A Short Introduction to Partial Differential Equations

• Autore: Novruzi Arian
• Pubblicazione: Cham : , : Springer, , 2024; ©2023
• Edizione: 1st ed.
• Descrizione fisica: 1 online resource (225 pages)
• Disciplina: 515.353
• Nota di contenuto: Intro -- Preface -- Contents -- 1 Notations and review -- 1.1 Continuous differentiable functions -- 1.2 Domains and Ck (∂Ω) spaces -- 1.2.1 Partition of unity -- 1.2.2 Domains and Ck (∂Ω) spaces -- 1.3 Review of some important results -- 1.3.1 Some results from Lp(Ω) spaces -- 1.3.2 Some results from (Functional) Analysis -- 1.3.3 An application: Ordinary Differential Equations -- Problems -- 2 Partial differential equations -- 2.1 Some prototypes of PDEs -- Problems -- 3 First-order PDEs: classical and weak solutions -- 3.1 Method of characteristics -- 3.2 Classical local solutions to first-order PDEs -- 3.2.1 Classical local solutions: flat boundary -- 3.2.1.1 Boundary and initial conditions -- 3.2.1.2 Classical local solutions -- 3.2.2 Classical local solutions: non-flat boundary -- 3.3 Conservation laws and weak solutions -- Problems -- 4 Second-order linear elliptic PDEs: maximum principle and classical solutions -- 4.1 Laplace equation and the method of separation of variables -- 4.2 Dirichlet problem in a ball -- 4.3 Maximum principle for Laplacian -- 4.4 Solution to the Dirichlet problem -- 4.4.1 Sub(super) harmonic functions and sub(super) solutions -- 4.4.2 Solution to the Dirichlet problem -- Problems -- 5 Distributions -- 5.1 Motivation -- 5.2 Distributions -- 5.2.1 Test functions -- 5.2.2 Distributions -- 5.2.3 Derivatives of distributions -- 5.3 Convolution of distributions and fundamental solutions -- 5.4 Tempered distributions and Fourier transform -- 5.4.1 Fourier transform -- 5.4.2 Tempered distributions and Fourier transform -- Problems -- 6 Sobolev spaces -- 6.1 Definitions and some first properties -- 6.1.1 Density of D in Wk,p -- 6.1.2 Some applications -- 6.2 Hs spaces and Fourier transform: Ws,p and Ws,p0 spaces -- 6.3 Continuous, compact, and dense embedding theorems in Hs(Ω) -- 6.3.1 Case Ω = RN -- 6.3.2 Case Ω RN. ; 6.4 Boundary traces in Sobolev spaces -- 6.5 Poincaré inequality -- 6.6 H−s(Ω) and W−s,q(Ω) spaces -- Problems -- 7 Second-order linear elliptic PDEs: weak solutions -- 7.1 Introduction -- 7.2 Existence and uniqueness of weak solutions -- 7.2.1 Preliminary results -- 7.2.2 Dirichlet problem -- 7.2.3 Neumann problem -- 7.3 Nonlinear second-order elliptic PDEs -- Problems -- 8 Second-order parabolic and hyperbolic PDEs -- 8.1 Heat and wave equations and the method of separation of variables -- 8.1.1 Heat equation and the method of separation of variables -- 8.1.2 Wave equation and the method of separation of variables -- 8.2 Some preliminary results -- 8.3 Weak solution to the heat equation -- 8.4 Weak solution to the wave equation -- Problems -- 9 Annex -- 9.1 Annex: Chapter 1 -- 9.1.1 Continuous differentiable functions -- 9.1.2 Some results from Lp(Ω) spaces -- 9.1.3 An application: Ordinary Differential Equations -- 9.2 Annex: Chapter 3 -- 9.2.1 Classical local solutions to first-order PDEs -- 9.2.2 Conservation laws and weak solutions -- 9.3 Annex: Chapter 4 -- 9.3.1 Dirichlet problem in a ball -- 9.3.2 Maximum principle for second-order linear elliptic PDEs -- 9.3.3 Solution to the Dirichlet problem -- 9.3.3.1 Sub(super) harmonic functions and sub(super) solutions -- 9.3.3.2 Some auxiliary results from Analysis -- 9.3.3.3 Proof of Theorem 4.4.7 -- 9.4 Annex: Chapter 5 -- 9.4.1 Some useful inequalities -- 9.4.2 More operations with distributions. Examples -- 9.4.3 Convergence of distributions. Distributions of finite order -- 9.4.4 Convolution of distributions -- 9.4.5 Tempered distributions and Fourier transform -- 9.4.6 Tempered distributions and convolution -- 9.5 Annex: Chapter 6 -- 9.5.1 Continuous and compact embeddings -- 9.5.2 Extension and density results in Sobolev spaces -- 9.5.3 Boundary traces in Sobolev spaces. ; 9.6 Annex: Chapter 7 -- 9.6.1 Regularity of weak solutions -- 9.6.1.1 Regularity in the interior -- 9.6.1.2 Regularity near the boundary.
• Titolo autorizzato: A Short Introduction to Partial Differential Equations
• ISBN: 3-031-39524-7
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910799230903321
• Lo trovi qui: Univ. Federico II
• Serie: CMS/CAIMS Books in Mathematics Series