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E-raamat: Introduction to Partial Differential Equations with MATLAB

(Fairfield University, Connecticut, USA)
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"Preface Many problems in the physical world can be modeled by partial differential equations, from applications as diverse as the flow of heat, the vibration of a ball, the propagation of sound waves, the diffusion of ink in a glass of water, electric and magnetic fields, the spread of algae along the ocean's surface, the fluctuation in the price of a stock option, and the quantum mechanical behavior of a hydrogen atom. However, as with any area of applied mathematics, the field of PDEs is interesting not only because of its applications, but because it has taken on a mathematical life of its own. The author has written this book with both ideas in mind, in the hope that the student will appreciate the usefulness of the subject and, at the same time, get a glimpse into the beauty of some of the underlying mathematics. This text is suitable for a two-semester introduction to partial differential equations and Fourier series for students who have had basic courses in multivariable calculus (through Stokes's and the Divergence Theorems) and ordinary differential equations. Over the years, the author has taught much of the material to undergraduate mathematics, physics and engineering students at Penn State and Fairfield Universities, as well as to engineering graduate students at Penn State and mathematics and engineering graduate students at Fairfield. It is assumed that the student has not had a course in real analysis. Thus, we treat pointwise convergence of Fourier series and do not talk about mean-square convergence until Chapter 8 (and, there, in terms of the Riemann, and not the Lebesgue, integral). Further, we feel that it is not appropriate to introduce so subtle an idea as uniform convergence in this setting, so we discuss it only in the Appendices"--

Arvustused

"This is an excellent textbook first, the book can be used by a person who has no interest in MATLAB at all, and, second, this book deserves to be considered byin fact, should be at the top of the list ofany professor looking for an undergraduate text in PDEs. there are several reasons why I view this book as being in the upper echelon of undergraduate PDE textbooks. One is the extremely high quality of exposition. Coleman writes clearly and cleanly, with a conversational tone and a high regard for motivation. He clearly has a great deal of experience teaching this subject and has learned what points are likely to cause confusion and therefore need expanded discussion. The author also employs the nice pedagogical feature of page-long preludes to each chapter, which not only summarize what the chapter will cover and how it fits into the general theme of things, but also typically provide some brief historical commentary as well. In general, the overall effect of this book is like listening to a discussion by a good professor in office hours. very highly recommended. I dont know when or if I will ever teach an undergraduate PDE course, but if I ever do, this book will certainly be on my short list of possible texts." Mark Hunacek, MAA Reviews, September 2013

Praise for the First Edition:"The strongest aspect of this text is the very large number of worked boundary value problem examples." SIAM

"This is a useful introductory text on PDEs for advanced undergraduate / beginning graduate students of applied mathematics, physics, or engineering sciences. a nice introductory text which certainly is of great use in preparing and delivering courses." Zentralblatt MATH

"Readers new to the subject will find Colemans appendix cataloguing important partial differential equations in their natural surroundings quite useful. Colemans more explicit, extended style would probably allow its use as an advanced graduate or reference text for UK engineers or physicists." Times Higher Education

"The book presents very useful material and can be used as a basic text for self-study of PDEs." EMS Newsletter

"Each chapter is introduced by a prelude that describes its content and gives historical background. Each section concludes with a set of exercises, many of which are marked MATLAB." CMS Notes "This is an excellent textbook first, the book can be used by a person who has no interest in MATLAB at all, and, second, this book deserves to be considered byin fact, should be at the top of the list ofany professor looking for an undergraduate text in PDEs. there are several reasons why I view this book as being in the upper echelon of undergraduate PDE textbooks. One is the extremely high quality of exposition. Coleman writes clearly and cleanly, with a conversational tone and a high regard for motivation. He clearly has a great deal of experience teaching this subject and has learned what points are likely to cause confusion and therefore need expanded discussion. The author also employs the nice pedagogical feature of page-long preludes to each chapter, which not only summarize what the chapter will cover and how it fits into the general theme of things, but also typically provide some brief historical commentary as well. In general, the overall effect of this book is like listening to a discussion by a good professor in office hours. very highly recommended. I dont know when or if I will ever teach an undergraduate PDE course, but if I ever do, this book will certainly be on my short list of possible texts." Mark Hunacek, MAA Reviews, September 2013

" a pick for any college-level collection strong in applied mathematics and nonlinear science, and provides a thorough assessment updated for the latest mathematical applications. From modeling problems ranging from heat flow to sound waves and algae spread to equations based on methods of solution and physical and mathematical applications, this reviews PDEs and their applications and is a pick for advanced math collections whose patrons have an basic knowledge of multivariable calculus and ODEs. Any working with MATLAB codes and problem-solving applications need this!" California Bookwatch, November 2013

Praise for the First Edition:"The strongest aspect of this text is the very large number of worked boundary value problem examples." SIAM

"This is a useful introductory text on PDEs for advanced undergraduate / beginning graduate students of applied mathematics, physics, or engineering sciences. a nice introductory text which certainly is of great use in preparing and delivering courses." Zentralblatt MATH

"Readers new to the subject will find Colemans appendix cataloguing important partial differential equations in their natural surroundings quite useful. Colemans more explicit, extended style would probably allow its use as an advanced graduate or reference text for UK engineers or physicists." Times Higher Education

"The book presents very useful material and can be used as a basic text for self-study of PDEs." EMS Newsletter

"Each chapter is introduced by a prelude that describes its content and gives historical background. Each section concludes with a set of exercises, many of which are marked MATLAB." CMS Notes

