Applied Partial Differential Equations with Fourier Series and Boundary Value Problems

<div> <h3>1. Heat Equation</h3> <ul> <li>1.1 Introduction</li> <li>1.2 Derivation of the Conduction of Heat in a One-Dimensional Rod</li> <li>1.3 Boundary Conditions</li> <li>1.4 Equilibrium Temperature Distribution</li> <li>1.5 Derivation of the Heat Equation in Two or Three Dimensions</li> </ul> <h3>2. Method of Separation of Variables</h3> <ul> <li>2.1 Introduction</li> <li>2.2 Linearity</li> <li>2.3 Heat Equation with Zero Temperatures at Finite Ends</li> <li>2.4 Worked Examples with the Heat Equation: Other Boundary Value Problems</li> <li>2.5 Laplace's Equation: Solutions and Qualitative Properties</li> </ul> <h3>3. Fourier Series</h3> <ul> <li>3.1 Introduction</li> <li>3.2 Statement of Convergence Theorem</li> <li>3.3 Fourier Cosine and Sine Series</li> <li>3.4 Term-by-Term Differentiation of Fourier Series</li> <li>3.5 Term-By-Term Integration of Fourier Series</li> <li>3.6 Complex Form of Fourier Series</li> </ul> <h3>4. Wave Equation: Vibrating Strings and Membranes</h3> <ul> <li>4.1 Introduction</li> <li>4.2 Derivation of a Vertically Vibrating String</li> <li>4.3 Boundary Conditions</li> <li>4.4 Vibrating String with Fixed Ends</li> <li>4.5 Vibrating Membrane</li> <li>4.6 Reflection and Refraction of Electromagnetic (Light) and Acoustic (Sound) Waves</li> </ul> <h3>5. Sturm-Liouville Eigenvalue Problems</h3> <ul> <li>5.1 Introduction</li> <li>5.2 Examples</li> <li>5.3 Sturm-Liouville Eigenvalue Problems</li> <li>5.4 Worked Example: Heat Flow in a Nonuniform Rod without Sources</li> <li>5.5 Self-Adjoint Operators and Sturm-Liouville Eigenvalue Problems</li> <li>5.6 Rayleigh Quotient</li> <li>5.7 Worked Example: Vibrations of a Nonuniform String</li> <li>5.8 Boundary Conditions of the Third Kind</li> <li>5.9 Large Eigenvalues (Asymptotic Behavior)</li> <li>5.10 Approximation Properties</li> </ul> <h3>6. Finite Difference Numerical Methods for Partial Differential Equations</h3> <ul> <li>6.1 Introduction</li> <li>6.2 Finite Differences and Truncated Taylor Series</li> <li>6.3 Heat Equation</li> <li>6.4 Two-Dimensional Heat Equation</li> <li>6.5 Wave Equation</li> <li>6.6 Laplace's Equation</li> <li>6.7 Finite Element Method</li> </ul> <h3>7. Higher Dimensional Partial Differential Equations</h3> <ul> <li>7.1 Introduction</li> <li>7.2 Separation of the Time Variable</li> <li>7.3 Vibrating Rectangular Membrane</li> <li>7.4 Statements and Illustrations of Theorems for the Eigenvalue Problem ∇²φ + λφ = 0</li> <li>7.5 Green's Formula, Self-Adjoint Operators and Multidimensional Eigenvalue Problems</li> <li>7.6 Rayleigh Quotient and Laplace's Equation</li> <li>7.7 Vibrating Circular Membrane and Bessel Functions</li> <li>7.8 More on Bessel Functions</li> <li>7.9 Laplace's Equation in a Circular Cylinder</li> <li>7.10 Spherical Problems and Legendre Polynomials</li> </ul> <h3>8. Nonhomogeneous Problems</h3> <ul> <li>8.1 Introduction</li> <li>8.2 Heat Flow with Sources and Nonhomogeneous Boundary Conditions</li> <li>8.3 Method of Eigenfunction Expansion with Homogeneous Boundary Conditions (Differentiating Series of Eigenfunctions)</li> <li>8.4 Method of Eigenfunction Expansion Using Green's Formula (With or Without Homogeneous Boundary Conditions)</li> <li>8.5 Forced Vibrating Membranes and Resonance</li> <li>8.6 Poisson's Equation</li> </ul> <h3>9. Green's Functions for Time-Independent Problems</h3> <ul> <li>9.1 Introduction</li> <li>9.2 One-dimensional Heat Equation</li> <li>9.3 Green's Functions for Boundary Value Problems for Ordinary Differential Equations</li> <li>9.4 Fredholm Alternative and Generalized Green's Functions</li> <li>9.5 Green's Functions for Poisson's Equation</li> <li>9.6 Perturbed Eigenvalue Problems</li> <li>9.7 Summary</li> </ul> <h3>10. Infinite Domain Problems: Fourier Transform Solutions of Partial Differential Equations</h3> <ul> <li>10.1 Introduction</li> <li>10.2 Heat Equation on an Infinite Domain</li> <li>10.3 Fourier Transform Pair</li> <li>10.4 Fourier Transform and the Heat Equation</li> <li>10.5 Fourier Sine and Cosine Transforms: The Heat Equation on Semi-Infinite Intervals</li> <li>10.6 Worked Examples Using Transforms</li> <li>10.7 Scattering and Inverse Scattering</li> </ul> <h3>11. Green's Functions for Wave and Heat Equations</h3> <ul> <li>11.1 Introduction</li> <li>11.2 Green's Functions for the Wave Equation</li> <li>11.3 Green's Functions for the Heat Equation</li> </ul> <h3>12. The Method of Characteristics for Linear and Quasilinear Wave Equations</h3> <ul> <li>12.1 Introduction</li> <li>12.2 Characteristics for First-Order Wave Equations</li> <li>12.3 Method of Characteristics for the One-Dimensional Wave Equation</li> <li>12.4 Semi-Infinite Strings and Reflections</li> <li>12.5 Method of Characteristics for a Vibrating String of Fixed Length</li> <li>12.6 The Method of Characteristics for Quasilinear Partial Differential Equations</li> <li>12.7 First-Order Nonlinear Partial Differential Equations</li> </ul> <h3>13. Laplace Transform Solution of Partial Differential Equations</h3> <ul> <li>13.1 Introduction</li> <li>13.2 Properties of the Laplace Transform</li> <li>13.3 Green's Functions for Initial Value Problems for Ordinary Differential Equations</li> <li>13.4 A Signal Problem for the Wave Equation</li> <li>13.5 A Signal Problem for a Vibrating String of Finite Length</li> <li>13.6 The Wave Equation and its Green's Function</li> <li>13.7 Inversion of Laplace Transforms Using Contour Integrals in the Complex Plane</li> <li>13.8 Solving the Wave Equation Using Laplace Transforms (with Complex Variables)</li> </ul> <h3>14. Dispersive Waves: Slow Variations, Stability, Nonlinearity, and Perturbation Methods</h3> <ul> <li>14.1 Introduction</li> <li>14.2 Dispersive Waves and Group Velocity</li> <li>14.3 Wave Guides</li> <li>14.4 Fiber Optics</li> <li>14.5 Group Velocity II and the Method of Stationary Phase</li> <li>14.7 Wave Envelope Equations (Concentrated Wave Number)</li> <li>14.7.1 Schrödinger Equation</li> <li>14.8 Stability and Instability</li> <li>14.9 Singular Perturbation Methods: Multiple Scales</li> </ul> </div>