§ Nonexistence of solutions for ODE and PDE
- ODE system, no bc: always solution by picard liendolf
- ODE system, with boundary cond:, can have no solution. Eg. , with boundary conditoin .
- PDE system, no bc: can still create no solutions!
- PDE system, with boundary cond: can have no solution because ODE is PDE.
§ Example 1 of PDE with no solutions
- Take a vector field on with . This vector field has concentric spirals.
- consider this vector field as a PDE, so we are looking for a function such that .
- No such potential function can exist, because this vector field allow us to extract work.
- Suppose such a potential exists. Then if I travel in a circle, according to the potential, net work is zero. But if I evaluate the integral, I will get work done. Thus, no soln exists!
- In general, asking if a differential form is exact is literally asking for a PDE to be solved!
- In this case, the form is also closed , since it's a 2D form on a 2D surface. This is an example of a closed form that is not exact.
- It's nice to see PDE theory and diffgeo connect
§ Example 2: use second axis as time
- Consider a PDE on a square . We will think of the first axis as space where the function is defined and the second axis as time where the function is perturbed.
- We start by saying . So the function at is constant, and at is linear.
- Next, we say that . This means that the function is not allowed to evolve through time.
- This is nonsensical, becase at , we expect a constant function to have become a linear function, but along the time axis, we say that no point in space can change.
- Thus, this DE has no solutions!
- We can use the extra dimensions available in a PDE to create "conflicting" data along different time axes.