I am studying PDE, their typologies and their characteristic for the use in CFD. This question is a follow up of that. I am reading 1, and several question raised to my mind (pg. 31):
- Regarding the Hyperbolic PDE: I understood that for the D'Alambert Eq. the region of the initial condition $[x_{i}-t_{i},x_{i}+t_{i}]$ that impacts the solution at $P(x_i,t_i)$, this because of the characteristic directions. It is difficult to understand why the solution in a point P depends on the whole area of the domain of dependence and not only from $[x_{i}-t_{i},x_{i}+t_{i}]$.
- For the Parabolic case: Given the Eq.
$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2} \tag{1}$
With intial condition $u=sin(\pi x)$ and boundary condition $u(0,t)=u(1,t)=0$ Eq. (1) the solution is:
$u(x,t)=sin(\pi x)exp(-\pi^2 t) \tag{2}$
So there is an exponential decay in time. The book says that the solution "march forward in time and diffuse in space." I don't understand the meaning of this phrase. How is the "marching" linked with the exponential decay in time? What does it mean that "diffuses" in space?
- Regarding the Elliptic one: Since the missing of the variable t in the PDE, would I not have a solution that variates in time, but only values that variates in space?
Thanks in advance
1 Fletcher, Clive A. J. (1998). Computational Techniques for Fluid Dynamics 1 .