# Heat Equation

## Contents

# Heat Equation#

## Python Notebooks#

Heat equation

Task 1: Integrate the heat equation for several days using a time step of hour and a heat conductivity of \(\nu = 1.2e^{-6}\) [m\(^2\) s\(^{-1}\)]. Plot the result. Once the code works, change the integration time. What happens if you integrate over a very long time?

Task 2: Rewrite the 1D heat equation (Task 1) using index arrays.

Task 3: Using the previous code, solve the Heat Equation using a temporal varying surface boundary condition. Use the following discretization: I = [0; 20 m], N = 40 grid points, \(\nu = 1.2e^{-6}\) [m\(^2\) s\(^{-1}\)], and a daily time step. Integrate the equation for several years, e.g. 5 years. Plot the result as a contour plot. Also plot temperature time series in several depths. Discuss the plot!