Solutions Manual to Accompany Beginning Partial Differential Equations

Peter V. O'Neil, PhD, is Professor Emeritus in the Department of Mathematics at The University of Alabama at Birmingham. Dr. O'Neil has over forty years of academic experience and is the recipient of the Lester R. Ford Award from the Mathematical Association of America. He is a member of the American Mathematical Society, the Society for Industrial and Applied Mathematics, and the American Association for the Advancement of Science.

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• Preface vii 1 First Ideas 11.1 Two Partial Differential Equations 11.2 Fourier Series 41.3 Two Eigenvalue Problems 121.4 A Proof of the Convergence Theorem 142 Solutions of the Heat Equation 152.1 Solutions on an Interval [0, L] 152.2 A Nonhomogeneous Problem 193 Solutions of the Wave Equation 253.1 Solutions on Bounded Intervals 253.2 The Cauchy Problem 323.2.1 d’Alembert’s Solution 323.2.2 The Cauchy Problem on a Half Line 363.2.3 Characteristic Triangles and Quadrilaterals 413.2.4 A Cauchy Problem with a Forcing Term 413.2.5 String with Moving Ends 423.3 The Wave Equation in Higher Dimensions 463.3.1 Vibrations in a Membrane with Fixed Frame 463.3.2 The Poisson Integral Solution 473.3.3 Hadamard’s Method of Descent 474 Dirichlet and Neumann Problems 494.1 Laplace’s Equation and Harmonic Functions 494.2 The Dirichlet Problem for a Rectangle 504.3 The Dirichlet Problem for a Disk 524.4 Properties of Harmonic Functions 574.4.1 Topology of Rn 574.4.2 Representation Theorems 584.4.3 The Mean Value Theorem and the Maximum Principle 604.5 The Neumann Problem 614.5.1 Uniqueness and Existence 614.5.2 Neumann Problem for a Rectangle 624.5.3 Neumann Problem for a Disk 634.6 Poisson’s Equation 644.7 An Existence Theorem for the Dirichlet Problem 655 Fourier Integral Methods of Solution 675.1 The Fourier Integral of a Function 675.2 The Heat Equation on the Real Line 705.3 The Debate Over the Age of the Earth 735.4 Burgers’ Equation 735.5 The Cauchy Problem for the Wave Equation 745.6 Laplace’s Equation on Unbounded Domains 766 Solutions Using Eigenfunction Expansions 796.1 A Theory of Eigenfunction Expansions 796.2 Bessel Functions 836.3 Applications of Bessel Functions 876.3.1 Temperature Distribution in a Solid Cylinder 876.3.2 Vibrations of a Circular Drum 876.4 Legendre Polynomials and Applications 907 Integral Transform Methods of Solution 977.1 The Fourier Transform 977.2 Heat and Wave Equations 1017.3 The Telegraph Equation 1047.4 The Laplace Transform 1068 First-Order Equations 1098.1 Linear First-Order Equations 1098.2 The Significance of Characteristics 1118.3 The Quasi-Linear Equation 114Series List 117