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**Heat equation on a half line!**

Hi,

I am now dealing with the heat equation on a half line, i.e., the heat equation is subject to one time-dependent boundary condition only at x=0 (the other boundary condition is zero at the infinity) and an initial condition.

I searched online, it seems that for the half line problem, only the sine transformation can solve the heat equation, but in that case, the final result is always zero at x=0 since when doing sine transformation, one should assume that the to-be-transformed function is odd, so the function is zero at x=0.

My question is, do you know any other techniques to solve the heat equation on the half line without using sine transform?

Thanks.