# One Dimensional Heat Explorer

This fun tool allows us to explore the flow of heat through a one-dimensional conductive medium by specifying its initial distribution and parameters of the heat equation with a source:

\begin{aligned} & u_t = D\,u_{xx}+f(x) & \: \: u(x,0)=u_0(x) \\ & \:\:\:\:\:\:u(0,t) = a & u(1,t) = b. \:\:\:\: \end{aligned}

\begin{aligned} & u_t = D\,u_{xx}+f(x) & \: \: u(x,0)=u_0(x) \\ & \:\:\:\:\:\:u(0,t) = a & u(1,t) = b. \:\:\:\: \end{aligned}
In this set of equations, $$u_t = D\,u_{xx}+f(x)$$ is a partial differential equation called the heat equation with rate of diffusion $D$ and positional heat source $f(x)$. The function $u_0$ describes the initial distribution of heat while the constants $a$ and $b$ indicate the fixed temperatures on the ends of the bar. Alternatively, we could indicate how heat flows through the ends of the bar by specifying $u_x(0,t)$ and $u_x(1,t)$. Using the checkboxes above, for example, you can specify that either derivative is zero indicating that the end is perfectly insulated.
\begin{aligned} u(x,t) &= a+(b-a)x + 2\sum_{k=1}^{\infty} \left( \frac{1-e^{-D k^2 \pi^2 t}}{D k^2 \pi^2} \int_0^1 f(\chi) \sin(k\pi \chi) d\chi \cdots \right. \\ &+ \left. e^{-D k^2 \pi^2 t} \left(\frac{(-1)^k b - a}{k\pi} + \int_0^1 u_0(\chi)\sin(k\pi \chi) d\chi \right) \right)\sin(k\pi x). \end{aligned}