Analysis and simulation of diffusion equation on star graph with/without temporal loads

We analyse classical and temporally loaded diffusion equations on a metric star graph and discuss possible real-world aspects. We establish the existence and uniqueness of the temporally loaded diffusion equation on a metric star graph using eigenfunction expansions and obtain some apriori estimates. Numerical schemes for considered problems are derived by finite difference approximations. The transmission conditions impose challenges by developing an additional difference scheme at the junction. Alongside, the approximation of loaded diffusion equation leads to loaded difference schemes, and appropriate solution is appointed using the superposition property of a system of linear algebraic equations to deal with loaded terms in the scheme. Thereafter, the unique solvability of difference schemes is derived. The unconditional stability of proposed schemes for classical and temporally loaded diffusion equations on metric star graphs is proved via the discrete energy method. The schemes are convergent with significant rate of O(Δt+max1≤i≤kΔxi2). The consideration of metric star graph for these problems is unprecedented and approaches used for solving loaded diffusion equation and in subsequent analyses are novel. The paper concludes theoretical findings of proposed schemes by showcasing compatible numerical results from several experiments over test problems results from several experiments over test problems.