Saturday, April 21, 2007

Diffusion with a source on a semi-infinite line

(Budak/Samarskii/Tikhonov, problem 3.7)

ut = a2uxx + 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

ut = a2uxx
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

ut = a2uxx
u2(x,0) = 0
u2(0,t) = μ(t)

u2(x,t) = ½ (a2/√π) ∫0t 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' 0t dt' { 1/√[a2(t-t')] [ e -(x-x')2/4a2(t-t') - e -(x+y)2/4a2(t-t') ] f(x',t') }

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