**Introduction**

The Heat Equation describes how temperature changes through a heated or cooled medium over time and space. In one dimension, the heat equation is

1D Heat Equation |

We will model a long bar of length 1 at an initial uniform temperature of 100 C, with one end kept at 100 C.

**Finite Difference Approximations**

The central and forward difference approximations for the 1st derivative wrt time and the 2nd derivative wrt space are

Forward and central difference approximations |

Substituting these relationships into the heat equation and rearranging gives an equation that describes the temperature u at position x along the bar and time t+Δt.

Equation 1 |

**Boundary Conditions**

Now lets define the initial and boundary conditions. The left-hand side (i.e. x=0) of the bar is kept at a fixed temperature of 100 C , while the initial temperature is 0 C.

We also need a boundary condition on the right hand side of the bar at x=1. The rate of change of temperature with respect to distance on right hand side of the bar is 0.

A central difference approximation to this boundary condition is

Rearranging this gives

Equation 2 |

Equation 3 |

Equation 4 |

Equation 4 describes the boundary condition on the right-hand side of the bar in a form that can implemented in Excel

**Excel Implementation**

To summarize, now we have

- Equation 1 - the finite difference approximation to the Heat Equation
- Equation 4 - the finite difference approximation to the right-hand boundary condition
- The boundary condition on the left u(1,t) = 100 C
- The initial temperature of the bar u(x,0) = 0 C

Rows represents the distance along the bar, with time increasing as you go down.

If you want the Mathcad implementation, then click here.

## 4 comments:

Very Elegant and useful work !!

Sorry I have a problem with your finite difference derivative approximation with respect to space (delta x). Could you please prove how this equation is obtained through differentiation of the function du/dt (when the function u is not physically-stated in the first instance)?

Hey! I've noticed when you make the diffusity too high of the timesteps too small or too large it starts to behave funny. May I ask why this is as I'm not learned enough to understand. Thanks.

*or

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