Spectral and Spectral Element Methods for Fractional PDEs by Mohsen Zayernouri B.Sc., Mechanical Engineering, Azad University, Iran, 2004 M.Sc., Mechanical Engineering, Tehran Polytechnic, Iran, 2006 Ph.D., Mechanical Engineering, University of Utah, USA, 2010 Sc.M., Applied Mathematics, Brown University, USA, 2012 A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Division of Applied Mathematics at Brown University PROVIDENCE, RHODE ISLAND May 2015 c Copyright 2015 by Mohsen Zayernouri This dissertation by Mohsen Zayernouri is accepted in its present form by The Division of Applied Mathematics as satisfying the dissertation requirement for the degree of Doctor of Philosophy. Date George Em Karniadakis, Ph.D., Advisor Recommended to the Graduate Council Date Mark Ainsworth, Ph.D., Reader Date Jan S. Hesthaven, Ph.D., Reader Date Mark M. Meerschaert, Ph.D., Reader Approved by the Graduate Council Date Peter M. Weber, Dean of the Graduate School iii Vitae Mohsen Zayernouri obtained his B.Sc. in mechanical engineering from Azad Uni- versity, Iran, where he was ranked first amongst the graduates in 2004. Next, he joined Tehran Polytechnic (Amirkabir University of Technology), where he acquired his M.Sc. in mechanical engineering as the top student in 2006. He received the best national M.Sc. award from Iranian Society of Mechanical Engineering (ISME), and then, he was elected into the National Foundation of Elite in Iran. Subsequently, he attended the University of Utah, USA, where he obtained his first Ph.D. in me- chanical engineering in 2010. Due to his great passion and interest in mathematics and scientific computing, he joined Brown University right after defending his Ph.D. thesis at Utah to seek a second Ph.D. in applied mathematics under the advice of Prof. George Em Karniadakis. The outcome of his research at Brown on developing spectral theories and high-order methods for fractional PDEs is a series of ten journal papers, provided in the list of references. iv Acknowledgements I would like to thank my advisor, Professor George Em Karniadakis, for the great amount of trust, unique advice, and endless encouragement. I am indebted to George for many valuable opportunities he gave me during the course of this work, also for generous sharing his research experience, which provides me an important reference for my future career. It was a privilege having such an exceptional committee of research and readers, consisting of Professor Mark Ainsworth, Professor Jan S. Hesthaven, and Professor Mark M. Meerschaert. I would like to sincerely thank them all for reading and correcting the thesis, also for their constructive feedback, which added a lot to the value of the present study. I have learned a lot about finite-difference methods, finite element methods, spec- tral methods, and spectral element methods from the excellent lectures, given by Professor Ainsworth, Professor Chi-Wang Shu, and Professor Johnny Guzmann, to whom my gratitude goes. I was also fortunate to learn about the theory of probabil- ity and stochastic partial differential equations from Prof. Boris Rozovsky who will remain as a great source of inspiration in my future carrier. In addition, I would like to acknowledge Professor Anastasios Matzavinos and Professor Marco L. Bittencourt for their support and collaboration during the preparation of the last two chapters of the dissertation. Moreover, many thanks are due to our wonderful staff, especially v to Ms. Madeline Brewster, Ms. Stephanie Han, and Ms. Jean Radican for being there whenever I needed help. I would like to express gratitude to many CRUNCHers, Mengdi Zhang, Dr. Changho Kim, Dr. Handy (Zhongqiang) Zhang, Paris Perdikaris, Minge Deng, Heyrim Cho, Dr. Minseok Choi, Dr. Xui Yang, Dr. Yue Yu, Seungjoon Lee, Yuhang, Dogkun Zhang, Ansel Blumers for all their help and many happy conservations. Moreover, I would like to thank my other friends, post-docs, senior researchers, and visitor scholars at CRUNCH group: Dr. Alireza Yazdani, Dr. Wanrong Cao, Dr. Daniele Venturi, Dr. Leopold Grinberg, Dr. Xuejin Li, Dr. Zhen Li, Dr. Fangying Song, Dr. Fanhai Zeng, Dr. Xuan Zhao, and Wei Cai for their help and interesting discussions. I would love to especially thank my wife, Dr. Maryam Naghibolhosseini, whose love, emotional support, and encouragement made me much stronger, happier, and more faithful throughout my study at Brown and to whom this work is dedicated. At last but certainly not least, I would like to express gratitude to my precious parents, lovely brothers, and wonderful friends for their constant love, support, and friendship. This work was supported by the Collaboratory on Mathematics for Mesoscopic Modeling of Materials (CM4) at PNNL funded by the Department of Energy, by OSD/MURI and by NSF/DMS. vi To my wife vii Contents Vitae iv Acknowledgments v 1 Introduction 1 1.0.1 Anomalous Diffusion . 