Description

Features

• NEW - New examples of Fourier series and exercises—Based on figures and designed to challenge students' ability to read and understand graphs of Fourier series.

• NEW - Complete proof of the Fourier Series Representation Theorem—A new section is added that contains a complete and clear proof of the Fourier series convergence theorem. Also added is a general result on the uniform convergence of Fourier series.

• NEW - New section on the application of Fourier series to the solution of forced vibrations of electrical and mechanical systems—The presentation goes beyond what is typically done in this area. It includes a method for suppressing the large oscillations of the system by analyzing the Fourier series solution.

• NEW - Expanded coverage in Chapter 3 (Partial Differential Equations in Rectangular Coordinates):

  • Includes the topics of characteristic lines and parallelograms, and intervals of dependence in d'Alembert's solution.

  • Treats boundary value problems on rectangular regions with Robin and Neumann conditions.

• NEW - New and expanded coverage of Partial Differential Equations in Polar and Cylindrical Coordinates (Chapter 4):

  • Covers boundary value problems on discs, wedges, and sectors in planes, with Robin and Neumann conditions.

  • Develops important advanced properties of Bessel functions, such as integral representations and asymptotic formulas; also introduces the method of stationary phase that is of interest to engineers, physicists, and applied mathematicians.

• NEW - New coverage of Sturm-Liouville Theory with Engineering Applications (Chapter 6):

  • Treats the biharmonic equation.

  • Discusses the vibrations and forced vibrations of plates.

• NEW - New material on the Fourier Transform and Its Applications (Chapter 7):

  • Treats generalized functions, derivatives of piecewise smooth functions, convolutions, and applications to the computation of Fourier transforms of piecewise smooth functions.

  • Solves the nonhomogeneous heat and wave equations and introduces the topics of fundamental solutions and weak solutions of partial differential equations. Detailed study of the solution of boundary value problems involving generalized functions, such as the Dirac delta function.

  • Introduces Duhamel's principle and solves nonhomogeneous equations, including the heat and wave equations.

• NEW - New chapter that deals with Green's functions and Conformal Mappings (Chapter 12).

  • The first section of this chapter (Section 12.1) starts with the basic properties of line integrals, proves Green's theorem from calculus, derives Green's formulas, and then proves fundamental results about harmonic functions and the uniqueness of solutions of Dirichlet problems.

  • Sections 12.2 and 12.3 prove Gauss's mean value property of harmonic functions and the maximum modulus principle, derive Green's functions and discuss their theoretical and physical significance.

  • Section 12.3 also treats the eigenfunction method for finding Green's functions.

  • In Section 12.4, the method of images is used to derive Green's functions.

  • Section 12.5 is a resourceful introduction to theory of analytic functions and their applications to partial differential equations. This section includes analytic functions, Cauchy-Riemann equations, harmonic conjugates and their physical interpretation, and many applications of analytic functions to the solution of Dirichlet problems.

  • Section 12.6 expands on the previous section and presents in detail the method of conformal mappings and its applications.

  • Sections 12.7 and 12.8 use conformal mappings to derive Green's functions. Neumann functions are also introduced and derived using conformal mappings.

  • The treatment of Green's functions in this book is self-contained and is written in the same easy-to-follow style as the rest of the book. The chapter is packed with interesting exercises and applications, and numerous illustrations.

• NEW - New Section on Review of Power Series (Appendix A). This section is added to the review of ordinary differential equations in Appendix A, in order to offer a self-contained treatment of the power series and Frobenius methods. It is ideal for users who have not covered this material in a previous ordinary differential equations course. It is also a resourceful reference for those who wish to have a quick review of these fundamental tools from ode's.

• Jumps right into PDEs—Presents review material of ODEs in Appendix A.

• Applied approach with proofs clearly marked or in appendices, which offers the option to present or omit proofs at the discretion of the instructor.

• Large number of exercises per section:

  • Includes detailed hints for the more advanced exercises.

  • Begins each set with a series of straightforward problems that reinforce basic concepts in that section; later exercises are more involved and lead to a deeper understanding of the concepts.

• The most computer-friendly PDE text on the market—Asks students to investigate problems using computer-generated graphics and to generate numerical data that cannot be computed by hand.

• Marginal comments and remarks throughout the text—Offers insightful remarks, keys to following the material, and formulas recalled for the students' convenience.

• Student and Instructor Solution Manuals—Available for download as pdf files from the author's website at www.math.missouri.edu/~nakhle

• Mathematica files—Available for download from the author's website at www.math.missouri.edu/~nakhle

• Includes material not covered in other texts—e.g., a chapter on quantum mechanics, and detailed coverage of conformal mappings.

• User can teach an optional theory oriented course on partial differential equations—This new edition incorporates many theoretical topics such as characteristic lines and characteristic triangles for the wave equation; extensive coverage of generalized functions; integral representations of Bessel functions, method of stationary phase; the biharmonic operator and its eigenfunctions; plate theory; spherical harmonics; weak and fundamental solutions of partial differential equations; Green's functions; conformal mappings and Green's functions; Neumann functions; maximum-minimum modulus principle; and a detailed theory of harmonic functions.

Table of Contents



 1. A Preview of Applications and Techniques.


 2. Fourier Series.


 3. Partial Differential Equations in Rectangular Coordinates.


 4. Partial Differential Equations in Polar and Cylindrical Coordinates.


 5. Partial Differential Equations in Spherical Coordinates.


 6. Sturm-Liouville Theory with Engineering Applications.


 7. The Fourier Transform and Its Applications.


 8. The Laplace and Hankel Transforms with Applications.


 9. Finite Difference Numerical Methods.


10. Sampling and Discrete Fourier Analysis with Applications to Partial Differential Equations.


11. An Introduction to Quantum Mechanics.


12. Green's Functions and Conformal Mappings.


Appendix A: Ordinary Differential Equations: Review of Concepts and Methods.


Appendix B: Tables of Transforms.


References.


Answers to Selected Exercises.


Index.

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Partial Differential Equations  (Mathematics)

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