Finite Difference Methods for Fractional Diffusion Equations

This book provides a self-contained introduction to finite difference methods for time-dependent space-fractional diffusion equations, emphasizing their theoretical properties and practical computational implementation. It collects results previously dispersed throughout the literature, presenting them within a coherent unified framework. In addition to covering numerical methods for fractional diffusion equations, their exact solutions, and their connection to Lévy flights, it also offers an accessible overview of fundamental concepts related to RiemannLiouville fractional derivatives.

By presenting a comprehensive treatment of the fundamental techniques of finite difference methods, the book lays a solid foundation for mastering the intricacies of finite differences for fractional differential equations. The final chapters address scenarios with boundary conditions, filling a gap in the existing literature. Each chapter concludes with exercises designed to help deepen the reader's understanding and prepare them for further specialized study.

Written from the perspective of a mathematician who enjoys physics and computation, the volume is intended as a starting point for any researcher who wants to enter into this exciting subject. It will appeal to graduate students and experts from different backgrounds who enjoy digging into mathematical, physical and computational ideas.

Provides an accessible introduction to finite difference methods for space-fractional diffusion equations Offers a solid theoretical framework that is accessible to researchers from diverse disciplinary backgrounds Includes a comprehensive treatment of problems on both infinite and bounded domains, filling a gap in the literature

Autorentext

Ercília Sousa is a Full Professor in the Department of Mathematics at the University of Coimbra, where she also completed her degree in pure mathematics. After receiving a DPhil from the University of Oxford, on numerical analysis, she joined the Research Center for Mathematics at the University of Coimbra. Her research interests are broadly in numerical analysis and partial differential equations, with a recent focus on numerical methods for fractional diffusion equations.

Inhalt

Prelude.- Fractional derivatives: fundamental concepts.- A governing equation for Lévy flights.- Consistency, stability and convergence of numerical methods.- Fractional differential equations in unbounded domains.- Fractional differential equations in bounded domains: absorbing boundaries.- Fractional differential equations in bounded domains: reflecting boundaries.- The road ahead.- References.