An Introduction to Nonlinear Partial Differential Equations, 2nd Edition

1. Partial Differential Equations.

1.1 Partial Differential Equations.

1.1.1 PDEs and Solutions.

1.1.2 Classification.

1.1.3 Linear vs. Nonlinear.

1.1.4 Linear Equations.

1.2 Conservation Laws.

1.2.1 One Dimension.

1.2.2 Higher Dimensions.

1.3 Constitutive Relations.

1.4 Initial and Boundary Value Problems.

1.5 Waves.

1.5.1 Traveling Waves.

1.5.2 Plane Waves.

1.5.3 Plane Waves and Transforms.

1.5.4 Nonlinear Dispersion.

2. First-Order Equations and Characteristics.

2.1 Linear First-Order Equations.

2.1.1 Advection Equation.

2.1.2 Variable Coefficients.

2.2 Nonlinear Equations.

2.3 Quasi-linear Equations.

2.3.1 The general solution.

2.4 Propagation of Singularities.

2.5 General First-Order Equation.

2.5.1 Complete Integral.

2.6 Uniqueness Result.

2.7 Models in Biology.

2.7.1 Age-Structure.

2.7.2 Structured predator-prey model.

2.7.3 Chemotherapy.

2.7.4 Mass structure.

2.7.5 Size-dependent predation.

3. Weak Solutions To Hyperbolic Equations.

3.1 Discontinuous Solutions.

3.2 Jump Conditions.

3.2.1 Rarefaction Waves.

3.2.2 Shock Propagation.

3.3 Shock Formation.

3.4 Applications.

3.4.1 Traffic Flow.

3.4.2 Plug Flow Chemical Reactors.

3.5 Weak Solutions: A Formal Approach.

3.6 Asymptotic Behavior of Shocks.

3.6.1 Equal-Area Principle.

3.6.2 Shock Fitting.

3.6.3 Asymptotic Behavior.

4. Hyperbolic Systems.

4.1 Shallow Water Waves; Gas Dynamics.

4.1.1 Shallow Water Waves.

4.1.2 Small-Amplitude Approximation.

4.1.3 Gas Dynamics.

4.2 Hyperbolic Systems and Characteristics.

4.2.1 Classification.

4.3 The Riemann Method.

4.3.1 Jump Conditions for Systems.

4.3.2 Breaking Dam Problem.

4.3.3 Receding Wall Problem.

4.3.4 Formation of a Bore.

4.3.5 Gas Dynamics.

4.4 Hodographs and Wavefronts.

4.4.1 Hodograph Transformation.

4.4.2 Wavefront Expansions.

4.5 Weakly Nonlinear Approximations.

4.5.1 Derivation of Burgers’ Equation.

5. Diffusion Processes.

5.1 Diffusion and Random Motion.

5.2 Similarity Methods.

5.3 Nonlinear Diffusion Models.

5.4 Reaction-Diffusion; Fisher’s Equation.

5.4.1 Traveling Wave Solutions.

5.4.2 Perturbation Solution.

5.4.3 Stability of Traveling Waves.

5.4.4 Nagumo’s Equation.

5.5 Advection-Diffusion; Burgers’ Equation.

5.5.1 Traveling Wave Solution.

5.5.2 Initial Value Problem.

5.6 Asymptotic Solution to Burgers’ Equation.

5.6.1 Evolution of a Point Source.

6. Reaction-Diffusion Systems.

6.1 Reaction-Diffusion Models.

6.1.1 Predator-Prey Model.

6.1.2 Combustion.

6.1.3 Chemotaxis.

6.2 Traveling Wave Solutions.

6.2.1 Model for the Spread of a Disease.

6.2.2 Contaminant transport in groundwater.

6.3 Existence of Solutions.

6.3.1 Fixed-Point Iteration.

6.3.2 Semi-Linear Equations.

6.3.3 Normed Linear Spaces.

6.3.4 General Existence Theorem.

6.4 Maximum Principles.

6.4.1 Maximum Principles.

6.4.2 Comparison Theorems.

6.5 Energy Estimates and Asymptotic Behavior.

6.5.1 Calculus Inequalities.

6.5.2 Energy Estimates.

6.5.3 Invariant Sets.

6.6 Pattern Formation.

7. Equilibrium Models.

7.1 Elliptic Models.

7.2 Theoretical Results.

7.2.1 Maximum Principle.

7.2.2 Existence Theorem.

7.3 Eigenvalue Problems.

7.3.1 Linear Eigenvalue Problems.

7.3.2 Nonlinear Eigenvalue Problems.

7.4 Stability and Bifurcation.

7.4.1 Ordinary Differential Equations.

7.4.2 Partial Differential Equations.

References.

Index.