Exercises

%- [x] analysis 0 (partially solved)
%- [x] analysis 1 (@2025-10-03)
%- [x] analysis 2 (@2025-10-10)
%- [x] analysis 3 (@2025-10-17)
% - [ ] understand method of characteristics geometrically (@2025-10-17)
%- [ ] analysis 4 (@2025-10-24)
%- [ ] analysis 5 (@2025-10-31)

Vorlesung

#timestamp 20250919

what is a well posed problem?

• Partial differential equation + condition (initial conditions, boundary conditions, etc.)

1. existence of a solution
2. uniqeness of the solution
3. stability w.r.t. the initial datum

All the tree $✓$ => well-posed
As soon as we don't know one of those => p.b. is ill-posed

example
consider the TRANSPORT equation

If $U\ge 0$, $U$ may represent the concentration of a pollutant at $t$ and position $x$

• similiar (EASIER) $U_t+U{U}_x=0$
• $C$ velocity of the river
• concentration of the pollutant at time 0 $g(x)$

Initial value problem

• $U(x,0)=g(x),x\in \mathbb{R}$ initial condition

example Wave equations in 1D vibrating string
boundary condition $U(0,t)=U(L,t)=0$
initial conditions $U(x,0)=f(x)$ (deflection), $U_t(x,0)=g(x)$ (speed)

Notice that the domain of the PDE is only defined in the interior of the domain because $U$ may not be differentiable at the boundary

definition strong solution
The solution of a PDE is strong (classical) iff all the derivatives of the solution that appear in the PDE exist and are continuous.
Otherwise, the solution is called weak.

• strong: $U_{tt}=\mathrm{\Delta }U$ have to be well-defined

=> there is no universal definition of a weak solution

classification and proerties of PDEs

definition Order of a PDE: order ogf the highest derivative of the unknown appearing within it
example $U_x+U^2{U}_{yy}=e^y{U}_{xy}$ $\to$ order 2
definition linear: terms (coefficients) can be dependent on any constant or unknown, but not on the function itself

#timestamp 2025-11-14

General formula, heat eq. non-hom. Neumann:

• if we go from Dirichlet to Neumann, replace $\sin \to \cos$ and add the $n=0$ term!!

=> only for heat/wave eqation is sin: Dirichlet, cos: Neumann