The final exam is cumulative, spanning all the material we covered in the first 5 chapters of the course text. It is highly recommended that you go over homework problems (solutions are on reserve in the black "Roberts" binder in the Math/Sci. Library), your class notes and the sample final exam linked below.

**Note:** You will be allowed one "cheat sheet," a single 8.5 x 11
piece of paper, front and back, full of whatever formulas, graphs, etc. you wish.

**Exam Review:** We will review for the final exam on Friday, May 6th from 2:00 - 4:00 pm
in Swords 359. Most likely, we will go over the sample final exam available
here ** Sample Final ** (pdf file.)
You can refer to previous exam review sheets for other study questions.
Please come prepared with specific questions.

The following concepts, definitions and equations are important material
for the exam:

- General PDE terminology: operators, linear operators, homogeneous and inhomogeneous linear equations, the order of a PDE, well-posed problems (existence, uniqueness and stability)
- Initial and Boundary Conditions: physical interpretations of initial conditions (position, velocity, etc.), describing conditions mathematically, boundary conditions (Dirichlet, Neumann, Robin, Periodic), physical interpretations of boundary conditions
- First-order Linear Equations: solving (geometric method or the method of characteristics, coordinate method), solving to satisfy an initial condition, solving inhomogeneous linear equations (use u_h + u_p), characteristic curves
- The Wave Equation: factoring an operator, general solution, D'Alembert's formula, physical interpretation of solutions, principle of causality, conservation of energy
- The Diffusion Equation: the Maximum (and minimum) Principle, properties of solutions (eg. invariance, well-posed for t > 0 , etc.), physical interpretation of solutions, general solution, the source function S(x,t), the error function Erf(x)
- Diffusion and Waves on the Half-line 0 < x < infty: odd and even extensions, how to solve a given equation with either Dirichlet or Neumann boundary conditions, unwinding D'Alembert's formula or the formula for the solution to the diffusion equation
- Waves with a Source: Using the u_h + u_p technique to solve u_tt - c^2 u_xx = f(x,t) and solving for the arbitrary functions given the initial conditions
- Boundary Problems: Separation of variables technique, solving PDE's with different boundary conditions (Dirichlet, Neumann, Periodic, Robin) as infinite series, finding eigenvalues and eigenfunctions, frequencies, using graphical analysis to find eigenvalues
- Fourier Series: Finding coefficients for the Fourier sine, cosine and full series; odd, even and periodic functions and their properties; odd, even and periodic extensions, inner product and orthogonality, the main convergence theorems, the method of subtraction
- Important Mathematical Concepts: Fundamental Theorem of Calculus, differentiating an integral, chain rule (multivariable version), convergence and divergence of infinite series, the three types of convergence (pointwise, uniform, mean-square L^2), solving basic ODE's
- Important Physical Concepts: F = ma, physical interpretations of the partial derivative, integrating to calculate mass, equilibrium solutions