Overview

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.

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9781108490993

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Reviews

'This interesting book provides an excellent and comprehensive presentation of recent major developments in nonlocal spectral theories and high-order numerical methods for fractional models. This monograph has been pedagogically made accessible for a wide spectrum of readers from mathematical/natural sciences and engineering communities, highlighting diverse applications such as surface/sub-surface anomalous transport, power-law rheology, multi-scale material failure, and fluid/scalar turbulence.' Francesco Mainardi, Retired Professor of Mathematical Physics, University of Bologna, Italy 'Written by four known specialists in numerical methods, this book covers a wide spectrum of advanced numerical methods suitable for fractional-order modelling of real processes, such as anomalous heat and mass transfer, rheology, and other complex phenomena. The style and the rigour of presentation makes this book a good source for researchers who need to apply such methods in various fields of science and engineering.' Igor Podlubny, Technical University of Kosice, Slovakia

Author Information

Mohsen Zayernouri is Associate Professor of Mechanical Engineering and Statistics and Probability at Michigan State University. He is the recipient of two Young Investigator Program (YIP) awards from the US Air Force Office of Scientific Research (AFOSR) and Army Research Office (ARO). He brings to bear advanced computational tools from applied mathematics and data sciences to develop stochastic modeling and predictive simulation tools for challenging physical and engineering problems, including: stochastic Lévy processes in turbulent flows and anomalous transport in multi-scale and disordered bio-materials. Li-Lian Wang is Professor of Applied Mathematics at Nanyang Technological University in Singapore. His research is focused on numerical analysis of fractional differential equations, spectral methods, triangular spectral-element methods, computational acoustics and electromagnetics, and simulations of metamaterials. He is co-author of the book 'Spectral Methods: Algorithms, Analysis and Applications.' Jie Shen is currently Chair Professor and Dean of the School of Mathematical Sciences at Eastern Institute of Technology, Ningbo, China. For more than two decades, he was Professor of Mathematics at Purdue University where he served as Director of the Center for Computational and Applied Mathematics from 2012 to 2022, and was named Distinguished Professor of Mathematics in 2023. He is an elected fellow of the American Mathematical Society and the Society for Industrial and Applied Mathematics, and lead author of the book 'Spectral Methods: Algorithms, Analysis and Applications.' George Em Karniadakis is Professor of Applied Mathematics at Brown University. He is a member of the US National Academy of Engineering and the author or co-author of six books. He is well known for his wide spectrum of work on high-dimensional stochastic modeling and multi-scale simulations of physical and biological systems. He is also a pioneer of spectral/hp-element methods for fluids in complex geometries, general polynomial chaos for uncertainty quantification, and the Sturm-Liouville theory for partial differential equations and fractional calculus.

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