Nonlocal boundary value problems in differential and difference settings (Q731570)

The author considers the partial differential equation \[ \frac{\partial u}{\partial t}=\frac{\partial }{\partial x}\left(k(x,t)\frac{\partial u}{\partial x}\right)-q(x,t)u+f(x,t), \quad 0<x<l, \; 0<t \leq T \] with the initial condition \[ u(x,0)=u_0(x), \quad 0 \leq x \leq l \] and with the nonlocal boundary conditions of the first kind \[ u_x(0,t)=0, \quad u(l,t)=\alpha(t)u(0,t),\quad 0 \leq t \leq T \] or of the second kind \[ u_x(0,t)=0, \quad u_x(l,t)=\alpha(t)u_x(0,t), \quad 0 \leq t \leq T. \] The both problems are approximated by finite difference schemes. The stability estimates are obtained for the differential as well as for the finite difference problems.