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E-grāmata: Partial Differential Equations: Analytical Methods and Applications

(Perm State University, Russia, and University of Louisville), , (Perm State University, Russia)
  • Formāts: 396 pages
  • Sērija : Textbooks in Mathematics
  • Izdošanas datums: 20-Nov-2019
  • Izdevniecība: CRC Press
  • ISBN-13: 9780429804410
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Partial Differential Equations: Analytical Methods and ApplicationsPalielināt
  • Formāts: 396 pages
  • Sērija : Textbooks in Mathematics
  • Izdošanas datums: 20-Nov-2019
  • Izdevniecība: CRC Press
  • ISBN-13: 9780429804410
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Partial Differential Equations: Analytical Methods and Applications covers all the basic topics of a Partial Differential Equations (PDE) course for undergraduate students or a beginners course for graduate students. It provides qualitative physical explanation of mathematical results while maintaining the expected level of it rigor.

This text introduces and promotes practice of necessary problem-solving skills. The presentation is concise and friendly to the reader. The "teaching-by-examples" approach provides numerous carefully chosen examples that guide step-by-step learning of concepts and techniques. Fourier series, Sturm-Liouville problem, Fourier transform, and Laplace transform are included. The books level of presentation and structure is well suited for use in engineering, physics and applied mathematics courses.

Highlights:











Offers a complete first course on PDEs





The texts flexible structure promotes varied syllabi for courses





Written with a teach-by-example approach which offers numerous examples and applications





Includes additional topics such as the Sturm-Liouville problem, Fourier and Laplace transforms, and special functions