Preface xi
Prelude to
Chapter 1
1(2)
1 Introduction
3(40)
1.1 What are Partial Differential Equations?
3(3)
1.2 PDEs We Can Already Solve
6(4)
1.3 Initial and Boundary Conditions
10(2)
1.4 Linear PDEs--Definitions
12(4)
1.5 Linear PDEs--The Principle of Superposition
16(3)
1.6 Separation of Variables for Linear, Homogeneous PDEs
19(6)
1.7 Eigenvalue Problems
25(18)
Prelude to
Chapter 2
41(2)
2 The Big Three PDEs
43(36)
2.1 Second-Order, Linear, Homogeneous PDEs with Constant Coefficients
43(1)
2.2 The Heat Equation and Diffusion
44(10)
2.3 The Wave Equation and the Vibrating String
54(5)
2.4 Initial and Boundary Conditions for the Heat and Wave Equations
59(7)
2.5 Laplace's Equation--The Potential Equation
66(5)
2.6 Using Separation of Variables to Solve the Big Three PDEs
71(8)
Prelude to
Chapter 3
77(2)
3 Fourier Series
79(50)
3.1 Introduction
79(1)
3.2 Properties of Sine and Cosine
80(9)
3.3 The Fourier Series
89(6)
3.4 The Fourier Series, Continued
95(9)
3.5 The Fourier Series--Proof of Pointwise Convergence
104(13)
3.6 Fourier Sine and Cosine Series
117(7)
3.7 Completeness
124(5)
Prelude to
Chapter 4
127(2)
4 Solving the Big Three PDEs on Finite Domains
129(34)
4.1 Solving the Homogeneous Heat Equation for a Finite Rod
129(9)
4.2 Solving the Homogeneous Wave Equation for a Finite String
138(9)
4.3 Solving the Homogeneous Laplace's Equation on a Rectangular Domain
147(6)
4.4 Nonhomogeneous Problems
153(10)
Prelude to
Chapter 5
161(2)
5 Characteristics
163(50)
5.1 First-Order PDEs with Constant Coefficients
163(11)
5.2 First-Order PDEs with Variable Coefficients
174(6)
5.3 The Infinite String
180(12)
5.4 Characteristics for Semi-Infinite and Finite String Problems
192(9)
5.5 General Second-Order Linear PDEs and Characteristics
201(12)
Prelude to
Chapter 6
211(2)
6 Integral Transforms
213(64)
6.1 The Laplace Transform for PDEs
213(7)
6.2 Fourier Sine and Cosine Transforms
220(10)
6.3 The Fourier Transform
230(12)
6.4 The Infinite and Semi-Infinite Heat Equations
242(12)
6.5 Distributions, the Dirac Delta Function and Generalized Fourier Transforms
254(12)
6.6 Proof of the Fourier Integral Formula
266(11)
Prelude to
Chapter 7
275(2)
7 Special Functions and Orthogonal Polynomials
277(52)
7.1 The Special Functions and Their Differential Equations
277(8)
7.2 Ordinary Points and Power Series Solutions; Chebyshev, Hermite and Legendre Polynomials
285(7)
7.3 The Method of Frobenius; Laguerre Polynomials
292(8)
7.4 Interlude: The Gamma Function
300(5)
7.5 Bessel Functions
305(12)
7.6 Recap: A List of Properties of Bessel Functions and Orthogonal Polynomials
317(12)
Prelude to
Chapter 8
327(2)
8 Sturm-Liouville Theory and Generalized Fourier Series
329(46)
8.1 Sturm-Liouville Problems
329(8)
8.2 Regular and Periodic Sturm-Liouville Problems
337(8)
8.3 Singular Sturm-Liouville Problems; Self-Adjoint Problems
345(9)
8.4 The Mean-Square or L2 Norm and Convergence in the Mean
354(7)
8.5 Generalized Fourier Series; Parseval's Equality and Completeness
361(14)
Prelude to
Chapter 9
373(2)
9 PDEs in Higher Dimensions
375(90)
9.1 PDEs in Higher Dimensions: Examples and Derivations
375(11)
9.2 The Heat and Wave Equations on a Rectangle; Multiple Fourier Series
386(16)
9.3 Laplace's Equation in Polar Coordinates: Poisson's Integral Formula
402(12)
9.4 The Wave and Heat Equations in Polar Coordinates
414(11)
9.5 Problems in Spherical Coordinates
425(14)
9.6 The Infinite Wave Equation and Multiple Fourier Transforms
439(17)
9.7 Postlude: Eigenvalues and Eigenfunctions of the Laplace Operator; Green's Identities for the Laplacian
456(9)
Prelude to
Chapter 10
463(2)
10 Nonhomogeneous Problems and Green's Functions
465(74)
10.1 Green's Functions for ODEs
465(19)
10.2 Green's Function and the Dirac Delta Function
484(16)
10.3 Green's Functions for Elliptic PDEs (I): Poisson's Equation in Two Dimensions
500(16)
10.4 Green's Functions for Elliptic PDEs (II): Poisson's Equation in Three Dimensions; the Helmholtz Equation
516(9)
10.5 Green's Functions for Equations of Evolution
525(14)
Prelude to
Chapter 11
537(2)
11 Numerical Methods
539(40)
11.1 Finite Difference Approximations for ODEs
539(12)
11.2 Finite Difference Approximations for PDEs
551(14)
11.3 Spectral Methods and the Finite Element Method
565(14)
A Uniform Convergence; Differentiation and Integration of Fourier Series 579(6)
B Other Important Theorems 585(6)
C Existence and Uniqueness Theorems 591(10)
D A Menagerie of PDEs 601(12)
E MATLAB Code for Figures and Exercises 613(14)
F Answers to Selected Exercises 627(20)
References 647(8)
Index 655
Matthew P. Coleman