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Partial Differential Equations: Mathematical Techniques for Engineers 1st ed. 2017 [Hardback]

  • Formāts: Hardback, 255 pages, height x width: 235x155 mm, weight: 5325 g, 9 Illustrations, color; 57 Illustrations, black and white; XIII, 255 p. 66 illus., 9 illus. in color., 1 Hardback
  • Sērija : Mathematical Engineering
  • Izdošanas datums: 05-May-2017
  • Izdevniecība: Springer International Publishing AG
  • ISBN-10: 3319552112
  • ISBN-13: 9783319552118
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Partial Differential Equations: Mathematical Techniques for Engineers 1st ed. 2017Palielināt
  • Formāts: Hardback, 255 pages, height x width: 235x155 mm, weight: 5325 g, 9 Illustrations, color; 57 Illustrations, black and white; XIII, 255 p. 66 illus., 9 illus. in color., 1 Hardback
  • Sērija : Mathematical Engineering
  • Izdošanas datums: 05-May-2017
  • Izdevniecība: Springer International Publishing AG
  • ISBN-10: 3319552112
  • ISBN-13: 9783319552118
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9783319552118.html
Keywords:
This monograph presents a graduate-level treatment of partial differential equations (PDEs) for engineers. The book begins with a review of the geometrical interpretation of systems of ODEs, the appearance of PDEs in engineering is motivated by the general form of balance laws in continuum physics. Four chapters are devoted to a detailed treatment of the single first-order PDE, including shock waves and genuinely non-linear models, with applications to traffic design and gas dynamics. The rest of the book deals with second-order equations. In the treatment of hyperbolic equations, geometric arguments are used whenever possible and the analogy with discrete vibrating systems is emphasized. The diffusion and potential equations afford the opportunity of dealing with questions of uniqueness and continuous dependence on the data, the Fourier integral, generalized functions (distributions), Duhamel's principle, Green's functions and Dirichlet and Neumann problems. The target audience primarily comprises graduate students in engineering, but the book may also be beneficial for lecturers, and research experts both in academia in industry.

Recenzijas

The book would be accessible to strong undergraduates with some multivariable calculus, basic linear algebra and ordinary differential equations. The author provides references at each stage to several of the standard texts, including those of Garabedian and John. This would be a good textbook for an introduction to PDEs or as a supplement to a more standard mathematical treatment. (William J. Satzer, MAA Reviews, maa.org, July, 2017)