The texts graphical material makes excellent use of modern software packages





Features numerous examples and applications which are suitable for readers studying the subject remotely or independently
Preface xi
1 Introduction
1(6)
1.1 Basic Definitions
1(3)
1.2 Examples
4(3)
2 First-Order Equations
7(14)
2.1 Linear First-Order Equations
7(5)
2.1.1 General Solution
7(3)
2.1.2 Initial Condition
10(2)
2.2 Quasilinear First-Order Equations
12(9)
2.2.1 Characteristic Curves
12(2)
2.2.2 Examples
14(7)
3 Second-Order Equations
21(8)
3.1 Classification of Second-Order Equations
21(1)
3.2 Canonical Forms
22(7)
3.2.1 Hyperbolic Equations
23(1)
3.2.2 Elliptic Equations
24(2)
3.2.3 Parabolic Equations
26(3)
4 The Sturm-Liouville Problem
29(14)
4.1 General Consideration
29(5)
4.2 Examples of Sturm-Liouville Problems
34(9)
5 One-Dimensional Hyperbolic Equations
43(56)
5.1 Wave Equation
43(2)
5.2 Boundary and Initial Conditions
45(3)
5.3 Longitudinal Vibrations of a Rod and Electrical Oscillations
48(4)
5.3.1 Rod Oscillations: Equations and Boundary Conditions
48(2)
5.3.2 Electrical Oscillations in a Circuit
50(2)
5.4 Traveling Waves: D'Alembert Method
52(5)
5.5 Cauchy Problem for Nonhomogeneous Wave Equation
57(3)
5.5.1 D'Alembert's Formula
57(1)
5.5.2 Green's Function
58(1)
5.5.3 Well-Posedness of the Cauchy Problem
59(1)
5.6 Finite Intervals: The Fourier Method for Homogeneous Equations
60(11)
5.7 The Fourier Method for Nonhomogeneous Equations
71(5)
5.8 The Laplace Transform Method: Simple Cases
76(2)
5.9 Equations with Nonhomogeneous Boundary Conditions
78(5)
5.10 The Consistency Conditions and Generalized Solutions
83(1)
5.11 Energy in the Harmonics
84(4)
5.12 Dispersion of Waves
88(5)
5.12.1 Cauchy Problem in an Infinite Region
88(3)
5.12.2 Propagation of a Wave Train
91(2)
5.13 Wave Propagation on an Inclined Bottom: Tsunami Effect
93(6)
6 One-Dimensional Parabolic Equations
99(40)
6.1 Heat Conduction and Diffusion: Boundary Value Problems
99(4)
6.1.1 Heat Conduction
99(1)
6.1.2 Diffusion Equation
100(1)
6.1.3 One-dimensional Parabolic Equations and Initial and Boundary Conditions
101(2)
6.2 The Fourier Method for Homogeneous Equations
103(8)
6.3 Nonhomogeneous Equations
111(3)
6.4 Green's Function and Duhamel's Principle
114(4)
6.5 The Fourier Method for Nonhomogeneous Equations with Nonhomogeneous Boundary Conditions
118(8)
6.6 Large Time Behavior of Solutions
126(3)
6.7 Maximum Principle
129(2)
6.8 The Heat Equation in an Infinite Region
131(8)
7 Elliptic Equations
139(48)
7.1 Elliptic Differential Equations and Related Physical Problems
139(2)
7.2 Harmonic Functions
141(1)
7.3 Boundary Conditions
142(4)
7.3.1 Example of an Ill-posed Problem
142(1)
7.3.2 Well-posed Boundary Value Problems
143(1)
7.3.3 Maximum Principle and its Consequences
144(2)
7.4 Laplace Equation in Polar Coordinates
146(1)
7.5 Laplace Equation and Interior BVP for Circular Domain
147(4)
7.6 Laplace Equation and Exterior BVP for Circular Domain
151(1)
7.7 Poisson Equation: General Notes and a Simple Case
151(3)
7.8 Poisson Integral
154(2)
7.9 Application of Bessel Functions for the Solution of Poisson Equations in a Circle
156(4)
7.10 Three-dimensional Laplace Equation for a Cylinder
160(4)
7.11 Three-dimensional Laplace Equation for a Ball
164(3)
7.11.1 Axisymmetric Case
164(1)
7.11.2 Non-axisymmetric Case
165(2)
7.12 BVP for Laplace Equation in a Rectangular Domain
167(2)
7.13 The Poisson Equation with Homogeneous Boundary Conditions
169(2)
7.14 Green's Function for Poisson Equations
171(5)
7.14.1 Homogeneous Boundary Conditions
171(4)
7.14.2 Nonhomogeneous Boundary Conditions
175(1)
7.15 Some Other Important Equations
176(11)
7.15.1 Helmholtz Equation
177(3)
7.15.2 Schrodinger Equation
180(7)
8 Two-Dimensional Hyperbolic Equations
187(40)
8.1 Derivation of the Equations of Motion
187(4)
8.1.1 Boundary and Initial Conditions
189(2)
8.2 Oscillations of a Rectangular Membrane
191(14)
8.2.1 The Fourier Method for Homogeneous Equations with Homogeneous Boundary Conditions
192(7)
8.2.2 The Fourier Method for Nonhomogeneous Equations with Homogeneous Boundary Conditions
199(4)
8.2.3 The Fourier Method for Nonhomogeneous Equations with Nonhomogeneous Boundary Conditions
203(2)
8.3 Small Transverse Oscillations of a Circular Membrane
205(22)
8.3.1 The Fourier Method for Homogeneous Equations with Homogeneous Boundary Conditions
