Analytical Solution of Partial Differential Equations

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Analytical Solution of Partial Differential Equations

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Analytical Solution ofPartial Dierential EquationsbyGordon C. Everstine29 April 2012Copyright c 19982012 by Gordon C. Everstine.All rights reserved.This book was typeset with LATEX2 (MiKTeX).PrefaceThese lecture notes are intended to supplement a one-semester graduate-level engineeringcourse at The George Washington University in classical techniques for the solution of linearpartial dierential equations that arise in engineering and physics. Included are Laplacesequation, Poissons equation, the heat equation, and the wave equation in Cartesian, cylin-drical, and spherical geometries. The emphasis is on separation of variables, Sturm-Liouvilletheory, Fourier series, eigenfunction expansions, and Fourier transform techniques. The mainprerequisite is a standard undergraduate calculus sequence including ordinary dierentialequations. However, much of what is needed involving the solution of ordinary dierentialequations is covered when the need arises. An elementary knowledge of complex variables isalso assumed in the discussion of the Fourier integral theorem and the Fourier transforms.In general, the mix of topics and level of presentation are aimed at upper-level undergrad-uates and rst-year graduate students in mechanical, aerospace, and civil engineering. Theapplications are drawn mostly from heat conduction (for parabolic and elliptic problems)and mechanics (for hyperbolic problems).Gordon EverstineGaithersburg, MarylandApril 2012iiiContents1 Review of Notation and Integral Theorems 11.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The Gradient Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 The Divergence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 The Dirac Delta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Derivation and Classication of PDEs 72.1 The Vibrating String . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Longitudinal Vibrations of Bars . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Heat Conduction in Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Elastodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5 Flexural Vibrations of Beams . . . . . . . . . . . . . . . . . . . . . . . . . . 162.6 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.7 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.8 Classication of Partial Dierential Equations . . . . . . . . . . . . . . . . . 212.9 Nature of Solutions of PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 Fourier Series Solutions 283.1 A One-Dimensional Heat Conduction Problem . . . . . . . . . . . . . . . . . 283.2 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.3 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.4 Alternatives for Coecient Evaluation . . . . . . . . . . . . . . . . . . . . . 363.5 Complete Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.6 Harmonic Analysis of Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.7 The Generalized Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . 444 Problems in Cartesian Coordinates 474.1 One-Dimensional Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.2 Two-Dimensional Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.2.1 The Transient Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 504.2.2 The Steady-State Problem with Nonzero Source . . . . . . . . . . . . 534.2.3 The Steady-State Problem with Nonzero Boundary Data . . . . . . . 545 Sturm-Liouville System 555.1 Examples of Sturm-Liouville Problems . . . . . . . . . . . . . . . . . . . . . 575.2 Boundary Value Problem Example . . . . . . . . . . . . . . . . . . . . . . . 596 Orthogonal Curvilinear Coordinates 626.1 Arc Length and Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646.2 Del Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646.3 Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666.4 Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666.5 Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666.6 Example: Cylindrical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 67iv7 Problems in Cylindrical Coordinates 687.1 Bessels Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737.2.1 The Transient Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 747.2.2 The Steady-State Problem with Nonzero Source . . . . . . . . . . . . 777.2.3 The Steady-State Problem with Nonzero Boundary Data . . . . . . . 788 Problems in Spherical Coordinates 808.1 Legendres Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 808.2 Interior Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 858.3 Exterior Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 879 Fourier Integral Transforms 889.1 Fourier Integral Theorem and Fourier Transforms . . . . . . . . . . . . . . . 889.2 Application to Integral Evaluation . . . . . . . . . . . . . . . . . . . . . . . . 919.3 Operational Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 929.4 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 949.5 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 979.6 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 999.7 Example 4: The Innite String and the dAlembert Solution . . . . . . . . . 100Bibliography 101Index 103List of Figures1 Two Vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 The Directional Derivative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Fluid Flow Through Small Parallelepiped. . . . . . . . . . . . . . . . . . . . 34 A Closed Volume Bounded by a Surface S. . . . . . . . . . . . . . . . . . . . 35 Pressure on Dierential Element of Surface. . . . . . . . . . . . . . . . . . . 46 The Dirac Delta Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Mass-Spring System