Finite-Difference Models of the Heat Equation

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Overview

This page has links to MATLAB code and documentation for finite-difference solutions the one-dimensional heat equation

$$\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$$

where $u$ is the dependent variable, $x$ and $t$ are the spatial and time dimensions, respectively, and $\alpha$ is the diffusion coefficient.

The zip archive contains implementations of the Forward-Time, Centered-Space (FTCS), Backward-Time, Centered-Space (BTCS) and Crank-Nicolson (CN) methods. In addition a more flexible version of the CN code is provided that makes it easy to solve problems with mixed boundary conditions, for example,

$$\left.\frac{\partial u}{\partial x}\right|_{x=0} = g + h u$$

where $g$ and $h$ are parameters of the boundary condition and can be time-dependent. All of the codes in this archive use a uniform mesh and a uniform diffusion coefficient, $\alpha$.

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Code archives

The following zip archives contain the MATLAB codes

• 1D Finite-difference models for solving the heat equation
• Code for direction solution of tri-diagonal systems of equations appearing in the the BTCS and CN models the 1D heat equation.

Code documentation

Forward-Time, Centered-Space in one space dimension

• FTCS in a Nutshell
• FTCS lecture slides

Backward-Time, Centered-Space in one space dimension

• BTCS in a Nutshell
• BTCS lecture slides

Crank-Nicolson in one space dimension

• Crank-Nicolson in a Nutshell
• Crank-Nicolson lecture slides

Implementing Boundary Conditions for Crank-Nicolson in one space dimension

• Alternative BC lecture slides

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