This simple script illustrates how a "march-in-time" solution works. The for loop handles the spatial physics, while the while loop advances the time. The plot generated will show the temperature profile decaying exponentially from the left wall towards the right, eventually reaching a steady state. Lesson 2: Two-Dimensional Steady-State Conduction Moving to two dimensions increases complexity. The governing equation is the Laplace equation: $$\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0$$
$$q = -k \frac{dT}{dx}$$
% Update boundary conditions T(1) = 100; % Left wall stays at 100C T(end) = T(end-1); % Insulation (zero gradient) max_iter = 10000
% Iterative Solver Parameters error_tol = 1e-4; max_iter = 10000; error = 1; iter = 0; error = 1
% Visualization imagesc(T); colorbar; title(['Temperature Distribution (Iter: ', num2str(iter), ')']); colormap('jet'); iter = 0