206(3)
8.3.2 Axisymmetric Oscillations of a Membrane
209(5)
8.3.3 The Fourier Method for Nonhomogeneous Equations with Homogeneous Boundary Conditions
214(2)
8.3.4 Forced Axisymmetric Oscillations
216(2)
8.3.5 The Fourier Method for Equations with Nonhomogeneous Boundary Conditions
218(9)
9 Two-Dimensional Parabolic Equations
227(34)
9.1 Heat Conduction within a Finite Rectangular Domain
227(10)
9.1.1 The Fourier Method for the Homogeneous Heat Equation (Free Heat Exchange)
230(3)
9.1.2 The Fourier Method for Nonhomogeneous Heat Equation with Homogeneous Boundary Conditions
233(4)
9.2 Heat Conduction within a Circular Domain
237(11)
9.2.1 The Fourier Method for the Homogeneous Heat Equation
238(3)
9.2.2 The Fourier Method for the Nonhomogeneous Heat Equation
241(5)
9.2.3 The Fourier Method for the Nonhomogeneous Heat Equation with Nonhomogeneous Boundary Conditions
246(2)
9.3 Heat Conduction in an Infinite Medium
248(2)
9.4 Heat Conduction in a Semi-Infinite Medium
250(11)
10 Nonlinear Equations
261(22)
10.1 Burgers Equation
261(3)
10.1.1 Kink Solution
261(1)
10.1.2 Symmetries of the Burger's Equation
262(2)
10.2 General Solution of the Cauchy Problem
264(3)
10.2.1 Interaction of Kinks
265(2)
10.3 Korteweg-de Vries Equation
267(10)
10.3.1 Symmetry Properties of the KdV Equation
267(1)
10.3.2 Cnoidal Waves
268(2)
10.3.3 Solitons
270(1)
10.3.4 Bilinear Formulation of the KdV Equation
271(1)
10.3.5 Hirota's Method
272(2)
10.3.6 Multisoliton Solutions
274(3)
10.4 Nonlinear Schrodinger Equation
277(6)
10.4.1 Symmetry Properties of NSE
277(1)
10.4.2 Solitary Waves
278(5)
A Fourier Series, Fourier and Laplace Transforms
283(26)
A.1 Periodic Processes and Periodic Functions
283(1)
A.2 Fourier Formulas
284(2)
A.3 Convergence of Fourier Series
286(2)
A.4 Fourier Series for Non-periodic Functions
288(1)
A.5 Fourier Expansions on Intervals of Arbitrary Length
289(1)
A.6 Fourier Series in Cosine or in Sine Functions
290(2)
A.7 Examples
292(2)
A.8 The Complex Form of the Trigonometric Series
294(1)
A.9 Fourier Series for Functions of Several Variables
295(1)
A.10 Generalized Fourier Series
296(2)
A.11 The Gibbs Phenomenon
298(1)
A.12 Fourier Transforms
299(4)
A.13 Laplace Transforms
303(3)
A.14 Applications of Laplace Transform for ODE
306(3)
B Bessel and Legendre Functions
309(32)
B.1 Bessel Equation
309(3)
B.2 Properties of Bessel Functions
312(3)
B.3 Boundary Value Problems and Fourier-Bessel Series
315(5)
B.4 Spherical Bessel Functions
320(2)
B.5 The Gamma Function
322(2)
B.6 Legendre Equation and Legendre Polynomials
324(4)
B.7 Fourier-Legendre Series in Legendre Polynomials
328(3)
B.8 Associated Legendre Functions
331(3)
B.9 Fourier-Legendre Series in Associated Legendre Functions
334(1)
B.10 Airy Functions
335(6)
C Sturm-Liouville Problem and Auxiliary Functions for One and Two Dimensions
341(8)
C.1 Eigenvalues and Eigenfunctions of 1D Sturm-Liouville Problem for Different Types of Boundary Conditions
341(2)
C.2 Auxiliary Functions
343(6)
D The Sturm-Liouville Problem for Circular and Rectangular Domains
349(26)
D.1 The Sturm-Liouville Problem for a Circle
349(3)
D.2 The Sturm-Liouville Problem for the Rectangle
352(3)
E The Heat Conduction and Poisson Equations for Rectangular Domains - Examples
355(1)
E.1 The Laplace and Poisson Equations for a Rectangular Domain with Nonhomogeneous Boundary Conditions - Examples
355(11)
E.2 The Heat Conduction Equations with Nonhomogeneous Boundary Conditions - Examples
366(9)
Bibliography 375(2)
Index 377
Victor Henner is a professor at the Department of Physics and Astronomy at the University of Louisville. He has Ph.Ds from the Novosibirsk Institute of Mathematics in Russia and Moscow State University. He co-wrote with Tatyana Belozerova Ordinary and Partial Differential Equations.

Tatyana Belozerova is a professor at Perm State University in Russia. Along with Ordinary and Partial Differential Equations, she co-wrote with Victor Henner Mathematical Methods in Physics: Partial Differential Equations, Fourier Series, and Special Functions.

Alexander Nepomnyashchy is a mathematics professor at Northwestern University and hails from the Faculty of Mathematics at Technion-Israel Institute of Technology. His research interests include non-linear stability theory and pattern formation.
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