Spectral and Spectral Element Methods for Fractional Ordinary and Partial Differential Equations

Fractional modeling is a new frontier of high-fidelity predictive modeling approaches in mathematics, science, and engineering. This introduction for graduate students and researchers is a guide to numerically solving fractional differential equations as tractable models for complex rheology, aging materials, and turbulence.

This comprehensive introduction to global spectral methods for fractional differential equations from leaders of this emerging field is designed to be accessible to graduate students and researchers across math, science, and engineering. The book begins by covering the foundational fractional calculus concepts needed to understand and model anomalous transport phenomena. The authors proceed to introduce a series of new spectral theories and new families of orthogonal and log orthogonal functions, then present corresponding spectral and spectral element methods for fractional differential equations. The book also covers the fractional Laplacian in unbounded and bounded domains and major developments in time-integration of fractional models. It ends by sampling the wide variety of real-world applications of fractional modeling, including concentration transport in surface/subsurface dynamics, complex rheology and material damage, and fluid turbulence and geostrophic transport.