In this blog, I am dedicated to providing free and accurate solutions and proofs for exercises in the textbook "Partial Differential Equations: Methods and Applications" whose author is Robert McOwen. However, due to limitations of my abilities, my solutions may not be perfect. Please feel free to point out and correct any grammatical or mathematical errors.
Chapter 1. First-Order Equations
Section 1.1 The Cauchy Problem for Quasilinear Equations
Section 1.2 Weak Solutions for Quasilinear Equations
Section 1.3 General Nonllinear Equations
Chapter 2. Principles for Higher-Order Equations
Section 2.1 The Cauchy Problem
Section 2.2 Second-Order Equations in Two Variables
Section 2.3 Linear Equations and Generalized Solutions
Chapter 3. The Wave Equation
Section 3.1 The One-Dimensional Wave Equation
Section 3.2 Higher Dimensions
Section 3.3 Energy Methods
Section 3.4 Lower-order Terms
Section 3.5 Applications to Light and Sound
Chapter 4. The Laplace Equation
Section 4.1 Introduction to the Laplace Equation
Section 4.2 Potential Theory and Greens' Functions
Section 4.3 Existence Theory
Section 4.4 Eigenvalues of Laplacian
Section 4.5 Applications to Vector Fields
Chapter 5. The Heat Equation
Section 5.1 The Heat Equation in a Bounded Domain
Section 5.2 The Pure Initial Value Problem
Section 5.3 Regularity and Similarity
Section 5.4 Application to Fluid Dynamics
Chapter 6. Linear Functional Analysis
Section 6.1 Function Spaces and Linear Operators
Section 6.2 Application to the Dirichlet Problem
Section 6.3 Duality and Compactness
Section 6.4 Sobolev Imbedding Theorems
Section 6.5 Generalizations and Refinements
Section 6.6 Unbounded Operators & Spectral Theory
Chapter 7. Differential Calculus Methods
Section 7.1 Calculus of Functionals and Variations
Section 7.2 Optimization with Constratins
Section 7.3 Calculus of Maps between Banach Spaces
Chapter 8. Linear Elliptic Theory
Section 8.1 Elliptic Operators on a Torus
Section 8.2 Estimates and Regularity on Domains
Section 8.3 Maximum Principles
Section 8.4 Solvability
Chapter 9. Two Additional Methods
Section 9.1 Schauder Fixed Ponint Theory
Section 9.2 Semigroups and Dynamics
Chapter 10. Systems of Conservation Laws
Section 10.1 Local Existence for Hyperbolic Systems
Section 10.2 Quasilinear Systems of Conservation Laws
Section 10.3 Systems of Two Conservation Laws
Chapter 11. Linear and Nonlinear Diffusion
Section 11.1 Parabolic Maximum Principles
Section 11.2 Local Existence and Regularity
Section 11.3 Global Behavior
Section 11.4 Applications to Navier-Stokes
Chapter 12. Linear and Nonlinear Waves
Section 12.1 Symmetric Hyperbolic Systems
Section 12.2 Linear Wave Dynamics
Section 12.3 Semilinear Wave Dynamics
Chapter 13. Nonlinear Elliptic Equations
Section 13.1 Perturbations and Bifurcations
Section 13.2 The Method of Sub- and Supersolutions
Section 13.3 The Variational Method
Section 13.4 Fixed Point Methods
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