This extremely readable book illustrates how mathematics applies directly to different fields of study. Focuses on problems that require physical to mathematical translations, by showing readers how equations have actual meaning in the real world. Covers fourier integrals, and transform methods, classical PDE problems, the Sturm-Liouville Eigenvalue problem, and much more. For readers interested in partial differential equations.

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This text is designed for a one-semester course in partial differential equations for the undergraduate student of engineering, physics, applied mathematics, social science, biology, and other sciences, for example, economics. The text covers the method of separation of variables, Fourier series, classical problems of physics and engineering, Sturm-Liouville eigenvalue problems, power series solutions of variable coefficient ordinary differential equations, and transform methods. Wherever possible, mathematical topics are motivated by physical laws or problems. As such, mathematical modeling of physical data and applications are stressed. Throughout the text, completely worked examples/counterexamples are used to develop mathematical concepts. This reduces the potential for the student to see mathematics as a set of magical steps, and it allows the student time to develop his/her own methodology for solving problems based on comprehension of the mathematical process. When a purely mathematical topic is developed, such as Fourier series, the approach taken is constructive in methodology by building on material the student has encountered in other courses. This provides the student a framework of connections allowing easy comprehension of the material, and it assists the instructor in developing the student's insight into higher mathematics.

Mathematical texts can be very intimidating to many students. Therefore, this text is designed to be truly readable and "student friendly." Whenever the text or parts of the text have been used in class, student end-of-course critiques indicated that the readability and usability of the text is in the 99*th* percentile.

The text is motivated by applications, which help the student in his/her studies in other areas of engineering and science. Many topics are introduced by using a physical model as opposed to a purely theoretical approach. For example, the section on the method of characteristics for first-order partial differential equations with constant coefficients is introduced by the physical example of a surfer catching a wave. Another example is the uniqueness of solution for the one-dimensional wave equation, which is developed by first considering conservation of energy for a vibrating string, a concept that most students should understand from either their first physics or calculus courses. Theoretical topics, such as Fourier series, are introduced by first discussing real vector spaces and the fact that different basis can be developed for *n*-dimensional space by considering an *n* x *n* matrix with *n* distinct eigenvalues and their corresponding eigenvectors.

The prerequisites for a student in a course using this text are the calculus sequence and elementary ordinary differential equations. An introduction to linear algebra would be helpful, but not necessary.

I have included a review of ordinary differential equations in the appendices. I have found this extremely valuable for many students. Also, for a more theoretical approach an appendix with proofs of selected theorems is provided.

**Course Outline**

A possible outline for a one semester course is the following:

Chapters 1 through 8, which is the core material. This provides for the development of the three classes of linear second-order partial differential equations, elliptic, parabolic and hyperbolic and the three types of boundary conditions, Dirichlet, Neumann, and Robin. Additionally, Chapters 1 through 8 gives a thorough discussion of the separation of variables technique, coverage of the relevant theorems of Fourier series and an introduction to the Sturm-Liouville boundary value problem. Once Chapters 1 through 8 are covered, there are several options. For a complete development of classical solution methods of second-order linear partial differential equations, I would suggest including Chapter 11, which develops the Fourier and Laplace transforms. For a wider set of applications, I would suggest including Chapters 9 and 10. It is also possible to chose selected topics from Chapters 9, 10, and 11 for a broad discussion of applications and technique.

Although the text is not directly tied to a mathematical software package, such as Mathematica, many of the exercises require the student to find partial sums of Fourier series. Also, students are required in the exercises to graph both the Fourier series representation of a function and a three-dimensional view of the solution of a partial differential equation for various partial sums. Thus, students should be familiar with some type of mathematical software package.

You may contact the author directly for Mathematica files and other complementary material related to the text by email at

**keanemj@gateway.net**

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