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Consider the following problem: $u_t = ku_{xx}$ on the semi-infinite strip $S = [0,\infty)\times [0,\infty)$, with the $zero$ initial condition $u(x,0) = 0,\, u(0,t)=g(t).$

I tried both the separation of variables and the fundamental solution methods, but both leads me to conclude the $u\equiv 0$, because of the zero initial condition. Usually, the initial conditions look something like: $$u(x,0) = f(x),$$ where $f(x)$ admits a Fourier Series. But in our case $f\equiv 0$ and I don't know how to circumvent this.

This is an old question from a PhD qual exam, so I highly doubt that the answer is just the identically zero function.

dezdichado
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    You can't have $u \equiv 0$ with $u(0,t)=g(t)$ unless $g \equiv 0$ as well. You need to weave the BC into the problem. – Ian Aug 22 '17 at 22:02
  • yes, but that's not what the problem I am having. The part $u(x,0) = 0$ is giving me troubles because, for example, the separation of variables method forces $u\equiv 0,$ which we clearly cannot have. – dezdichado Aug 22 '17 at 22:31
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    The separation of variables approach for the heat equation is just to represent the initial condition as a superposition of eigenfunctions and then claim that this is the evolution. That can't see the BC, so you can't do that as is. One way to do things is to consider the problem that $v(x,t)=u(x,t)-g(t)$ solves: $\frac{\partial v}{\partial t}=\frac{\partial u}{\partial t} - g'(t)=\frac{\partial^2 u}{\partial x^2} - g'(t)=\frac{\partial^2 v}{\partial x^2} - g'(t),v(0,t)=0$. You can then turn this into a family of second order ODEs using temporal Laplace transforms. – Ian Aug 22 '17 at 22:35

1 Answers1

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Separation of variables gives $$ \frac{T'}{kT} = -s^2 = \frac{X''}{X} \\ T(0)=0. $$ So $T(t)= e^{-ks^2 t}-1$ and $X(x) = A(s)\cos(s x)+B(s)\sin(sx)$, leading to $$ u(x,t) = \int_{0}^{\infty} (e^{-ks^2t}-1)(A(s)\cos(sx)+B(s)\sin(sx))ds. $$ $u(x,0)=0$ works out. The condition $u(0,t)=g(t)$ allows $B\equiv 0$, provided $A$ satisfies $$ g(t) = \int_{0}^{\infty}(e^{-ks^2 t}-1)A(s)ds. $$

Disintegrating By Parts
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