Analysis of fractional differential equations in higher dimensional complex system with proportional delay, discrete term, and nonlinear coupling: applications in a biological model and porous media

This manuscript considers a general coupled system of n-equations with the Caputo-Fabrizio derivative. We establish the existence theory for the system using fixed-point analysis. Sufficient conditions are derived to ensure the existence and uniqueness of the solution. Additionally, a real-life example of a two-species household model is demonstrated through our analysis. Furthermore, we present graphical solutions corresponding to various fractional orders by employing an advanced numerical algorithm. Three distinct sets of fractional order values are used in this context. An additional example of a fractional porous media equation is also provided to demonstrate the proposed method, which can be effectively solved using the outlined approach.