Part I Background
1 Vector Fields and Ordinary Differential Equations
3(22)
1.1 Introduction
3(1)
1.2 Curves and Surfaces in Rn
4(5)
1.2.1 Cartesian Products, Affine Spaces
4(2)
1.2.2 Curves in Rn
6(2)
1.2.3 Surfaces in R3
8(1)
1.3 The Divergence Theorem
9(3)
1.3.1 The Divergence of a Vector Field
9(1)
1.3.2 The Flux of a Vector Field over an Orientable Surface
10(1)
1.3.3 Statement of the Theorem
11(1)
1.3.4 A Particular Case
11(1)
1.4 Ordinary Differential Equations
12(13)
1.4.1 Vector Fields as Differential Equations
12(1)
1.4.2 Geometry Versus Analysis
13(1)
1.4.3 An Example
14(2)
1.4.4 Autonomous and Non-autonomous Systems
16(1)
1.4.5 Higher-Order Equations
17(1)
1.4.6 First Integrals and Conserved Quantities
18(3)
1.4.7 Existence and Uniqueness
21(1)
1.4.8 Food for Thought
22(2)
References
24(1)
2 Partial Differential Equations in Engineering
25(26)
2.1 Introduction
25(1)
2.2 What is a Partial Differential Equation?
26(1)
2.3 Balance Laws
27(9)
2.3.1 The Generic Balance Equation
28(3)
2.3.2 The Case of Only One Spatial Dimension
31(3)
2.3.3 The Need for Constitutive Laws
34(2)
2.4 Examples of PDEs in Engineering
36(15)
2.4.1 Traffic Flow
36(1)
2.4.2 Diffusion
37(1)
2.4.3 Longitudinal Waves in an Elastic Bar
38(1)
2.4.4 Solitons
39(1)
2.4.5 Time-Independent Phenomena
40(1)
2.4.6 Continuum Mechanics
41(6)
References
47(4)
Part II The First-Order Equation
3 The Single First-Order Quasi-linear PDE
51(24)
3.1 Introduction
51(2)
3.2 Quasi-linear Equation in Two Independent Variables
53(3)
3.3 Building Solutions from Characteristics
56(5)
3.3.1 A Fundamental Lemma
56(1)
3.3.2 Corollaries of the Fundamental Lemma
57(1)
3.3.3 The Cauchy Problem
58(2)
3.3.4 What Else Can Go Wrong?
60(1)
3.4 Particular Cases and Examples
61(10)
3.4.1 Homogeneous Linear Equation
61(1)
3.4.2 Non-homogeneous Linear Equation
62(2)
3.4.3 Quasi-linear Equation
64(7)
3.5 A Computer Program
71(4)
References
74(1)
4 Shock Waves
75(14)
4.1 The Way Out
75(1)
4.2 Generalized Solutions
76(2)
4.3 A Detailed Example
78(4)
4.4 Discontinuous Initial Conditions
82(7)
4.4.1 Shock Waves
82(3)
4.4.2 Rarefaction Waves
85(3)
References
88(1)
5 The Genuinely Nonlinear First-Order Equation
89(26)
5.1 Introduction
89(1)
5.2 The Monge Cone Field
90(2)
5.3 The Characteristic Directions
92(4)
5.4 Recapitulation
96(2)
5.5 The Cauchy Problem
98(1)
5.6 An Example
99(2)
5.7 More Than Two Independent Variables
101(4)
5.7.1 Quasi-linear Equations
101(3)
5.7.2 Non-linear Equations
104(1)
5.8 Application to Hamiltonian Systems
105(10)
5.8.1 Hamiltonian Systems
105(1)
5.8.2 Reduced Form of a First-Order PDE
106(1)
5.8.3 The Hamilton-Jacobi Equation
107(1)
5.8.4 An Example
108(4)
References
112(3)
Part III Classification of Equations and Systems
6 The Second-Order Quasi-linear Equation
115(16)
6.1 Introduction
115(2)
6.2 The First-Order PDE Revisited
117(1)
6.3 The Second-Order Case
118(3)
6.4 Propagation of Weak Singularities
121(6)
6.4.1 Hadamard's Lemma and Its Consequences
121(2)
6.4.2 Weak Singularities
123(2)
6.4.3 Growth and Decay
125(2)
6.5 Normal Forms
127(4)
References
130(1)
7 Systems of Equations
131(26)
7.1 Systems of First-Order Equations
131(9)
7.1.1 Characteristic Directions
131(2)
7.1.2 Weak Singularities
133(1)
7.1.3 Strong Singularities in Linear Systems
134(1)
7.1.4 An Application to the Theory of Beams
135(2)
7.1.5 Systems with Several Independent Variables
137(3)
7.2 Systems of Second-Order Equations
140(17)
7.2.1 Characteristic Manifolds
140(2)
7.2.2 Variation of the Wave Amplitude
142(2)
7.2.3 The Timoshenko Beam Revisited
144(3)
7.2.4 Air Acoustics
147(3)
7.2.5 Elastic Waves
150(3)
References
153(4)
Part IV Paradigmatic Equations
8 The One-Dimensional Wave Equation
157(26)
8.1 The Vibrating String
157(1)
8.2 Hyperbolicity and Characteristics
158(1)
8.3 The d'Alembert Solution
159(1)
8.4 The Infinite String
160(3)
8.5 The Semi-infinite String
163(5)
8.5.1 D'Alembert Solution
163(2)
8.5.2 Interpretation in Terms of Characteristics
165(2)
8.5.3 Extension of Initial Data
167(1)
8.6 The Finite String
168(6)
8.6.1 Solution
168(3)
8.6.2 Uniqueness and Stability
171(2)
8.6.3 Time Periodicity
173(1)
8.7 Moving Boundaries and Growth
174(1)
8.8 Controlling the Slinky?
175(2)
8.9 Source Terms and Duhamel's Principle
177(6)
References
182(1)
9 Standing Waves and Separation of Variables
183(26)
9.1 Introduction
183(1)
9.2 A Short Review of the Discrete Case
184(5)
9.3 Shape-Preserving Motions of the Vibrating String
189(3)
9.4 Solving Initial-Boundary Value Problems by Separation of Variables
192(6)
9.5 Shape-Preserving Motions of More General Continuous Systems
198(11)
9.5.1 String with Variable Properties
198(3)
9.5.2 Beam Vibrations
201(2)
9.5.3 The Vibrating Membrane
203(5)
References
208(1)
10 The Diffusion Equation
209(30)
10.1 Physical Considerations
209(5)
10.1.1 Diffusion of a Pollutant
209(3)
10.1.2 Conduction of Heat
212(2)
10.2 General Remarks on the Diffusion Equation
214(1)
10.3 Separating Variables
215(1)
10.4 The Maximum-Minimum Theorem and Its Consequences
216(3)
10.5 The Finite Rod
219(2)
10.6 Non-homogeneous Problems
221(2)
10.7 The Infinite Rod
223(2)
10.8 The Fourier Series and the Fourier Integral
225(3)
10.9 Solution of the Cauchy Problem
228(2)
10.10 Generalized Functions
230(4)
10.11 Inhomogeneous Problems and Duhamel's Principle
234(5)
References
238(1)
11 The Laplace Equation
239(14)
11.1 Introduction
239(1)
11.2 Green's Theorem and the Dirichlet and Neumann Problems
240(3)
11.3 The Maximum-Minimum Principle
243(1)
11.4 The Fundamental Solutions
244(2)
11.5 Green's Functions
246(2)
11.6 The Mean-Value Theorem
248(1)
11.7 Green's Function for the Circle and the Sphere
249(4)
References
252(1)
Index 253