Textbook:"Partial Differential Equations: An Introduction" by Walter A Strauss.

Syllabus:

Chapter 1: Where PDEs Come From
   1.1  Waht is a Partial Differential Equation?
   1.2  First-order Linear Equations
   1.3  Flows, Vibrations, and Diffusions
   1.4  Initial and Boundary Conditions
   1.5  Well-Posed Problems
   1.6  Types of Second-order Equations

Chapter 2:  Waves and Diffusions
   2.1  The Wave Equation
   2.2  Causality and Energy
   2.3  The Diffusion Equation
   2.4  Diffusion on the Whole Line
   2.5  Comparison of Waves and Diffusions'

Chapter 3: Reflections and Sources
   3.1  Diffusion on the half-line
   3.3  Diffusion with a Source

Chapter 4:  Boundary Problems
   4.1  Seperation of Variables, the Dirichlet Condition
   4.2  The Neumann Condition
   4.3  The Robin Condition

Chapter 5:  Fourier Series
   5.1  The Coefficients
   5.2  Even, Odd, Periodic, and Complex Functions
   5.3  Orthonality and General Fourier Series
   5.4  Completeness

Chapter 6:  Harmonic Functions
   6.1  Laplace's Equation
   6.2  Rectangle and Cubes
   6.3  Poisson's Formula
   6.4  Circles, Wedges, and Annuli

Chapter 10: Boundaries in the Plane and in Space
   10.1  Fourier's Method, Revisited
   10.2  Vibrations of a Drumhead

It the time allows -
Chapter 13:  PDE Problems from Physics
   13.3  Scattering