Table of Contents Normal 0 false false false Heat Equation 1.1 Introduction 1.2 Derivation of the Conduction of Heat in a One-Dimensional Rod 1.3 Boundary Conditions 1.4 Equilibrium Temperature Distribution 1.4.1 Prescribed Temperature 1.4.

2 Insulated Boundaries 1.5 Derivation of the Heat Equation in Two or Three Dimensions Method of Separation of Variables 2.1 Introduction 2.2 Linearity 2.3 Heat Equation with Zero Temperatures at Finite Ends 2.3.1 Introduction 2.3.

2 Separation of Variables 2.3.3 Time-Dependent Equation 2.3.4 Boundary Value Problem 2.3.5 Product Solutions and the Principle of Superposition 2.3.

6 Orthogonality of Sines 2.3.7 Formulation, Solution, and Interpretation of an Example 2.3.8 Summary 2.4 Worked Examples with the Heat Equation: Other Boundary Value Problems 2.4.1 Heat Conduction in a Rod with Insulated Ends 2.

4.2 Heat Conduction in a Thin Circular Ring 2.4.3 Summary of Boundary Value Problems 2.5 Laplace''s Equation: Solutions and Qualitative Properties 2.5.1 Laplace''s Equation Inside a Rectangle 2.5.

2 Laplace''s Equation for a Circular Disk 2.5.3 Fluid Flow Past a Circular Cylinder (Lift) 2.5.4 Qualitative Properties of Laplace''s Equation Fourier Series 3.1 Introduction 3.2 Statement of Convergence Theorem 3.3 Fourier Cosine and Sine Series 3.

3.1 Fourier Sine Series 3.3.2 Fourier Cosine Series 3.3.3 Representing f(x) by Both a Sine and Cosine Series 3.3.4 Even and Odd Parts 3.

3.5 Continuous Fourier Series 3.4 Term-by-Term Differentiation of Fourier Series 3.5 Term-By-Term Integration of Fourier Series 3.6 Complex Form of Fourier Series Wave Equation: Vibrating Strings and Membranes 4.1 Introduction 4.2 Derivation of a Vertically Vibrating String 4.3 Boundary Conditions 4.

4 Vibrating String with Fixed Ends 4.5 Vibrating Membrane 4.6 Reflection and Refraction of Electromagnetic (Light) and Acoustic (Sound) Waves 4.6.1 Snell''s Law of Refraction 4.6.2 Intensity (Amplitude) of Reflected and Refracted Waves 4.6.

3 Total Internal Reflection Sturm-Liouville Eigenvalue Problems 5.1 Introduction 5.2 Examples 5.2.1 Heat Flow in a Nonuniform Rod 5.2.2 Circularly Symmetric Heat Flow 5.3 Sturm-Liouville Eigenvalue Problems 5.

3.1 General Classification 5.3.2 Regular Sturm-Liouville Eigenvalue Problem 5.3.3 Example and Illustration of Theorems 5.4 Worked Example: Heat Flow in a Nonuniform Rod without Sources 5.5 Self-Adjoint Operators and Sturm-Liouville Eigenvalue Problems 5.

6 Rayleigh Quotient 5.7 Worked Example: Vibrations of a Nonuniform String 5.8 Boundary Conditions of the Third Kind 5.9 Large Eigenvalues (Asymptotic Behavior) 5.10 Approximation Properties Finite Difference Numerical Methods for Partial Differential Equations 6.1 Introduction 6.2 Finite Differences and Truncated Taylor Series 6.3 Heat Equation 6.

3.1 Introduction 6.3.2 A Partial Difference Equation 6.3.3 Computations 6.3.4 Fourier-von Neumann Stability Analysis 6.

3.5 Separation of Variables for Partial Difference Equations and Analytic Solutions of Ordinary Difference Equations 6.3.6 Matrix Notation 6.3.7 Nonhomogeneous Problems 6.3.8 Other Numerical Schemes 6.

3.9 Other Types of Boundary Conditions 6.4 Two-Dimensional Heat Equation 6.5 Wave Equation 6.6 Laplace''s Equation 6.7 Finite Element Method 6.7.1 Approximation with Nonorthogonal Functions (Weak Form of the Partial Differential Equation) 6.

7.2 The Simplest Triangular Finite Elements Higher Dimensional Partial Differential Equations 7.1 Introduction 7.2 Separation of the Time Variable 7.2.1 Vibrating Membrane: Any Shape 7.2.2 Heat Conduction: Any Region 7.

2.3 Summary 7.3 Vibrating Rectangular Membrane 7.4 Statements and Illustrations of Theorems for the Eigenvalue Problem ∇2φ + λφ = 0 7.5 Green''s Formula, Self-Adjoint Operators and Multidimensional Eigenvalue Problems 7.6 Rayleigh Quotient and Laplace''s Equation 7.6.1 Rayleigh Quotient 7.

