DIFFERENTIAL EQUATIONS

Description

Cover Half Title Series Page Title Page Copyright Page Contents 1. Introduction to the Maple DEtools 1.1. Analytical Solutions and Their Plotting 1.2. Direction Fields and Integral Curves 1.3. Computer Lab 1.4. Supplementary Maple Programs 1.4.1. Implicit and Explicit Solutions 1.4.2. Numerical Solutions 2. First-Order Differential Equations 2.1. Linear Differential Equations 2.2. Project: Mixing Problems 2.2.1. Project1: One Tank 2.2.2. Project2: Two Tanks 2.3. Separable Differential Equations 2.4. Exact Equations 3. Numerical Methods for First-Order Equations 3.1. Picard’s Iteration Method and Semi-Batch Reactor 3.2. An Existence and Uniqueness Theorem 3.3 Picard Iteration Method 3.4. Computer Lab 3.5. Numerical Procedures and Fermentation Kinetics 3.6. The Euler Method 3.7. Higher-Order Methods 3.8. Maple Procedures 3.9. Computer Lab 3.10. Supplementary Maple Programs 3.10.1. The Order of Convergence 4. Differential Equations with Constant Coefficients 4.1. Second-Order Equations with Constant Coefficients 4.2. Variation of Parameters 4.2.1. The Wronskian 4.3. The Method of Undetermined Coefficients 4.4. Higher-Order, Homogeneous Equations 4.4.1. Polynomial Solutions 4.5. Nonhomogeneous Linear Equations 4.5.1. Undetermined Coefficients 4.5.2. Variation of Parameters 4.5.3. Further Remarks on the Variation of Parameters Method 5. Applications of Second-Order Linear Equations 5.1. Simple Harmonic Motion 5.2. General Solutions 5.3. Method of Undetermined Coefficients 5.4. Additional Useful Commands 5.5. Computer Lab 5.6. Supplementary Maple Programs 5.6.1. The Phenomenon of Beats 5.6.2. The Phenomenon of Resonance 5.7. Particular Solutions 5.8. Computer Lab 5.9. Supplementary Maple Programs 5.9.1. Resonance Curves 5.9.2. An Example 6. Two-Point Boundary Value Problems, Catalytic Reactors and Boundary-Layer Phenomena 6.1. Analytical Solutions 6.2. Finite-Difference Methods 6.2.1. Finite-Difference Procedure for the Two-Point BVP: 6.3. Computer Lab 6.4. Supplementary Maple Programs 6.4.1. An Exact Asymptotic Expansion 7. Eigenvalue Problems 7.1. Sturm-Liouville Problems 7.2. Numerical Approximations 7.3. The Newton-Raphson Method 7.4. Computer Lab 7.5. Supplementary Mapple Programs 7.5.1. An Eigenvalue Equation 8. Power Series Methods for Solving Differential Equations 8.1. Nonlinear Differential Equations 8.2. Regular-Singular Points 8.3. Programs for Finding Solutions 8.4. Projects 9. Nonlinear Autonomous Systems 9.1. The Taylor Series Method 9.2. The Phase Plane 9.3. Linear Systems 9.4. Useful Maple Commands 9.5. Computer Lab 9.6. Supplementary Maple Programs 9.6.1. Taylor Series Expansion 9.6.2. The Damped Pendulum 10. Integral Transforms 10.1. The Laplace Transform of Elementary Functions 10.2. Solving Differential Equations with the Laplace Transform 10.3. Fourier Transforms 11. Partial Differential Equations 11.1. Elementary Methods 11.2. The First-Order Partial Differential Equation 11.3. The Heat Equation 11.4. The Vibrating String 11.4.1. Separation of Variables with Mapple 11.5. The Laplace Equation 12. Transmutations 12.1. The Method of Ascent 12.2. Orthogonal Systems of Functions 12.3. Acoustic Propagation Bibliography Index