A study on the existence of unique stable approximate solutions: fractal-fractional-based model of an energy supply–demand system

We investigate a fractional energy supply–demand system (ES–DS) model using power-law-type kernels and advanced operators called fractal-fractional operators with a couple of fractal and fractional orders. It is established that for the fractal-fractional model of the ES–DS, a solution exists and it is unique. One of the principal innovations is to conduct the stability of the fractal-fractional-based ES–DS model. Moreover, after some theorems on the existence theory, we upgrade the Adams–Bashforth (AB) method in the context of the fractal-fractional-based operators and simulate the graphs for varied data on the fractal and fractional orders when we explored the stability requirements in four variants to approach the required numerical solutions. Next, by removing the fractal order, we transform our fractal-fractional-based ES–DS model to a fractional Caputo-type model, and then, a numerical methodology known as the Taylor operational matrix method is applied to simulate new graphs and compare them to previous fractal-fractional-based model.