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Invitation to Partial Differential Equations [Hardback]

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  • Formāts: Hardback, 319 pages, height x width: 254x178 mm, weight: 792 g
  • Sērija : Graduate Studies in Mathematics
  • Izdošanas datums: 01-Jan-2022
  • Izdevniecība: American Mathematical Society
  • ISBN-10: 0821836404
  • ISBN-13: 9780821836408
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Invitation to Partial Differential EquationsPalielināt
  • Formāts: Hardback, 319 pages, height x width: 254x178 mm, weight: 792 g
  • Sērija : Graduate Studies in Mathematics
  • Izdošanas datums: 01-Jan-2022
  • Izdevniecība: American Mathematical Society
  • ISBN-10: 0821836404
  • ISBN-13: 9780821836408
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9780821836408.html
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This book is based on notes from a beginning graduate course on partial differential equations. Prerequisites for using the book are a solid undergraduate course in real analysis. There are more than 100 exercises in the book. Some of them are just exercises, whereas others, even though they may require new ideas to solve them, provide additional important information about the subject.

It is a great pleasure to see this book - written by a great master of the subject - finally in print. This treatment of a core part of mathematics and its applications offers the student both a solid foundation in basic calculations techniques in the subject, as well as a basic introduction to the more general machinery, e.g., distributions, Sobolev spaces, etc., which are such a key part of any modern treatment. As such this book is ideal for more advanced undergraduates as well as mathematically inclined students from engineering or the natural sciences.

Recenzijas

Shubin has a lovely intuitive writing style which provides a gentle introduction to this beautiful subject. Many good exercises (and solutions) are provided!"" - Rafe Mazzeo, Stanford University

""This text provides an excellent semester's introduction to classical and modern topics in linear PDE, suitable for students with a background in advanced calculus and Lebesgue integration. The author intersperses treatments of the Laplace, heat, and wave equations with developments of various functional analytic tools, particularly distribution theory and spectral theory, introducing key concepts while deftly avoiding heavy technicalities."" - Michael Taylor, University of North Carolina, Chapel Hill

