Diffusion with a source on a semi-infinite line

(Budak/Samarskii/Tikhonov, problem 3.7)

u_t = a²u_xx + f(x,t)
u(x,0) = φ(x)
u(0,t) = μ(t)
0 ≤ x < ∞ and 0 ≤ t < ∞

The solution of the problem can be represented as a sum
u(x,t) = u₁(x,t) + u₂(x,t) + u₃(x,t)
where u₁(x,t) represents the effect of the initial condition u(x,o) = φ(x), u₂(x,t) represents the effect of the boundary condition u(0,t) = μ(t), and u₃(x,t) represents the effect of the nonhomogeneous term f(x,t).

Solution for u₁

u_t = a²u_xx
u₁(x,0) = φ(x)
u₁(0,t) = 0

u₁(x,t) = ½ (1/√π) ∫₀^∞ 1/√(a²t) [ e ^{-(x-x')2/4a2t} - e ^{-(x+x')2/4a2t} ] φ(x') dx'

Solution for u₂

u_t = a²u_xx
u₂(x,0) = 0
u₂(0,t) = μ(t)

u₂(x,t) = ½ (a²/√π) ∫₀^t x/[a²(t-t')]^3/2 e^{-x2/4a2(t-t')} μ(t') dt'

Solution for u₃

u₃(x,t) = ½ (1/√π) ∫₀^∞ dx' ∫₀^t dt' { 1/√[a²(t-t')] [ e ^{-(x-x')2/4a2(t-t')} - e ^{-(x+y)2/4a2(t-t')} ] f(x',t') }