MATH 480, Review for Midterm Test 1

Test coverage

1. Introduction
   1. Notation for PDE. Functions of several variables, partial derivatives, examples (lecture notes)
   2. Nonlinear and linear PDE, linear homogeneous equations. Linear superposition principle (0.1.2)
   3. Example: the one-dimensional heat equation. Initial and boundary conditions. Steady solutions (0.1.4)
   4. Separation of variables. Real and complex separated solutions (0.2.1-0.2.3)
   5. Separated solutions with boundary conditions (0.2.4)
   6. One-dimensional heat equation: solving the initial-boundary value problem (lecture notes)
2. Orthogonal Functions and Fourier Series
   1. Orthogonal functions. Inner product. (0.3.1)
   2. Orthogonal projection. Fourier coefficients. Least square approximation of functions(0.3.2)
   3. Infinite orthogonal sets and general Fourier series (0.3.3)
   4. Bessel's inequality and Parseval's equality (0.3.3)
   5. Trigonometric Fourier series on (-L,L). Partial sums. Periodic extension of functions (1.1.1, lecture notes)
   6. Symmetry considerations. Even and odd functions. Sine and cosine series (1.1.2)
   7. Convergence of Fourier series. Piecewise-smooth functions. Dirichlet kernel. Riemann-Lebesgue lemma (1.2)
   8. Uniform convergence and the Gibbs phenomenon (1.3.1, 1.3.3, 1.3.4)
   9. Differentiation and integration of Fourier series (1.3.5, 1.3.6)
   10. Parseval's theorem and mean-square error (1.4)

Important concepts

• PDE; order of PDE, linear/nonlinear, linear homogeneous
• Boundary condition; initial conditions
• Linear operator
• Linear superposition principle
• Real and complex separated solutions
• Fourier coefficients. Orthogonal projection in a function space
• Orthogonal and orthonormal sets (systems) of functions
• Abstract Fourier series, Bessel's inequality, Parseval's equality
• Least squares approximation and mean-square error
• Periodic extension, even and odd extensions
• Piecewise smooth functions
• Mean-square convergence
• Uniform and non-uniform convergence
• Gibbs phenomenon; Gibbs overshoot

Theoretical material

• Linear superposition principle (linear aglebra principle with PDE notation; proofs are easy!)
• Theorem about separation of variables (Proposition 0.2.2)
• Bessel's inequalty and Mean-square error (Proposition 0.3.4)
• Cauchy-Schwarz inequality (proof is useful and easy!)
• Lemma about the Dirichlet kernel (proof uses complex geometric progression)
• Proof of convergence thorem (general structure and details from 1.2.3 and exercises)
• Differentiation of Fourier series (Proposition 1.3.3)
• Parseval's theorem (Theorem 1.2)

Important types of problems

• Solving simple PDEs by integration (Homework 1)
• Changes of variables and reducing harder PDEs to easier/known ones (Homework 1)
• Steady state solutions: reducing PDE problems to ODE problems (Homework 2)
• Separation of variables! (Homework 2 and 3, check other problems in 0.2.3 and 0.2.4!)
• Working with orthogonal functions: demonstrating orthogonality, finding projections, Fourier coefficients, producing orthogonal sets via Gram-Schmidt (Homework 3)
• Computing Fourier coefficients (Fourier series) analytically for simple examples (Homework 4)
• Using convergence properties of Fourier series to find sums of numerical series (Homework 4)
• Manipulating Fourier series (differentiation, integration, identities) (Homework 5)
• Using Parseval's theorem to estimate mean-square error (Homework 5)

Necessary ODE background

• Linear, second order equations with constant coefficients (based on solutions of quadratic equation!)
• Separable equations
• Linear, first order equations with nonconstant coefficients (integrating factor)
• Euler's equations with non-constant coefficients