|M.Sc Student||Paneah Peter|
|Subject||Nonlocal Problems for Linear Second Order Hyperbolic|
|Department||Department of Applied Mathematics||Supervisor||Professor Yehuda Pinchover|
Let P be a second-order linear hyperbolic differential operator defined on the rectangle D=[0,a] x [0,b].
Consider the problem
P u = f in D
ò K1(x) u(x,y) dx = j(y) on [0,b]
ò K2(y) u(x,y) dy = f(x) on [0,a]
where K1(x) and K2(y) are fixed integrable kernels and f, f and j are the given data.
In general, this problem is not uniquely solvable. We obtain necessary and sufficient conditions for the unique solvability of the problem when Pu =uxy.
We reduce the general problem to an integral equation. We obtain sufficient conditions on kernels K1(x) and K2(y) for the unique solvability of this integral equation. We show that if K1(x) and K2(y) do not change their sign on the intervals [0,a] and [0,b] correspondingly then the problem is uniquely solvable locally.
Moreover, under additional assumptions, the problem is Fredholm and its index is equal to zero in any rectangle.
An example is given, demonstrating that if any of these conditions is not satisfied, then the problem might not be Fredholm. Finally, we consider another nonlocal problem with associated conditions given by a distribution (a sum of weighted delta-functions). We show that the above results remain valid in this situation too. If the above sum consists of only one term, we obtain the generalized Goursat problem, and we show its unique solvability in an arbitrary domain.