ABSTRACT
In general, a nonhomogeneous linear differential equation of the parabolic type with variable coefficients in one dimension can be written as
∂w
∂t − Lx[w] = Φ(x, t), (17.1.1.1)
where
Lx[w] ≡ a(x, t) ∂ 2w
∂x2 + b(x, t)
∂w
∂x + c(x, t)w, a(x, t) > 0. (17.1.1.2)
Consider the nonstationary boundary value problem for Eq. (17.1.1.1) with an initial condition of general form
w = f(x) at t = 0, (17.1.1.3)
and arbitrary nonhomogeneous linear boundary conditions
α1 ∂w
∂x + β1w = g1(t) at x = x1, (17.1.1.4)
α2 ∂w
∂x + β2w = g2(t) at x = x2. (17.1.1.5)
By appropriately choosing the coefficients α1, α2, β1, and β2 in (17.1.1.4) and (17.1.1.5), we obtain the first, second, third, and mixed boundary value problems for Eq. (17.1.1.1).