3 2 Fractional Sturm-Liouville Eigen-Problems 7 2.1 Background . 8 2.2 Definitions . 11 2.3 Part I: Regular Fractional Sturm-Liouville Problems of Kind I & II . 13 2.3.1 Regular Boundary-Value Problem Definition . 14 2.3.2 Analytical Eigensolutions to RFSLP-I & -II . 16 2.3.3 Properties of the Eigensolution to RFSLP-I & -II . 30 2.4 Part II: Singular Fractional Sturm-Liouville Problems of Kind I & II . 33 2.4.1 Properties of the Eigen-solutions to SFSLP-I&-II . 43 2.5 Numerical Approximation . 49 2.5.1 Numerical Tests . 52 3 Tempered Fractional Sturm-Liouville Eigen-Problems 55 3.1 Background . 57 3.2 Definitions . 59 3.3 Well-posedness . 63 3.4 Regular TFSLPs of Kind I & II . 66 3.4.1 Regular Tempered Eigen-Problems . 69 3.4.2 Explicit Eigensolutions to the regular TFSLP-I & -II . 70 3.4.3 Properties of the Eigenfunctions of the regular TFSLP-I & -II 75 3.5 Singular Tempered Fractional Problems . 76 3.5.1 Properties of the Eigen-solutions to the singular TFSLP-I&-II 79 3.6 Approximability of the Tempered Eigenfunctions . 81 viii (i) α,β;µ 3.6.1 Spectral Approximation using Singular Tempered Basis Pn (x), µ (0; 1) ............................. 83 3.6.2 Numerical2 Approximation . 85 3.6.3 Stability and Convergence Analysis . 89 4 Petrov-Galerkin Spectral Method and Discontinuous Galerkin Method for Fractional ODEs 93 4.1 Background . 94 4.1.1 Finite Difference Methods (FDM) . 95 4.1.2 Spectral Methods (SMs) . 96 4.1.3 Spectral/hp Element Methods . 98 4.2 Notation and Definitions . 99 4.3 Petrov-Galerkin (PG) Spectral Method . 100 4.3.1 Basis Functions . 101 4.3.2 Test Functions . 102 4.3.3 PG Spectral Method for the FIVP . 104 4.3.4 PG Spectral Method for the FFVP . 106 4.4 Discontinuous Methods . 109 4.4.1 Discontinuous Spectral Method (DSM; Single-Element) . 110 4.4.2 Discontinuous Spectral Element Method (DSEM; Multi-Element)116 4.4.3 Numerical Tests for DSEM . 122 4.5 Discussion . 126 5 Fractional Delay Differential Equations 130 5.1 Background . 131 5.2 Notation and Problem Definition . 136 5.3 Petrov-Galerkin Spectral Method: Continuous & Single-Domain . 137 5.3.1 Space of Basis Functions . 138 5.3.2 Space of Test Functions . 139 5.3.3 Stability and Error Analysis . 141 5.3.4 Implementation of the PG Spectral Method . 148 5.3.5 Numerical Examples for PG Spectral Method . 151 5.4 Discontinuous Galerkin (DG) Schemes . 157 5.4.1 Discontinuous Spectral Method (DSM; Single-Domain) . 158 5.4.2 Discontinuous Spectral Element Method (DSEM; Multi-Element)163 5.4.3 Numerical Examples for DSEM scheme . 168 5.5 Discussion . 170 6 Spectral Element Methods for Fractional Advection Equation 174 6.1 Background . 176 6.2 Problem Definition . 180 6.3 PG-DG Method: SM-in-Time & DSEM-in-Space . 181 6.3.1 Basis Functions . 182 6.3.2 Test Functions . 185 6.3.3 Implementation of SM-DSEM Scheme . 189 ix 6.4 Time-integration using SM-DSEM when τ = 1 . 197 6.5 DG-DG Method: DSEM-in-Time & DSEM-in-Space . 199 6.5.1 Basis and Test Function Spaces in DSEM-DSEM Scheme . 200 6.5.2 Implementation of DSEM-DSEM Scheme . 200 6.6 Discussion . 206 7 Fractional Spectral Collocation Method 210 7.1 Background . 211 7.2 Notation and Definitions . 212 7.3 Fractional Lagrange interpolants . 214 7.3.1 Fractional differentiation matrix Dσ, 0 < σ < 1 . 217 7.3.2 Fractional differentiation matrix D1+σ, 0 < σ < 1 . 220 7.3.3 Collocation/interpolation points . 222 7.4 Numerical Tests . 228 7.4.1 Steady-state Problems . 229 7.4.2 Time-dependent FPDEs . 234 7.5 Discussion . 240 8 Variable-Order Fractional PDEs 244 8.1 Background . 245 8.2 Preliminaries . 249 8.3 Problem Definition . 252 8.4 Fractional Lagrange Interpolants (FLIs) . 254 ∗ RL 8.4.1 Construction of FLI when x x . 255 ∗ @D ≡ D 8.4.2 Central FLIs when x @jxj of Riesz Type . 257 8.5 Fractional Differentiation MatricesD ≡ . 258 ∗ 8.5.1 x of Left-Sided Riemann-Liouville Type . 259 ∗D 8.5.2 x of Right-Sided Riemann-Liouville Type . 261 ∗D 8.5.3 x of Riesz Type . 263 D RL τ 8.5.4 Temporal Differentiation Matrix Dt . 266 C τ 8.5.5 Temporal Differentiation Matrix Dt . 267 8.6 Numerical Tests . 268 ∗ RL 8.6.1 Linear FPDEs with x x . 269 8.6.2 Linear FPDEs with RieszD ≡ DerivativesD . 273 8.6.3 A Penalty Method for FPDEs . 277 8.6.4 Nonlinear FPDEs . 280 9 A Unified Petrov-Galerkin Spectral Method for FPDEs 283 9.1 Background . 285 9.2 Preliminaries on Fractional Calculus .