The Method of Lines

Partial differential equations can be approximated as ODE’s using a technique known as the Method of Lines (MOL). The heat transfer of a metal slab that has a temperature as a boundary condition on one end can be modeled using a PDE. The solution to the problem, which is to determine the temperature profile of the slab, is a function of space and time. The spatial profile changes along the length of the slab, and thus every location of the slab has a different temperature as a function of time.

The PDE for a 1-D slab is represented by

 

By discretizing the spatial derivative as nodes of equal distance, the behavior of the slab can be approximated by an ODE as follows:

 

δ is the distance between each node. By choosing a discrete number N over a length L, the nodal distance can be calculated as L/N. The temperature profile at each node depends on the preceding and succeeding nodes (and the current node) in order to “march” from one end of the slab to the other. Thus, at each instant of time, a series of ODE’s can be written to represent the discretized version of the PDE.