4. (a) Write down the fundamental solution of the diffusion equation ut kur, on Rx (0,...

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4. (a) Write down the fundamental solution of the diffusion equation ut kur, on Rx (0, ∞). U = You do not need to give any justification for your answer. (2 points) (b) Use the method of reflection to find a solution for the diffusion equation on the half line [0, ∞) with a Neumann boundary condition in 0: U t = kux, on (0, ∞) x (0, ∞) u(,0) = 6 on [0, ∞) (1) ux (0,.) = 0 on (0, ∞). (4 points) (c) Use the energy method to show that there is at most one solution of (1) with u(x, t) → 0 for x→∞. (4 points) 4. (a) Write down the fundamental solution of the diffusion equation ut kur, on Rx (0, ∞). U = You do not need to give any justification for your answer. (2 points) (b) Use the method of reflection to find a solution for the diffusion equation on the half line [0, ∞) with a Neumann boundary condition in 0: U t = kux, on (0, ∞) x (0, ∞) u(,0) = 6 on [0, ∞) (1) ux (0,.) = 0 on (0, ∞). (4 points) (c) Use the energy method to show that there is at most one solution of (1) with u(x, t) → 0 for x→∞. (4 points)