The One-Dimensional Heat Equation: Analytical Solution, Finite-Difference Schemes and Applications to Heat Conduction

Abstract

Background. The heat equation is the prototype of parabolic partial differential equations and underlies models of heat conduction, diffusion, financial mathematics and image processing.

Methods. We derive the one-dimensional heat equation from Fourier's law and energy conservation, construct its analytical solution by separation of variables, and discretise it with three finite-difference schemes: the explicit forward-time centred-space (FTCS) scheme, the implicit backward Euler scheme and the Crank–Nicolson scheme. Consistency and stability are analysed by von Neumann's method, and the schemes are tested on a problem with a known exact solution.

Results. The measured convergence orders agree with theory: FTCS and Crank–Nicolson are second order in the mesh width (observed order 2.00), whereas backward Euler is first order in time (observed order 1.02–1.08). FTCS is stable for r ≤ 0.5 and diverges for r = 0.6 (maximum amplitude about 2e+08 after 67 steps with discontinuous initial data), while the implicit schemes remain stable for r = 4.

Conclusions. For smooth problems Crank–Nicolson offers the best accuracy per unit of computational work; for non-smooth data, backward Euler is the safer choice. A worked application to the cooling of a steel rod shows how the analytical solution gives the characteristic cooling time of about 35 minutes.