Heat Equation

Finally, the initial boundary value problem for the heat equation

\[\frac{\partial}{\partial t} u(x,t) + \frac{\partial^2}{\partial u^2} u(x,t) = 0 \quad (\ast)\]

on a metric graph Γ under Neumann-Kirchhoff conditions is approximated.

Consider a lollipop graph that can be constructed using the predefined method metric_lollipop_graph

Γ = metric_lollipop_graph()
plot_graph_3d(Γ)

As initial condition, we choose a model initial condition that has compact support on randomly chosen, edge of Γ and is zero elsewhere. A routine to assemble this initial condition is implemented in

u0 = model_initial_condition(Γ)

5-element Vector{Function}:
 #17 (generic function with 1 method)
 #17 (generic function with 1 method)
 #16 (generic function with 1 method)
 #17 (generic function with 1 method)
 #17 (generic function with 1 method)

The solution of $(\ast)$ can be simulated for $t \in [0,T]$ by calling

T = 1
animate_diffusion(Γ, u0, T)

Lets go for fractional diffusion on a tree!

Γ = metric_tree_graph()
u0 = model_initial_condition(Γ)
T = 3
animate_diffusion(Γ, u0, T, α = 0.1)