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I have come across the following Heat equation IBVP but I am not quite sure how to solve it:

$$v_t = kv_{xx} \ \ \ \ \ \ ( 1 < x < \infty, \ \ 0 < t < \infty) ,$$ $$ v(x,0) = \delta (x - x_0) \ \ \ \ \ \ \mathrm{for} \ \ t = 0, \ \ x_0 > 1 $$ $$ v(1,t) = 0 \ \ \ \ \ \ \mathrm{for} \ \ x = 1 $$ $$ v(\infty,t) = 0 \ \ \ \ \ \ \mathrm{for} \ \ x = \infty. $$

Due to the Dirac delta initial condition, I was thinking some form of fourier like transformation is needed, but I am not sure given the boundary conditions if that will work.

Is there a way to solve this problem ? Thanks.

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Start from the 1D heat kernel: $$v(x,t) = \frac{1}{\sqrt{4\pi k t}} e^{-x^2/(4 k t)}.$$ First, shift it so that the initial conditions line up: $$v(x,t) = \frac{1}{\sqrt{4\pi k t}} e^{-(x-x_0)^2/(4 k t)}.$$ This does not satisfy your Dirichlet boundary conditions; however you can fix it by applying the reflection principle:

$$v(x,t) = \frac{1}{\sqrt{4\pi k t}} e^{-(x-x_0)^2/(4 k t)}-\frac{1}{\sqrt{4\pi k t}} e^{-(x-[2-x_0])^2/(4 k t)}.$$

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  • $\begingroup$ Firstly thank you very much for helping me with this. Can you please provide me a link to the reflection principle you are using. Sorry I am rather new to pdes and the only reflection principle I am aware of is the one associated with Brownian motion. $\endgroup$ – Comic Book Guy May 29 '16 at 0:53
  • $\begingroup$ Check out section 5.3 of math.tut.fi/~piche/pde/notes05.pdf $\endgroup$ – user7530 May 29 '16 at 0:59

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