7
votes
Accepted
How to generate a random function with conditions?
You can simulate functions $u$ vanishing at $\infty$ using the formula
$$u(x)=u_N(x):=\sum_{n=0}^N \xi_n l_n(x),$$
where $N$ is a natural number, the $\xi_n$'s are independent standard normal (or ...
4
votes
What are dissipative PDEs?
Q: Is there a rigorous mathematical definition or an easy characterization of a dissipative PDE?
Dissipative Operators and Hyperbolic Systems of Partial Differential Equations by R.S. Phillips gives a ...
2
votes
Accepted
Inconsistency in determinability of the solution of a linear first order PDE
Pedestrian answer.
In practice $h>0$ is a small number. In what follows, it will be enough that $h\in(0,1)$. Because of
$$\left(e^{(\frac1h-1)x}u_1\right)'=\frac1h e^{(\frac1h-1)x}f,$$
we must have
...
2
votes
Inconsistency in determinability of the solution of a linear first order PDE
It may help to review quickly why the backwards Euler method works for finite dimensional ODEs, before moving to the infinite dimensional case.
Finite Dimensional Backwards Euler
Consider the ...
2
votes
What are dissipative PDEs?
In some sense, the so-called "entropy" does not have a unified definition in the mathematics community and sometimes the notion of "entropy" is identified (loosely speaking) as ...
2
votes
Where can I find the paper by Tappert and Hardin on split-step Fourier transform method?
The original-original was two talks in 1973 at a Boston SIAM conference, just a descriptive paragraph for each. About a year later there was a more formal talk for Ocean acoustics specialists given ...
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