Mathematics

For a one-year, graduate-level course in Partial Differential Equations.

Designed to bridge the gap between introductory texts in partial differential equations and the current literature in research journals, this text introduces students to the basics of classical PDEs and to a wide variety of more modern methods—especially the use of functional analysis—which has characterized much of the recent development of PDEs. Throughout, the results are almost completely self-contained.

• NEW - New applications—e.g., How the wave equation applies to light (via Maxwell's equations) and sound (via Euler's equations); how Laplace's equation gives results on the decomposition of vector fields; and how the heat equation may be used to study slow viscous incompressible fluids and Brownian motion.

  • Gives the reader modern practical applications of PDEs.

• NEW - A new section on unbounded operators and spectral theory—Ch. 6.

  • Provides students with essential background for results in later chapters.

• NEW - Appendix on Physics —Derived from basic physical principles, most of the physically important equations in the book.

  • Shows students the modeling process used to obtain the specific equations.

• NEW - Approximately 15-20% new/revised exercises.

  • Gives instructors fresh material for assignments and gives students abundant opportunity to work through and test their understanding of material and engages them more actively in the learning experience.

• From basic material through modern methods—Chs. 1-5 present the classical material; Chs. 6-13 cover the modern methods, with tools from functional analysis.

  • Provides students and instructors with a single source for the foundational material necessary to understand the current PDE literature.

• Equal treatment of elliptic, hyperbolic, and parabolic theory—Chs. 3-5 cover basic aspects of linear hyperbolic, elliptic, and parabolic theory; Chs.  10-13 cover more general and nonlinear aspects of parabolic, hyperbolic, and elliptic theory.

  • Provides students and instructors with broad coverage of the topic. Most other texts offer only a single-theory focus.

• Applications to equations that are important in physics and engineering—Both on the basic and more advanced level.

  • Makes the theory understandable and relevant.

• Many worked examples—Some of which are “revisited” more than once.

• Many figures and illustrations.

  • Helps visually oriented learners better grasp concepts.

• A full chapter on “Hints & Solutions for Selected Exercises”—Features detailed hints, and sometimes full solutions, for more than half of the exercises that are given.

  • Allows instructors to assign either problems for which the solution or a hint is given, or problems for which none is given. However, students may find the solutions or hints for similar problems of some use.

• Extensive references to other literature—Both at the end of the text, and at the end of each chapter.

  • Provides both students and instructors with additional resources to obtain further information.

• New applications—e.g., How the wave equation applies to light (via Maxwell's equations) and sound (via Euler's equations); how Laplace's equation gives results on the decomposition of vector fields; and how the heat equation may be used to study slow viscous incompressible fluids and Brownian motion.

  • Gives the reader modern practical applications of PDEs.

• A new section on unbounded operators and spectral theory—Ch. 6.

  • Provides students with essential background for results in later chapters.

• Appendix on Physics —Derived from basic physical principles, most of the physically important equations in the book.

  • Shows students the modeling process used to obtain the specific equations.

• Approximately 15-20% new/revised exercises.

  • Gives instructors fresh material for assignments and gives students abundant opportunity to work through and test their understanding of material and engages them more actively in the learning experience.

Introduction.


 1. First-Order Equations.


 2. Principles for Higher-Order Equations.


 3. The Wave Equation.


 4. The Laplace Equation.


 5. The Heat Equation.


 6. Linear Functional Analysis.


 7. Differential Calculus Methods.


 8. Linear Elliptic Theory.


 9. Two Additional Methods.


10. Systems of Conservation Laws.


11. Linear and Nonlinear Diffusion.


12. Linear and Nonlinear Waves.


13. Nonlinear Elliptic Equations.


Appendix on Physics.


Hints and Solutions for Selected Exercises.


References.


Index.


Index of Symbols.

Partial Differential Equations  (Mathematics)

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