A monotonic finite-difference scheme for a parabolic problem with nonlocal conditions
The authors prove convergence, maximum principles, and establish the order of finite difference schemes for solving the following parabolic problem in one space dimension: \[ \varrho(x,t)\frac{\partial u}{\partial t} = \frac{\partial}{\partial x}\left(p(x)\frac{\partial u}{\partial x}\right)- q(x,t) u + f(x,t),\quad (x,t)\in Q_T=(0,1)\times (0,T], \] with initial condition \[ u(x,0)=u^0(x),\quad x\in [0,1], \] and non--local boundary conditions: \[ u(l,t)= \gamma_l \left(\widetilde{\alpha}_l(t) u(\widetilde{x}_l(t),t) + \int_0^1 \widetilde{\beta}_l(x,t) u(x,t) dx \right) + f_l(t), \] for \(t\in (0,T]\), \(l=0,1\). Results for the corresponding stationary problem are also given.
scientific article; zbMATH DE number 1916132