle problem, we analyze a nonlinear degenerate parabolic problem whose diffusion coefficient is the Heaviside function of the distance of the solution itself from a given target function. More precisely, we show that this model behaves as an evolutive variational inequality having the target as an obstacle: under suitable hypotheses, starting from an initial state above the target the solution evolves in time towards an asymptotic solution, eventually getting in contact with part of the target itself. At last, referring to the third one, that is a time-fractional type model (a Caputo time fractional degenerate diffusion equation) that can find a wider use, for example, from biology to mechanics, to superslow diffusion in porous media, till financial type phenomena, we prove to be equivalent to the fractional parabolic obstacle problem, showing that its solution evolves for any α ∈ (0, 1) to the same stationary state, the solution of the classic elliptic obstacle problem. The only thing which changes with α is the convergence speed. For all these models we provide numerical simulations.

Finite difference methods for degenerate diffusion equations and fractional diffusion equations / Alberini, Carlo. - (2021 Feb 19).

Finite difference methods for degenerate diffusion equations and fractional diffusion equations

ALBERINI, CARLO
2021

Abstract

This thesis focuses on the study of three mathematical models consisting of degenerate diffusion equations and fractional diffusion equations arising from different study needs. The first one is taken from the study of self-organized criticality phenomena arising from recent papers by Barbu. About the second one, that is connected to the obstacle problem, we analyze a nonlinear degenerate parabolic problem whose diffusion coefficient is the Heaviside function of the distance of the solution itself from a given target function. More precisely, we show that this model behaves as an evolutive variational inequality having the target as an obstacle: under suitable hypotheses, starting from an initial state above the target the solution evolves in time towards an asymptotic solution, eventually getting in contact with part of the target itself. At last, referring to the third one, that is a time-fractional type model (a Caputo time fractional degenerate diffusion equation) that can find a wider use, for example, from biology to mechanics, to superslow diffusion in porous media, till financial type phenomena, we prove to be equivalent to the fractional parabolic obstacle problem, showing that its solution evolves for any α ∈ (0, 1) to the same stationary state, the solution of the classic elliptic obstacle problem. The only thing which changes with α is the convergence speed. For all these models we provide numerical simulations.
File allegati a questo prodotto
File Dimensione Formato  
Tesi_dottorato_Alberini.pdf

accesso aperto

Tipologia: Tesi di dottorato
Licenza: Creative commons
Dimensione 3.81 MB
Formato Adobe PDF
3.81 MB Adobe PDF Visualizza/Apri PDF

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1503464
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus ND
  • ???jsp.display-item.citation.isi??? ND
social impact