6.2 Time-Dependent Heat Equation and Laplace''s Equation 7.7 Vibrating Circular Membrane and Bessel Functions 7.7.1 Introduction 7.7.2 Separation of Variables 7.7.

3 Eigenvalue Problems (One Dimensional) 7.7.4 Bessel''s Differential Equation 7.7.5 Singular Points and Bessel''s Differential Equation 7.7.6 Bessel Functions and Their Asymptotic Properties (near z = 0) 7.7.

7 Eigenvalue Problem Involving Bessel Functions 7.7.8 Initial Value Problem for a Vibrating Circular Membrane 7.7.9 Circularly Symmetric Case 7.8 More on Bessel Functions 7.8.1 Qualitative Properties of Bessel Functions 7.

8.2 Asymptotic Formulas for the Eigenvalues 7.8.3 Zeros of Bessel Functions and Nodal Curves 7.8.4 Series Representation of Bessel Functions 7.9 Laplace''s Equation in a Circular Cylinder 7.9.

1 Introduction 7.9.2 Separation of Variables 7.9.3 Zero Temperature on the Lateral Sides and on the Bottom or Top 7.9.4 Zero Temperature on the Top and Bottom 7.9.

5 Modified Bessel Functions 7.10 Spherical Problems and Legendre Polynomials 7.10.1 Introduction 7.10.2 Separation of Variables and One-Dimensional Eigenvalue Problems 7.10.3 Associated Legendre Functions and Legendre Polynomials 7.

10.4 Radial Eigenvalue Problems 7.10.5 Product Solutions, Modes of Vibration, and the Initial Value Problem 7.10.6 Laplace''s Equation Inside a Spherical Cavity Nonhomogeneous Problems 8.1 Introduction 8.2 Heat Flow with Sources and Nonhomogeneous Boundary Conditions 8.

3 Method of Eigenfunction Expansion with Homogeneous Boundary Conditions (Differentiating Series of Eigenfunctions) 8.4 Method of Eigenfunction Expansion Using Green''s Formula (With or Without Homogeneous Boundary Conditions) 8.5 Forced Vibrating Membranes and Resonance 8.6 Poisson''s Equation Green''s Functions for Time-Independent Problems 9.1 Introduction 9.2 One-dimensional Heat Equation 9.3 Green''s Functions for Boundary Value Problems for Ordinary Differential Equations 9.3.

1 One-Dimensional Steady-State Heat Equation 9.3.2 The Method of Variation of Parameters 9.3.3 The Method of Eigenfunction Expansion for Green''s Functions 9.3.4 The Dirac Delta Function and Its Relationship to Green''s Functions 9.3.

5 Nonhomogeneous Boundary Conditions 9.3.6 Summary 9.4 Fredholm Alternative and Generalized Green''s Functions 9.4.1 Introduction 9.4.2 Fredholm Alternative 9.

4.3 Generalized Green''s Functions 9.5 Green''s Functions for Poisson''s Equation 9.5.1 Introduction 9.5.2 Multidimensional Dirac Delta Function and Green''s Functions 9.5.

3 Green''s Functions by the Method of Eigenfunction Expansion and the Fredholm Alternative 9.5.4 Direct Solution of Green''s Functions (One-Dimensional Eigenfunctions) 9.5.5 Using Green''s Functions for Problems with Nonhomogeneous Boundary Conditions 9.5.6 Infinite Space Green''s Functions 9.5.

7 Green''s Functions for Bounded Domains Using Infinite Space Green''s Functions 9.5.8 Green''s Functions for a Semi-Infinite Plane (y > 0) Using Infinite Space Green''s Functions: The Method of Images 9.5.9 Green''s Functions for a Circle: The Method of Images 9.6 Perturbed Eigenvalue Problems 9.6.1 Introduction 9.

6.2 Mathematical Example 9.6.3 Vibrating Nearly Circular Membrane 9.7 Summary Infinite Domain Problems: Fourier Transform Solutions of Partial Differential Equations 10.1 Introduction 10.2 Heat Equation on an Infinite Domain 10.3 Fourier Transform Pair 10.

3.1 Motivation from Fourier Series Identity 10.3.2 Fourier Transform 10.3.3 Inverse Fourier Transform of a Gaussian 10.4 Fourier Transform and the Heat Equation 10.4.

1 Heat Equation 10.4.2 Fourier Transforming the Heat Equation: Transforms of Derivatives 10.4.3 Convolution Theorem 10.4.4 Summary of Properties of the Fourier Transform 10.5 Fourier Sine and Cosine Transforms: The Heat Equation on Semi-Infinite Intervals 10.

5.1 Introduction 10.5.2 Heat Equation on a Semi-Infinite Interval I 10.5.3 Fourier Sine and Cosine Transforms 10.5.4 Transforms of Derivatives 10.

5.5 Heat Equation on a Semi-Infinite Interval II 10.5.6 Tables of Fourier Sine and Cosine Transforms