Solutions Manual to Accompany Beginning Partial Differential Equations (eBook)

Peter V. O'Neil (Autor)

eBook Download: EPUB

2014 | 3. Auflage
John Wiley & Sons (Verlag)
9781118880586 (ISBN)

Infos

As the Solutions Manual, this book is meant to accompany the main title, Beginning of Partial Differential Equations, Third Edition. The Third Edition features a challenging, yet accessible, introduction to partial differential equations, and provides a solid introduction to partial differential equations, particularly methods of solution based on characteristics, separation of variables, as well as Fourier series, integrals, and transforms.  Thoroughly updated with novel applications such as Poe’s pendulum and Kepler’s problem in astronomy, the book begins with first-order linear and quasi-linear PDEs and the role of characteristics in the existence and uniqueness of solutions. Canonical forms are discussed for the linear second-order equation, along with the Cauchy problem, existence and uniqueness of solutions, and characteristics as carriers of discontinuities in solutions. Fourier series, integrals, and transforms are followed by their rigorous application to wave and diffusion equations as well as to Dirichlet and Neumann problems. In addition, solutions are viewed through physical interpretations of PDEs. The book concludes with a transition to more advanced topics, including the proof of an existence theorem for the Dirichlet problem and an introduction to distributions.  New topical coverage includes novel applications, such as Poe’s pendulum and Kepler’s problem in astronomy.  Solutions using the Laplace transform have been added.  The book continues to be appropriate for those who need to emphasize a rigorous mathematical treatment and those who need quick access to methods and applications.  The first group is addressed by including details of proofs in Chapter Eight, sections of which can be included at any point in course lectures.  The second group is addressed with a detailed chapter organization that allows for a rapid transition from method to solution to application in the beginning chapters.

Autor

Peter V. O'Neil, PhD, is Professor Emeritus in the Department of Mathematics at The University of Alabama at Birmingham. Dr. O'Neil has over forty years of academic experience and is the recipient of the Lester R. Ford Award from the Mathematical Association of America. He is a member of the American Mathematical Society, the Society for Industrial and Applied Mathematics, and the American Association for the Advancement of Science.

Inhalt

Preface vii

1 First Ideas 1

1.1 Two Partial Differential Equations 1

1.2 Fourier Series 4

1.3 Two Eigenvalue Problems 12

1.4 A Proof of the Convergence Theorem 14

2 Solutions of the Heat Equation 15

2.1 Solutions on an Interval [0, L] 15

2.2 A Nonhomogeneous Problem 19

3 Solutions of the Wave Equation 25

3.1 Solutions on Bounded Intervals 25

3.2 The Cauchy Problem 32

3.2.1 d'Alembert's Solution 32

3.2.2 The Cauchy Problem on a Half Line 36

3.2.3 Characteristic Triangles and Quadrilaterals 41

3.2.4 A Cauchy Problem with a Forcing Term 41

3.2.5 String with Moving Ends 42

3.3 The Wave Equation in Higher Dimensions 46

3.3.1 Vibrations in a Membrane with Fixed Frame 46

3.3.2 The Poisson Integral Solution 47

3.3.3 Hadamard's Method of Descent 47

4 Dirichlet and Neumann Problems 49

4.1 Laplace's Equation and Harmonic Functions 49

4.2 The Dirichlet Problem for a Rectangle 50

4.3 The Dirichlet Problem for a Disk 52

4.4 Properties of Harmonic Functions 57

4.4.1 Topology of Rn 57

4.4.2 Representation Theorems 58

4.4.3 The Mean Value Theorem and the Maximum Principle 60

4.5 The Neumann Problem 61

4.5.1 Uniqueness and Existence 61

4.5.2 Neumann Problem for a Rectangle 62

4.5.3 Neumann Problem for a Disk 63

4.6 Poisson's Equation 64

4.7 An Existence Theorem for the Dirichlet Problem 65

5 Fourier Integral Methods of Solution 67

5.1 The Fourier Integral of a Function 67

5.2 The Heat Equation on the Real Line 70

5.3 The Debate Over the Age of the Earth 73

5.4 Burgers' Equation 73

5.5 The Cauchy Problem for the Wave Equation 74

5.6 Laplace's Equation on Unbounded Domains 76

6 Solutions Using Eigenfunction Expansions 79

6.1 A Theory of Eigenfunction Expansions 79

6.2 Bessel Functions 83

6.3 Applications of Bessel Functions 87

6.3.1 Temperature Distribution in a Solid Cylinder 87

6.3.2 Vibrations of a Circular Drum 87

6.4 Legendre Polynomials and Applications 90

7 Integral Transform Methods of Solution 97

7.1 The Fourier Transform 97

7.2 Heat and Wave Equations 101

7.3 The Telegraph Equation 104

7.4 The Laplace Transform 106

8 First-Order Equations 109

8.1 Linear First-Order Equations 109

8.2 The Significance of Characteristics 111

8.3 The Quasi-Linear Equation 114

Series List 117

Produkt-Details

• Erscheint lt. Verlag: 25.9.2014
• Reihe/Serie: Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts
• Reihe/Serie: Wiley Series in Pure and Applied Mathematics
• Reihe/Serie: Wiley Series in Pure and Applied Mathematics
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Technik
• Schlagworte: Applied Mathematics in Engineering • Applied Mathematics in Science • Applied Mathmatics in Engineering • Cauchy • convergence • Differential Equations • Differentialgleichung • Differentialgleichungen • Equation • forcing • Half • Heat • Intervals • line • Mathematics • Mathematik • Mathematik in den Ingenieurwissenschaften • Mathematik in den Naturwissenschaften • moving • Nonhomogeneous • Partial • Partielle Differentialgleichung • Problem • Proof • Solutions • Term • Theorem • Wave
• ISBN-13: 9781118880586 / 9781118880586