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

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.

'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

• 1. Fractional calculus and anomalous transport; 2. Spectral expansions and related approximations; 3. Global schemes for fractional ODEs (FODEs); 4. Global schemes for fractional PDEs (FPDEs); 5. Integral fractional Laplacian in unbounded domains; 6. Fractional Laplacian in bounded domains; 7. Time-integration of fractional models; 8. Applications of anomalous transport and fractional modeling; References; Index.