1. Higher-Order PDEs
1.1. Higher-Order Linear Partial Differential Equations
1.1.1. Equation of transverse vibration of elastic rod
1.1.2. Biharmonic equation
1.2. Higher-Order Nonlinear Partial Differential Equations
1.2.1. Korteweg-de Vries equation
1.2.2. Equation of a steady laminar boundary layer on a flat plate
1.2.3. Bousinesq equation
1.2.4. Equation of motion of a viscous fluid
2. Approximate and Numerical Methods
2.1. Parabolic PDE
2.2. Hyperbolic PDE
2.3. Eliptic PDE
3. First-Order PDEs
3.1. General Form of First-Order PDE
3.2. Quasilinear Equations
3.2.1. General form of first-order quasilinear PDE
3.2.2. Characteristic system
3.2.3. General solution
3.3. Cauchy Problem: Two Formulations
3.3.1. Generalized Cauchy problem
3.3.2. Classical Cauchy problem
3.3.3. Existence and uniqueness theorem
3.3.4. Procedure of solving the Cauchy problem
4. Second-Order PDEs
4.1. Linear, Semilinear, and Nonlinear Second-Order PDEs
4.1.1. Linear second-order PDEs and their properties
4.1.2. Principle of linear superposition
4.1.3. Semilinear and nonlinear second-order PDEs
4.2. Some Linear Equations Encountered in Applications
4.2.1. Heat equation (a parabolic equation)
4.2.2. Wave equation (a hyperbolic equation)
4.2.3. Laplace equation (an elliptic equation)
4.3. Classification of Second-Order Partial Differential Equations
4.3.1. Types of equations
4.3.2. Characteristic equations
4.3.3. Canonical form of parabolic equations (case b^2 - ac = 0)
4.3.4. Two canonical forms of hyperbolic equations (case b^2 - ac > 0)
4.3.5. Canonical form of elliptic equations (case b^2 - ac < 0)
4.4. Basic Problems for PDEs of Mathematical Physics
4.4.1. Cauchy problem and boundary value problems for parabolic equations
4.4.2. Cauchy problem and boundary value problems for hyperbolic equations
4.4.3. Boundary value problems for elliptic equations
4.5. Some Nonlinear Equations
4.5.1. Nonlinear equation
4.5.2. Kolmogorov-Petrovskii-Piskunov equation
4.5.3. Burgers equation
4.5.4. Nonlinear wave equation
4.5.5. Nonlinear Klein-Gordon equation
4.5.6. Nonlinear Laplace equation
4.5.7. Monge-Ampere equation
4.6. Simplest Types of Exact Solutions of Nonlinear PDEs
4.6.1. Preliminary remarks
4.6.2. Traveling-wave solutions
4.6.3. Self-similar solutions