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Beginning Partial Differential Equations

Peter V. O'Neil
Publication Date: 
Number of Pages: 
Pure and Applied Mathematics: A Wiley Series of Texts, Monographs, and Tracts
[Reviewed by
Jeff Ibbotson
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What is the best progression of ideas for a student learning PDEs for the first time? This book takes the approach of focusing on two major equations first: the Heat Equation and Wave Equation are the second order examples of choice. The author uses them to good effect to highlight many of the important features of partial differential equations. Fourier series are introduced early and used to handle boundary value problems on finite domains. Separation of variables is the major tool.

The first part of the book attacks every possible permutation of boundary conditions for finite domains. The emphasis is concrete throughout. A few theorems are quoted but never proven. In fact, the orthogonality conditions \[ \int_{-L}^{L} \cos\left(\frac{n\pi x}{L}\right) \cos\left(\frac{k\pi x}{L}\right)\,dx = L\delta_{n,k} \] are not proven nor even posed as an exercise! Is the assumption that students will recall them from their second semester Calculus classes?

The author does indicate clearly that Fourier series do not always converge but the only cases considered involve piecewise differentiable functions and their Fourier series representations. The author does take time to introduce the Dirichlet Kernel and he does prove that the Fourier series of piecewise continuous functions converge to the average of left and right limits of such a function, but that’s the extent of the classical analysis that appears in the book. There is no mention of uniform convergence and little overlap with the ideas of the theory of real variables. And perhaps that’s OK — the author’s stated goal is to produce series and integral representations for solutions of PDEs. That’s the sole medium for tackling all other issues in this approach.

Strewn throughout the early chapters are a number of nice discussions of things like harmonic functions and the mean value property, Hadamard’s method of descent for the wave equation, characteristic quadrilaterals, the Telegraph equation, Bessel functions and a chapter on Fourier transforms.

There are also two applications to science and literature. The first reviews the well-known dispute over the age of the Earth in Victorian times. Lord Kelvin became one of the first scientists to estimate the age of the earth by drawing conclusions from the solution form for the three-dimensional heat equation. The story has become well-known by now. The second application is much less known and involves Edgar Allen Poe’s story “The Pit and the Pendulum”. The hapless victim sees the sharpened blade of a pendulum come closer as the rod is artificially lengthened. Does this lengthening make the blade swing faster? This is what the poor innocent in Poe’s story reports but careful analysis of some Bessel functions suggest otherwise.

I enjoyed perusing O’Neil’s book. A beginner in the field of PDEs will learn quite a number of juicy facts concerning the flow of heat and the transmission of waves. While a next step will undoubtedly involve more rigor in the use of analytic tools, this first course will catch the attention of those with a curiosity for studying physical processes using differential equations.

Jeff Ibbotson is the Smith Teaching Chair at Phillips Exeter Academy. He spends much of his time reading, playing ping pong and raising beagles.

1 First Ideas 1

1.1 Two Partial Differential Equations 1

1.2 Fourier Series 10

1.3 Two Eigenvalue Problems 28

1.4 A Proof of the Fourier Convergence Theorem 30

2. Solutions of the Heat Equation 39

2.1 Solutions on an Interval (0, L) 39

2.2 A Nonhomogeneous Problem 64

2.3 The Heat Equation in Two space Variables 71

2.4 The Weak Maximum Principle 75

3. Solutions of the Wave Equation 81

3.1 Solutions on Bounded Intervals 81

3.2 The Cauchy Problem 109

3.3 The Wave Equation in Higher Dimensions 137

4. Dirichlet and Neumann Problems 147

4.1 Laplace’s Equation and Harmonic Functions 147

4.2 The Dirichlet Problem for a Rectangle 153

4.3 The Dirichlet Problem for a Disk 158

4.4 Properties of Harmonic Functions 165

4.5 The Neumann Problem 187

4.6 Poisson’s Equation 197

4.7 Existence Theorem for a Dirichlet Problem 200

5. Fourier Integral Methods of Solution 213

5.1 The Fourier Integral of a Function 213

5.2 The Heat Equation on a Real Line 220

5.3 The Debate over the Age of the Earth 230

5.4 Burger’s Equation 233

5.5 The Cauchy Problem for a Wave Equation 239

5.6 Laplace’s Equation on Unbounded Domains 244

6. Solutions Using Eigenfunction Expansions 253

6.1 A Theory of Eigenfunction Expansions 253

6.2 Bessel Functions 266

6.3 Applications of Bessel Functions 279

6.4 Legendre Polynomials and Applications 288

7. Integral Transform Methods of Solution 307

7.1 The Fourier Transform 307

7.2 Heat and Wave Equations 318

7.3 The Telegraph Equation 332

7.4 The Laplace Transform 334

8 First-Order Equations 341

8.1 Linear First-Order Equations 342

8.2 The Significance of Characteristics 349

8.3 The Quasi-Linear Equation 354

9 End Materials 361

9.1 Notation 361

9.2 Use of MAPLE 363

9.3 Answers to Selected Problems 370

Index 434