Partial differential equations : an introduction to analytical and numerical methods / / Wolfgang Arendt, Karsten Urban ; translated by James B. Kennedy

• Autore: Arendt Wolfgang <1950->
• Pubblicazione: Cham, Switzerland : , : Springer, , [2023]; ©2023
• Descrizione fisica: 1 online resource (463 pages)
• Disciplina: 515.353
• Soggetto topico: Differential equations, Partial
• Persona (resp. second.): UrbanKarsten; KennedyJames B. <1932->
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Intro -- Foreword by the Translator -- Preface -- Acknowledgments -- About the Authors -- About the Translator -- Contents -- List of figures -- 1 Modeling, or where do differential equations come from -- 1.1 Mathematical modeling -- 1.1.1 Modeling with partial differential equations -- 1.1.2 Modeling is only the first step -- 1.2 Transport processes -- 1.2.1 Conservation laws -- 1.2.2 From a conservation law to a differential equation -- 1.2.3 The linear transport equation -- 1.2.4 The convection-reaction equation -- 1.2.5* Burgers' equation -- 1.3 Diffusion -- 1.4 The wave equation -- 1.5 The Black-Scholes equation -- 1.6 Let's get higher dimensional -- 1.6.1 Transport processes -- 1.6.2 Diffusion processes -- 1.6.3 The wave equation -- 1.6.4 Laplace's equation -- 1.7* But there's more -- 1.7.1 The KdV equation -- 1.7.2 Geometric differential equations -- 1.7.3 The plate equation -- 1.7.4 The Navier-Stokes equations -- 1.7.5 Maxwell's equations -- 1.7.6 The Schrödinger equation -- 1.8 Classification of partial differential equations -- 1.9* Comments -- 1.10 Exercises -- 2 Classification and characteristics -- 2.1 Characteristics of initial value problems on ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper R) /StPNE pdfmark [/StBMC pdfmarkRps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark -- 2.1.1 Homogeneous problems -- 2.1.2 Inhomogeneous problems -- 2.1.3* Burgers' equation -- 2.2 Equations of second order -- 2.3* Nonlinear equations of second order -- 2.4* Equations of higher order and systems -- 2.5 Exercises -- 3 Elementary methods -- 3.1 The one-dimensional wave equation -- 3.1.1 D'Alembert's formula on ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper R times double struck upper R) /StPNE pdfmark [/StBMC pdfmarkR Rps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark. ; 3.1.2 The wave equation on an interval -- 3.2 Fourier series -- 3.3 Laplace's equation -- 3.3.1 The Dirichlet problem on the unit square -- 3.3.2 The Dirichlet problem on the disk -- 3.3.3 The elliptic maximum principle -- 3.3.4 Well-posedness of the Dirichlet problem for the square and the disk -- 3.4 The heat equation -- 3.4.1 Separation of variables -- 3.4.2 The parabolic maximum principle -- 3.4.3 Well-posedness of the parabolic initial-boundary value problem on the interval -- 3.4.4 The heat equation in ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper R Superscript d) /StPNE pdfmark [/StBMC pdfmarkRdps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark -- 3.5 The Black-Scholes equation -- 3.6 Integral transforms -- 3.6.1 The Fourier transform -- 3.6.2* The Laplace transform -- 3.7 Outlook -- 3.8 Exercises -- 4 Hilbert spaces -- 4.1 Inner product spaces -- 4.2 Orthonormal bases -- 4.3 Completeness -- 4.4 Orthogonal projections -- 4.5 Linear and bilinear forms -- 4.5.1* Extensions and generalizations -- 4.6 Weak convergence -- 4.7 Continuous and compact operators -- 4.8 The spectral theorem -- 4.9* Comments on Chapter 4 -- 4.10 Exercises -- 5 Sobolev spaces and boundary value problems in dimension one -- 5.1 Sobolev spaces in one variable -- 5.2 Boundary value problems on the interval -- 5.2.1 Dirichlet boundary conditions -- 5.2.2 Neumann boundary conditions -- 5.2.3 Robin boundary conditions -- 5.2.4 Mixed and periodic boundary conditions -- 5.2.5 Non-symmetric differential operators -- 5.2.6* A variational approach to singularly perturbed problems and the transport equation -- 5.3* Comments on Chapter 5 -- 5.4 Exercises -- 6 Hilbert space methods for elliptic equations -- 6.1 Mollifiers -- 6.2 Sobolev spaces on ΩRd -- 6.3 The space H10 (Ω) -- 6.4 Lattice operations on H1(Ω). ; 6.5 The Poisson equation with Dirichlet boundary conditions -- 6.6 Sobolev spaces and Fourier transforms -- 6.7 Local regularity -- 6.8 Inhomogeneous Dirichlet boundary conditions -- 6.9 The Dirichlet problem -- 6.10 Elliptic equations with Dirichlet boundary conditions -- 6.11 H2-regularity -- 6.12* Comments on Chapter 6 -- 6.13 Exercises -- 7 Neumann and Robin boundary conditions -- 7.1 Gauss's theorem -- 7.2 Proof of Gauss's theorem -- 7.3 The extension property -- 7.4 The Poisson equation with Neumann boundary conditions -- 7.5 The trace theorem and Robin boundary conditions -- 7.6* Comments on Chapter 7 -- 7.7 Exercises -- 8 Spectral decomposition and evolution equations -- 8.1 A vector-valued initial value problem -- 8.2 The heat equation: Dirichlet boundary conditions -- 8.3 The heat equation: Robin boundary conditions -- 8.4 The wave equation -- 8.5 Inhomogeneous parabolic equations -- 8.6* Space/time variational formulations -- 8.7* Comments on Chapter 8 -- 8.8 Exercises -- 9 Numerical methods -- 9.1 Finite differences for elliptic problems -- 9.1.1 FDM: the one-dimensional case -- 9.1.2 FDM: the two-dimensional case -- 9.2 Finite elements for elliptic problems -- 9.2.1 The Galerkin method -- 9.2.2 Triangulation and approximation on triangles -- 9.2.3 Affine functions on triangles -- 9.2.4 Norms on triangles -- 9.2.5 Transformation into a reference element -- 9.2.6 Interpolation for finite elements -- 9.2.7 Finite element spaces -- 9.2.8 The Poisson problem on polygons -- 9.2.9 The stiffness matrix and the linear system of equations -- 9.2.10 Numerical experiments -- 9.3* Extensions and generalizations -- 9.3.1 The Petrov-Galerkin method -- 9.3.2 Further extensions -- 9.4 Parabolic problems -- 9.4.1 Finite differences -- 9.4.2 Finite elements -- 9.4.3* Error estimates via space/time variational formulations -- 9.5 The wave equation. ; 9.5.1 Finite differences -- 9.5.2 Finite elements -- 9.6* Comments on Chapter 9 -- 9.7 Exercises -- 10 Maple®, or why computers can sometimes help -- 10.1 Maple® -- 10.1.1 Elementary examples -- 10.1.2 Solutions via Fourier transforms -- 10.1.3 Laplace transforms -- 10.1.4 It can also be done numerically -- 10.1.5 Calculating function values -- 10.2 Exercises -- Appendix -- A.1 Banach spaces and linear operators -- A.2 The space C(K) -- A.3 Integration -- A.4 More details on the Black-Scholes equation -- A.4.1 Basics of stochastics -- A.4.2 Black-Scholes model -- A.4.3 The fair price -- References -- Index of names -- Index of symbols -- Index.
• Titolo autorizzato: Partial differential equations
• ISBN: 3-031-13379-X
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910637713003321
• Lo trovi qui: Univ. Federico II
• Serie: Graduate texts in mathematics ; ; 294.