Time dependent Partial Differential Equations and Their Numerical Solution In these notes we study time dependent partial differential equations and their numerical solution. The analytic and the numerical theory are developed in parallel. For example we discuss well posed linear and nonlinear problems linear and nonlinear stability of difference approximations and error estimates. Special emphasis is given to boundary conditions and their discretization. We develop a rather general theory of admissible boundary conditions based on energy estimates or Laplace transform techniques. These results are fundamental for the mathematical and numerical treatment of large classes of applications like Newtonian and non Newtonian flows two phase flows and geophysical problems