Investigation Of Financial Bubble Mathematical Model Under Fractal-Fractional Caputo Derivative

In this study, we proposed a novel approach for modeling the dynamics of a three-agent financial bubble using the fractal-fractional (FF) derivative of the Caputo sense. This new concept was developed to deal with the complex geometry of any dynamical system, and it utilizes both the fractional derivative for the order and the fractal term for the order of the independent variables. The model was investigated using the conformable order derivative of the Caputo operator, with a focus on the fractal dimension and fractional order. The existence and uniqueness of the solution were tested using a FF global derivative, and the approximate root of the system was calculated using the numerically iterative technique of the Newton polynomial. To verify the accuracy of the approximate root scheme, we applied the power singular law with two fractional parameters in the developed numerical technique. The curve representation of the system was also verified by applying the data for the fractals and different conformable orders. Additionally, we tested the sensitivities of the fractional parameters and dynamical system parameters by varying the parameter values. This allowed us to gain a better understanding of how changes in these parameters affect the system’s behavior and stability. As a result, this study presents an innovative and effective approach for modeling the dynamics of financial bubbles using the FF derivative of the Caputo sense. The results of this research contribute to the ongoing efforts to develop more accurate and comprehensive models of complex systems in economics and finance.