1. Introduction2. Classification and Characteristics3. Qualitative Behavior of Solutions to Elliptic Equations4. Qualitative Behavior of Solutions to Evolution Equations5. First-Order Equations6. Eigenfunction Expansions and Integral Transforms: Theory7. Eigenfunction Expansions and Integral Transforms: Applications8. Green's Functions9. Difference Methods for Parabolic Equations10. Difference and Characteristic Methods for Parabolic Equations11. Difference Methods for Hyperbolic Equations12. Difference Methods for Elliptic Equations13. Variational Formulation of Boundary Value Problems14. The Finite Element Method: An Introduction