I am looking at the heat equation and in my notes it says the initial temperature distribution $u(0,x)=u_0(x)$.
what does this mean?
What does $u_0(x)$ represent?
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In the heat equation $u(t,x)$ is a function of the two variable $t$ and $x$ that represents the temperature at time $t$ in the point $x$. So $u(0,x)=f(x)$ is the function that represents the initial temperature at point $x$, and it is a function of $x$ only. Your notation simply indicates this $f(x)$ with the symbol $u_0(x)$.