Basic Partial Differential Equations

Methods of solution for partial differential equations (PDEs) used in mathematics, science, and engineering are clarified in this self-contained source. The reader will learn how to use PDEs to predict system behaviour from an initial state of the system and from external influences, and enhance the success of endeavours involving reasonably smooth, predictable changes of measurable quantities. This text enables the reader to not only find solutions of many PDEs, but also to interpret and use these solutions. It offers 6000 exercises ranging from routine to challenging. The palatable, motivated proofs enhance understanding and retention of the material. Topics not usually found in books at this level include but examined in this text:- the application of linear and nonlinear first-order PDEs to the evolution of population densities and to traffic shocks - convergence of numerical solutions of PDEs and implementation on a computer - convergence of Laplace series on spheres - quantum mechanics of the hydrogen atom - solving PDEs on manifoldsThe text requires some knowledge of calculus but none on differential equations or linear algebra. MuPDF error: syntax error: invalid key in dict MuPDF error: syntax error: invalid key in dict MuPDF error: syntax error: invalid key in dict MuPDF error: syntax error: invalid key in dict MuPDF error: syntax error: invalid key in dict MuPDF error: syntax error: invalid key in dict MuPDF error: syntax error: invalid key in dict MuPDF error: syntax error: invalid key in dict MuPDF error: syntax error: invalid key in dict Cover 2 Half Title 2 Title 4 Copyright 5 Table of Contents 6 Preface 10 1. Review and Introduction 20 1.1 A Review of Ordinary Differential Equations 21 1.2 Generalities About PDEs 42 1.3 General Solutions and Elementary Techniques 63 2. First–Order PDEs 76 2.1 First–Order Linear PDEs (Constant Coefficients) 77 2.2 Variable Coefficients 93 2.3 Higher Dimensions, Quasi—linearity, Applications 111 2.4 Supplement on Nonlinear First—Order PDEs (Optional) 130 3. The Heat Equation 140 3.1 Derivation of the Heat Equation and Solutions of the Standard Initial/Boundary–Value problems 141 3.2 Uniqueness and the Maximum Principle 159 3.3 Time–Independent Boundary Conditions 176 3.4 Time–Dependent Boundary Conditions 191 4. Fourier Series and Sturm–Liouville Theory 206 4.1 Orthogonality and the Definition of Fourier Series 207 4.2 Convergence Theorems for Fourier Series 226 4.3 Sine and Cosine Series and Applications 256 4.4 Sturm–Liouville Problems 277 5. The Wave Equation 300 5.1 The Wave Equation – Derivation and Uniqueness 301 5.2 The D'Alembert Solution of the wave equation 318 5.3 Inhomogeneous Boundary Conditions and Wave Equations 339 6. Laplace's Equation 358 6.1 General Orientation 360 6.2 The Dirichlet Problem for the rectangle 370 6.3 The Dirichlet Problem for Annuli and Disks 385 6.4. The Maximum Principle and Uniqueness for the Dirichlet Problem 404 6.5 Complex Variable Theory with Applications 417 7. Fourier Transforms 434 7.1 Complex Fourier Series 438 7.2 Basic Properties of Fourier Transforms 450 7.3 The Inversion Theorem and Parseval's Equality 466 7.4 Fourier Transform Methods for PDEs 477 7.5 Applications to Problems on Finite and Semi–Infinite Intervals 501 8. Numerical Solutions. An Introduction 522 8.1 The O Symbol and Approximation of Derivatives 523 8.2 The Explicit Difference Method and the Heat Equation 534 8.3 Difference Equations and Round–off Errors 552 8.4 An Overview of Some Other Numerical Methods 567 9. PDEs in Higher Dimensions 578 9.1 Higher–Dimensional PDEs – Rectangular Coordinates 580 9.2 The Eigenfunction Viewpoint 596 9.3 PDEs in Spherical Coordinates 610 9.4 Spherical Harmonics, Laplace Series and Applications 627 9.5 Special Functions and Applications 655 9.6 Solving PDEs on Manifolds 673 Appendix 696 A.1 The Classification Theorem 696 A.2 Fubini's Theorem 700 A.3 Leibniz's Rule 702 A.4 The Maximum/Minimum Theorem 710 A.5 Table of Fourier Transforms 712 A.6 Bessel Functions 713 References 716 Selected Answers 730 Index of Notation 750 Index 756 "Methods of solution for partial differential equations (PDEs) used in mathematics, science, and engineering are clarified in this self-contained source. The reader will learn how to use PDEs to predict system behaviour from an initial state of the system and from external influences, and enhance the success of endeavours involving reasonably smooth, predictable changes of measurable quantities. This text enables the reader to not only find solutions of many PDEs, but also to interpret and use these solutions. It offers 6000 exercises ranging from routine to challenging. The palatable, motivated proofs enhance understanding and retention of the material. Topics not usually found in books at this level include but examined in this text:the application of linear and nonlinear first-order PDEs to the evolution of population densities and to traffic shocksconvergence of numerical solutions of PDEs and implementation on a computerconvergence of Laplace series on spheresquantum mechanics of the hydrogen atomsolving PDEs on manifoldsThe text requires some knowledge of calculus but none on differential equations or linear algebra."--Provided by publisher Methods of solution for partial differential equations (PDEs) used in mathematics, science, and engineering are clarified in this self-contained source. The reader will learn how to use PDEs to predict system behaviour from an initial state of the system and from external influences, and enhance the success of endeavours involving reasonably smooth, predictable changes of measurable quantities. This text enables the reader to not only find solutions of many PDEs, but also to interpret and use these solutions. It offers 6000 exercises ranging from routine to challenging. The palatable, motivated proofs enhance understanding and retention of the material. Topics not usually found in books at this level include but examined in this text: the application of linear and nonlinear first-order PDEs to the evolution of population densities and to traffic shocks convergence of numerical solutions of PDEs and implementation on a computer convergence of Laplace series on spheres quantum mechanics of the hydrogen atom solving PDEs on manifolds The text requires some knowledge of calculus but none on differential equations or linear algebra. "Basic Biophysics for Biology presents the fundamental physical and chemical principles required to understand much of modern biology. The author has made extensive use of illustrations rather than a mathematical approach to establish connections between macroscopic-world models and submicroscopic phenomena. Topics covered include the nucleus, atomic and molecular structure, the principles of thermodynamics, free energy, catalysis, diffusion, and heat flow. Students and professionals in general biology, physiology, genetics, and radiation biology will appreciate this carefully prepared, non-mathematical volume."--Provided by publisher