Fractional Dissipative PDEs

Abstract

In this chapter we provide an introduction to fractional dissipative partial differential equations (PDEs) with a focus on trying to understand their dynamics. The class of PDEs we focus on are reaction-diffusion equations but we also provide an outlook on closely related classes of PDEs. To simplify the exposition, we only discuss the cases of fractional time derivatives and fractional space derivatives in the PDE separately. As our main tools, we describe analytical as well as numerical methods, which are generically necessary to study nonlinear dynamics. We start with the analytical study of steady states and local linear stability for fractional time derivatives. Then we extend this view to a global perspective and consider time-fractional PDEs and gradient flows. Next, we continue to steady states, linear stability analysis and bifurcations for space-fractional PDEs. As a final analytical consideration we discuss existence and stability of traveling waves for space-fractional PDEs. In the last parts, we provide numerical discretization schemes for fractional (dissipative) PDEs and we utilize these techniques within numerical continuation in applied examples of fractional reaction-diffusion PDEs. We conclude with a brief summary and outlook on open questions in the field.

book chapter

References in corpus (25)

• The World of the Complex Ginzburg-Landau Equation
• The fundamental solution of the space-time fractional diffusion equation
• Nonlocal diffusion and applications
• From Diffusion to Anomalous Diffusion: A Century after Einstein's Brownian Motion
• A short FE implementation for a 2d homogeneous Dirichlet problem of a Fractional Laplacian
• Aspects of an adaptive finite element method for the fractional Laplacian: a priori and a posteriori error estimates, efficient implementation and multigrid solver
• Localized states in the conserved Swift-Hohenberg equation with cubic nonlinearity
• Non-Markovian Random Walks and Non-Linear Reactions: Subdiffusion and Propagating Fronts
• Time-stepping error bounds for fractional diffusion problems with non-smooth initial data
• A comparative study on nonlocal diffusion operators related to the fractional Laplacian
• Traveling waves for a bistable equation with nonlocal-diffusion
• The Swift-Hohenberg equation with a nonlocal nonlinearity
• A posteriori error analysis for approximations of time-fractional subdiffusion problems
• Weighted analytic regularity for the integral fractional Laplacian in polygons
• Numerical continuation for fractional PDEs: sharp teeth and bloated snakes
• Validity of amplitude equations for non-local non-linearities
• Optimal operator preconditioning for pseudodifferential boundary problems
• Rational spectral methods for PDEs involving fractional Laplacian in unbounded domains
• Exponential Convergence of hp FEM for the Integral Fractional Laplacian in Polygons
• Reaction-subdiffusion systems and memory: spectra, Turing instability and decay estimates
• Regularity structures and the dynamical ${\Phi }_3^4$ model
• Infinite time horizon spatially distributed optimal control problems with pde2path algorithms and tutorial examples
• Bistable transition fronts in R^N
• Differential geometric bifurcation problems in pde2path -- algorithms and tutorial examples
• Analysis and numerics of traveling waves for asymmetric fractional reaction-diffusion equations

Cited by in corpus (2)

• On the Fractional Dynamics of Kinks in sine-Gordon Models
• Two-dimensional fractional discrete NLS equations: dispersion relations, rogue waves, fundamental and vortex solitons