Partial Differential Equations: Topics in Fourier Analysis, Second Edition

Partial Differential Equations: Topics in Fourier Analysis, Second Edition explains how to use the Fourier transform and heuristic methods to obtain significant insight into the solutions of standard PDE models. It shows how this powerful approach is valuable in getting plausible answers that can then be justified by modern analysis. Using Fourier analysis, the text constructs explicit formulas for solving PDEs governed by canonical operators related to the Laplacian on the Euclidean space. After presenting background material, it focuses on: Second-order equations governed by the Laplacian on Rn; the Hermite operator and corresponding equation; and the sub-Laplacian on the Heisenberg group Designed for a one-semester course, this text provides a bridge between the standard PDE course for undergraduate students in science and engineering and the PDE course for graduate students in mathematics who are pursuing a research career in analysis. Through its coverage of fundamental examples of PDEs, the book prepares students for studying more advanced topics such as pseudo-differential operators. It also helps them appreciate PDEs as beautiful structures in analysis, rather than a bunch of isolated ad-hoc techniques. New to the Second Edition: Three brand new chapters covering several topics in analysis not explored in the first edition Complete revision of the text to correct errors, remove redundancies, and update outdated material Expanded references and bibliography New and revised exercises. Designed for a one-semester course, this text provides a bridge between the standard PDE course for undergraduate students in science and engineering and the PDE course for graduate students in mathematics who are pursuing a research career in analysis. Cover 1 Half Title 2 Title Page 4 Copyright Page 5 Contents 6 Preface 8 1. The Multi-Index Notation 12 2. The Gamma Function 18 3. Convolutions 26 4. Fourier Transforms 36 5. Tempered Distributions 48 6. The Heat Kernel 58 7. The Free Propagator 68 8. The Newtonian Potential 76 9. The Bessel Potential 82 10. Global Hypoellipticity in the Schwartz Space 86 11. The Poisson Kernel 94 12. The Bessel–Poisson Kernel 100 13. Wave Kernels 106 14. The Heat Kernel of the Hermite Operator 116 15. The Green Function of the Hermite Operator 124 16. Global Regularity of the Hermite Operator 134 17. The Heisenberg Group 140 18. The Sub-Laplacian and the Twisted Laplacians 150 19. Convolutions on the Heisenberg Group 156 20. Wigner Transforms and Weyl Transforms 160 21. Spectral Analysis of Twisted Laplacians 166 22. Heat Kernels Related to the Heisenberg Group 172 23. Green Functions Related to the Heisenberg Group 178 24. Theta Functions and the Riemann Zeta-Function 182 25. The Twisted Bi-Laplacian 190 26. Complex Powers of the Twisted Bi-Laplacian 198 Bibliography 202 Index 206 solutions,of,PDEs;,methods,based,on,Fourier,analysis;,Fourier,transform;,Second-order,equations,governed,by,the,Laplacian;,Hermite,operator;,sub-Laplacian,on,the,Heisenberg,group;,PDEs,important,in,physics solutions of PDEs,methods based on Fourier analysis,Fourier transform,Second-order equations governed by the Laplacian,Hermite operator,sub-Laplacian on the Heisenberg group,PDEs important in physics "Partial Differential Equations: Topics in Fourier Analysis, Second Edition explains how to use the Fourier transform and heuristic methods to obtain significant insight into the solutions of standard PDE models. It shows how this powerful approach is valuable in getting plausible answers that can then be justified by modern analysis. Using Fourier analysis, the text constructs explicit formulas for solving PDEs governed by canonical operators related to the Laplacian on the Euclidean space. After presenting background material, it focuses on: second-order equations governed by the Laplacian on Rn; the Hermite operator and corresponding equation; and the sub-Laplacian on the Heisenberg group designed for a one-semester course. This text provides a bridge between the standard PDE course for undergraduate students in science and engineering and the PDE course for graduate students in mathematics who are pursuing a research career in analysis. Through its coverage of fundamental examples of PDEs, the book prepares students for studying more advanced topics such as pseudo-differential operators. It also helps them appreciate PDEs as beautiful structures in analysis, rather than a bunch of isolated ad-hoc techniques. New to the second edition: complete revision of the text to correct errors, removal of redundancies, and updates to outdated material. Also includes expanded references and bibliography, new and revised exercises, and three brand new chapters covering several topics in analysis not explored in the first edition"-- Provided by publisher