Numerical study for the fractional model of banks' competition using two efficient computational methods

In this paper, we examine and analyze how Egyptian banks compete for profits. Four fractional differential equations make up this model. As a first step toward mitigating its sudden negative impact, we present the appropriate optimal control technique for bank profits during crises (for instance, the Covid-19 crisis). The proposed system is then solved using the spectral collocation method (SCM). This approach efficiently solves the model (with Caputo-sense of fractional derivative) by approximating the solution using Gegenbauer wavelet polynomials (GWPs). In addition, we consider the same model but with another type of derivative known as Caputo-Fabrizio (CF), and use its properties to convert the proposed model into a system of fractional integral equations which are numerically estimated with the help of the Simpson's-\(1/3\) rule as an efficient numerical technique for integration. By contrasting the findings of the given techniques with those of the RK4 method, we can confirm their efficiency and accuracy. The results show that the techniques are efficient tools for simulating the solution to such a problem. The sudden drop in bank profits that occurred in 2020 as a result of the Covid-19 crisis may have been offset by the control mechanism in place. Special attention is given to studying the effect of some parameters in the system to provide a complete numerical simulation of the system which can be used in making decisions related to banking operations.