The heat equation solution using the MOL

The classic heat-transfer PDE starts with the consideration of a fictitious slab of length 1 unit, initially maintained at a uniform temperature of 100 units at time t=0. The surface temperature is abruptly changed to 0 units such that for all t>=0, the temperature at location x=0 is 0, thus defining the boundary condition at x=0. An insulated boundary condition is supplied at the other end, with x=1. The PDE that defines this heat transfer problem, along with the initial and boundary conditions can be stated as follows:

The boundary conditions are:

The initial conditions are:

            

The discretization begins by dividing the slab of length 1 unit into equal sub-divisions, say N=10. Thus, the slab is divided into sections of length δ=0.1 unit each. With 10 divisions, the slab thus gets i+1 nodes, starting with i=1. At every instant of time, the condition at x=0 is provided as zero. Using the Method of Lines, the ODE’s then become

The boundary condition at x=0 is

                                       

Using the finite difference method, the boundary condition at node i=11 can be approximated as follows:

From the above expression, u11 can be calculated. Having converted the PDE into a series of ODE’s, any numerical integration technique can be used for the solution.