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El. knyga: Introduction to Partial Differential Equations

  • Formatas: EPUB+DRM
  • Serija: Universitext
  • Išleidimo metai: 12-Jan-2017
  • Leidėjas: Springer International Publishing AG
  • Kalba: eng
  • ISBN-13: 9783319489360
Kitos knygos pagal šią temą:
Introduction to Partial Differential EquationsDidesnis paveikslėlis
  • Formatas: EPUB+DRM
  • Serija: Universitext
  • Išleidimo metai: 12-Jan-2017
  • Leidėjas: Springer International Publishing AG
  • Kalba: eng
  • ISBN-13: 9783319489360
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This modern take on partial differential equations does not require knowledge beyond vector calculus and linear algebra. The author focuses on the most important classical partial differential equations, including conservation equations and their characteristics, the wave equation, the heat equation, function spaces, and Fourier series, drawing on tools from analysis only as they arise.  Within each section the author creates a narrative that answers the five questions: What is the scientific problem we are trying to understand?How do we model that with PDE?What techniques can we use to analyze the PDE?How do those techniques apply to this equation?What information or insight did we obtain by developing and analyzing the PDE? The text stresses the interplay between modeling and mathematical analysis, providing a thorough source of problems and an inspiration for the development of methods. 

1. Introduction.- 2. Preliminaries.- 3. Conservation Equations and Characteristics.- 4. The Wave Equation.- 5. Separation of Variables.- 6. The Heat Equation.- 7. Function Spaces.- 8. Fourier Series.- 9. Maximum Principles.- 10. Weak Solutions.- 11. Variational Methods.- 12. Distributions.- 13. The Fourier Transform.- A. Appendix: Analysis Foundations.- References.- Notation Guide.- Index.

Recenzijos

The book under review is intended for an introductory course for students. The author gives a balanced presentation that includes modern methods, without requiring prerequisites beyond vector calculus and linear algebra. Concepts and definitions from analysis are introduced only as they are needed in the text. (Dian K. Palagachev, zbMATH 1364.35001, 2017)

1 Introduction
1(8)
1.1 Partial Differential Equations
1(1)
1.2 Example: d'Alembert's Wave Equation
2(1)
1.3 Types of Equations
3(2)
1.4 Well Posed Problems
5(1)
1.5 Approaches
6(3)
2 Preliminaries
9(16)
2.1 Real Numbers
9(1)
2.2 Complex Numbers
10(2)
2.3 Domains in Rn
12(1)
2.4 Differentiability
13(2)
2.5 Ordinary Differential Equations
15(3)
2.6 Vector Calculus
18(5)
2.7 Exercises
23(2)
3 Conservation Equations and Characteristics
25(20)
3.1 Model Problem: Oxygen in the Bloodstream
25(2)
3.2 Lagrangian Derivative and Characteristics
27(5)
3.3 Higher-Dimensional Equations
32(3)
3.4 Quasilinear Equations
35(6)
3.5 Exercises
41(4)
4 The Wave Equation
45(30)
4.1 Model Problem: Vibrating String
45(2)
4.2 Characteristics
47(4)
4.3 Boundary Problems
51(2)
4.4 Forcing Terms
53(6)
4.5 Model Problem: Acoustic Waves
59(2)
4.6 Integral Solution Formulas
61(6)
4.7 Energy and Uniqueness
67(2)
4.8 Exercises
69(6)
5 Separation of Variables
75(22)
5.1 Model Problem: Overtones
76(1)
5.2 Helmholtz Equation
76(5)
5.3 Circular Symmetry
81(6)
5.4 Spherical Symmetry
87(6)
5.5 Exercises
93(4)
6 The Heat Equation
97(14)
6.1 Model Problem: Heat Flow in a Metal Rod
97(4)
6.2 Scale-Invariant Solution
101(2)
6.3 Integral Solution Formula
103(4)
6.4 Inhomogeneous Problem
107(2)
6.5 Exercises
109(2)
7 Function Spaces
111(20)
7.1 Inner Products and Norms
111(3)
7.2 Lebesgue Integration
114(2)
7.3 LP Spaces
116(3)
7.4 Convergence and Completeness
119(3)
7.5 Orthonormal Bases
122(3)
7.6 Self-adjointness
125(3)
7.7 Exercises
128(3)
8 Fourier Series
131(24)
8.1 Series Solution of the Heat Equation
131(3)
8.2 Periodic Fourier Series
134(4)
8.3 Pointwise Convergence
138(3)
8.4 Uniform Convergence
141(2)
8.5 Convergence in L2
143(2)
8.6 Regularity and Fourier Coefficients
145(6)
8.7 Exercises
151(4)
9 Maximum Principles
155(22)
9.1 Model Problem: The Laplace Equation
155(6)
9.2 Mean Value Formula
161(4)
9.3 Strong Principle for Subharmonic Functions
165(2)
9.4 Weak Principle for Elliptic Equations
167(3)
9.5 Application to the Heat Equation
170(4)
9.6 Exercises
174(3)
10 Weak Solutions
177(28)
10.1 Test Functions and Weak Derivatives
177(3)
10.2 Weak Solutions of Continuity Equations
180(7)
10.3 Sobolev Spaces
187(3)
10.4 Sobolev Regularity
190(4)
10.5 Weak Formulation of Elliptic Equations
194(2)
10.6 Weak Formulation of Evolution Equations
196(6)
10.7 Exercises
202(3)
11 Variational Methods
205(34)
11.1 Model Problem: The Poisson Equation
206(1)
11.2 Dirichlet's Principle
207(1)
11.3 Coercivity and Existence of a Minimum
208(6)
11.4 Elliptic Regularity
214(3)
11.5 Eigenvalues by Minimization
217(7)
11.6 Sequential Compactness
224(3)
11.7 Estimation of Eigenvalues
227(7)
11.8 Euler-Lagrange Equations
234(3)
11.9 Exercises
237(2)
12 Distributions
239(22)
12.1 Model Problem: Coulomb's Law
239(3)
12.2 The Space of Distributions
242(3)
12.3 Distributional Derivatives
245(3)
12.4 Fundamental Solutions
248(4)
12.5 Green's Functions
252(5)
12.6 Time-Dependent Fundamental Solutions
257(2)
12.7 Exercises
259(2)
13 The Fourier Transform
261(16)
13.1 Fourier Transform
261(6)
13.2 Tempered Distributions
267(4)
13.3 The Wave Kernel
271(2)
13.4 The Heat Kernel
273(1)
13.5 Exercises
273(4)
Appendix A Analysis Foundations 277(4)
References 281(2)
Index 283
David Borthwick, Department of Mathematics and Computer Science, Emory University,  Atlanta, GA 30322
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