Beginning Partial Differential Equations

PETER V. O’NEIL, PHD, is Professor Emeritus in the Department of Mathematics at the University of Alabama at Birmingham. He has over forty years of experience in teaching and writing and is the recipient of the Lester R. Ford Award from the Mathematical Association of America. Dr. O’Neil is also a member of the American Mathematical Society, the Mathematical Association of America, the Society for Industrial and Applied Mathematics, and the American Association for the Advancement of Science.

• 1 First Ideas 1 1.1 Two Partial Differential Equations 11.2 Fourier Series 101.3 Two Eigenvalue Problems 281.4 A Proof of the Fourier Convergence Theorem 302. Solutions of the Heat Equation 392.1 Solutions on an Interval (0, L) 392.2 A Nonhomogeneous Problem 642.3 The Heat Equation in Two space Variables 712.4 The Weak Maximum Principle 753. Solutions of the Wave Equation 813.1 Solutions on Bounded Intervals 813.2 The Cauchy Problem 1093.3 The Wave Equation in Higher Dimensions 1374. Dirichlet and Neumann Problems 1474.1 Laplace’s Equation and Harmonic Functions 1474.2 The Dirichlet Problem for a Rectangle 1534.3 The Dirichlet Problem for a Disk 1584.4 Properties of Harmonic Functions 1654.5 The Neumann Problem 1874.6 Poisson’s Equation 1974.7 Existence Theorem for a Dirichlet Problem 2005. Fourier Integral Methods of Solution 2135.1 The Fourier Integral of a Function 2135.2 The Heat Equation on a Real Line 2205.3 The Debate over the Age of the Earth 2305.4 Burger’s Equation 2335.5 The Cauchy Problem for a Wave Equation 2395.6 Laplace’s Equation on Unbounded Domains 2446. Solutions Using Eigenfunction Expansions 2536.1 A Theory of Eigenfunction Expansions 2536.2 Bessel Functions 2666.3 Applications of Bessel Functions 2796.4 Legendre Polynomials and Applications 2887. Integral Transform Methods of Solution 3077.1 The Fourier Transform 3077.2 Heat and Wave Equations 3187.3 The Telegraph Equation 3327.4 The Laplace Transform 3348 First-Order Equations 3418.1 Linear First-Order Equations 3428.2 The Significance of Characteristics 3498.3 The Quasi-Linear Equation 3549 End Materials 3619.1 Notation 3619.2 Use of MAPLE 3639.3 Answers to Selected Problems 370Index 434

“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.”  (Mathematical Association of America, 15 February 2015)“This book is one of the textbooks that provide an introduction to basic methods and applications of partial differential equations for students of mathematics, physics and engineering.”  (Zentralblatt MATH, 1 October 2014)