Foreword xi
Preface xiii
Selected notational conventions xvii
Chapter 1 Linear differential operators
1(14)
§1.1 Definition and examples
1(1)
§1.2 The total and the principal symbols
2(2)
§1.3 Change of variables
4(1)
§1.4 The canonical form of second-order operators with constant coefficients
5(2)
§1.5 Characteristics. Ellipticity and hyperbolicity
7(1)
§1.6 Characteristics and the canonical form of second-order operators and second-order equations for n = 2
8(3)
§1.7 The general solution of a homogeneous hyperbolic equation with constant coefficients for n = 2
11(1)
§1.8 Appendix. Tangent and cotangent vectors
12(2)
§1.9 Problems
14(1)
Chapter 2 One-dimensional wave equation
15(28)
§2.1 Vibrating string equation
15(5)
§2.2 Unbounded string. The Cauchy problem. D'Alembert's formula
20(4)
§2.3 A semibounded string. Reflection of waves from the end of the string
24(2)
§2.4 A bounded string. Standing waves. The Fourier method (separation of variables method)
26(7)
§2.5 Appendix. The calculus of variations and classical mechanics
33(6)
§2.6 Problems
39(4)
Chapter 3 The Sturm-Liouville problem
43(14)
§3.1 Formulation of the problem
43(1)
§3.2 Basic properties of eigenvalues and eigenfunctions
44(3)
§3.3 The short-wave asymptotics
47(3)
§3.4 The Green's function and completeness of the system of eigenfunctions
50(4)
§3.5 Problems
54(3)
Chapter 4 Distributions
57(42)
§4.1 Motivation of the definition. Spaces of test functions
57(6)
§4.2 Spaces of distributions
63(4)
§4.3 Topology and convergence in the spaces of distributions
67(3)
§4.4 The support of a distribution
70(4)
§4.5 Differentiation of distributions and multiplication by a smooth function
74(13)
§4.6 A general notion of the transposed (adjoint) operator. Change of variables. Homogeneous distributions
87(4)
§4.7 Appendix. Minkowski inequality
91(3)
§4.8 Appendix Completeness of distribution spaces
94(2)
§4.9 Problems
96(3)
Chapter 5 Convolution and Fourier transform
99(28)
§5.1 Convolution and direct product of regular functions
99(2)
§5.2 Direct product of distributions
101(3)
§5.3 Convolution of distributions
104(3)
§5.4 Other properties of convolution. Support and singular support of a convolution
107(2)
§5.5 Relation between smoothness of a fundamental solution and that of solutions of the homogeneous equation
109(4)
§5.6 Solutions with isolated singularities. A removable singularity theorem for harmonic functions
113(1)
§5.7 Estimates of derivatives of a solution of a hypoelliptic equation
114(2)
§5.8 Fourier transform of tempered distributions
116(4)
§5.9 Applying the Fourier transform to find fundamental solutions
120(1)
§5.10 Liouville's theorem
121(2)
§5.11 Problems
123(4)
Chapter 6 Harmonic functions
127(24)
§6.1 Mean-value theorems for harmonic functions
127(2)
§6.2 The maximum principle
129(2)
§6.3 Dirichlet's boundary value problem
131(2)
§6.4 Hadamard's example
133(3)
§6.5 Green's function for the Laplacian
136(4)
§6.6 Holder regularity
140(3)
§6.7 Explicit formulas for Green's functions
143(4)
§6.8 Problems
147(4)
Chapter 7 The heat equation
151(20)
§7.1 Physical meaning of the heat equation
151(2)
§7.2 Boundary value problems for the heat and Laplace equations
153(2)
§7.3 A proof that the limit function is harmonic
155(1)
§7.4 A solution of the Cauchy problem for the heat equation and Poisson's integral
156(5)
§7.5 The fundamental solution for the heat operator. Duhamel's formula
161(3)
§7.6 Estimates of derivatives of a solution of the heat equation
164(1)
§7.7 Holmgren's principle. The uniqueness of solution of the Cauchy problem for the heat equation
165(3)
§7.8 A scheme for solving the first and second initial-boundary value problems by the Fourier method
168(2)
§7.9 Problems
170(1)
Chapter 8 Sobolev spaces. A generalized solution of Dirichlet's problem
171(22)
§8.1 Spaces HS(Q)
171(6)
§8.2 Spaces Hs(n)
177(4)
§8.3 Dirichlet's integral. The Friedrichs inequality
181(2)
§8.4 Dirichlet's problem (generalized solutions)
183(7)
§8.5 Problems
190(3)
Chapter 9 The eigenvalues and eigenfunctions of the Laplace operator
193(22)
§9.1 Symmetric and selfadjoint operators in Hilbert space
193(4)
§9.2 The Friedrichs extension
197(4)
§9.3 Discreteness of spectrum for the Laplace operator in a bounded domain
201(1)
§9.4 Fundamental solution of the Helmholtz operator and the analyticity of eigenfunctions of the Laplace operator at the interior points. Bessel's equation
202(7)
§9.5 Variational principle. The behavior of eigenvalues under variation of the domain. Estimates of eigenvalues
209(3)
§9.6 Problems
212(3)
Chapter 10 The wave equation
215(26)
§10.1 Physical problems leading to the wave equation
215(5)
§10.2 Plane, spherical, and cylindric waves
220(2)
§10.3 The wave equation as a Hamiltonian system
222(6)
§10.4 A spherical wave caused by an instant flash and a solution of the Cauchy problem for the three-dimensional wave equation
228(6)
§10.5 The fundamental solution for the three-dimensional wave operator and a solution of the nonhomogeneous wave equation
234(2)
§10.6 The two-dimensional wave equation (the descent method)
236(3)
§10.7 Problems
239(2)
Chapter 11 Properties of the potentials and their computation
241(16)
§11.1 Definitions of potentials
242(2)
§11.2 Functions smooth up to F from each side, and their derivatives
244(7)
§11.3 Jumps of potentials
251(2)
§11.4 Calculating potentials
253(3)
§11.5 Problems
256(1)
Chapter 12 Wave fronts and short-wave asymptotics for hyperbolic equations
257(32)
§12.1 Characteristics as surfaces of jumps
257(5)
§12.2 The Hamilton-Jacobi equation. Wave fronts, bicharacteristics, and rays
262(7)
§12.3 The characteristics of hyperbolic equations
269(2)
§12.4 Rapidly oscillating solutions. The eikonal equation and the transport equations
271(10)
§12.5 The Cauchy problem with rapidly oscillating initial data
281(6)
§12.6 Problems
287(2)
Chapter 13 Answers and hints. Solutions
289(22)
Bibliography 311(4)
Index 315
Maxim Braverman, Northeastern University, Boston, MA, USA.

Robert McOwen, Northeastern University, Boston, MA, USA.

Peter Topalov, Northeastern University, Boston, MA, USA.