Hello,snipez90 said:Dang I thought this was going to be an actual PDE question. There is really no need to graph nor is there reason for mathematical proof unless you are clueless about the exponential function. Remember for pointwise convergence we only consider what happens by fixing an x in the set under consideration and then letting n approach infinity. Now fix x = 0, what is the limiting function here? Now fix an arbitrary x > 0, what happens when you let n go to infinity then?
Pointwise convergence in PDEs refers to the behavior of a sequence of functions at a specific point. It means that as the sequence of functions approaches a limit, the values of the functions at that point also approach a limit. This is in contrast to uniform convergence, where the entire function approaches a limit uniformly.
Pointwise convergence and uniform convergence differ in the way the convergence is evaluated. Pointwise convergence considers the behavior of a sequence of functions at a specific point, while uniform convergence considers the behavior of the entire function over a given interval. Uniform convergence is a stronger form of convergence, as it requires the functions to approach a limit uniformly, while pointwise convergence only requires the values at a specific point to approach a limit.
Pointwise convergence is important in PDEs because it helps determine the behavior of the solution to the PDE at a specific point. It allows us to analyze the behavior of the solution as the parameters of the PDE change, and to identify any points where the solution may not behave as expected.
Yes, a sequence of functions can converge pointwise but not uniformly in PDEs. This occurs when the functions approach a limit at a specific point, but the convergence is not uniform over the entire function. In other words, the values of the functions at that point may approach a limit, but the behavior of the rest of the function may be erratic or not approach a limit at all.
Pointwise convergence can be tested by evaluating the limit of the sequence of functions at a specific point. If the limit exists and is equal to the value of the function at that point, then the sequence of functions converges pointwise. This can be done for each point in the domain of the function to determine if the convergence is pointwise for the entire function.