Finite-Difference Schemes for a Nonlinear Parabolic Problem with Nonlocal Boundary Conditions

Abstract

Consider the nonlinear parabolic equation

$$ \frac{{\partial u}} {{\partial t}} = \frac{\partial } {{\partial x}}(p(x)\frac{{\partial u}} {{\partial x}}) - q(x,t)u + f(u,x,t), $$

((9.1))

for (x, t) ∈ Q _T = (0,1) × (0, T], 0 < T ≤ ∞, subject to the initial condition

$$ u(x,0) = u_0 (x), x \in [0,1] $$

and the nonlocal boundary conditions

$$ u(0,t) = \gamma _0 (\alpha _0 (t)u(c_0 (t),t) + \int_0^1 {\beta _0 (x,t)u(x,t)dx) + g_0 (t),} $$

$$ u(1,t) = \gamma _1 (\alpha _1 (t)u(c_1 (t),t) + \int_0^1 {\beta _1 (x,t)u(x,t)dx) + g_1 (t).} $$