Exploring the solutions of a financial bubble model via a new fractional derivative

This study presents a fractional mathematical model that explains how behavioral and social contagion in the market, can explain the bubble, collapse, and stability phases of financial bubbles. We study the proposed model via a new fractional derivative in the framework of the Caputo derivative involving a modified generalized Mittag-Leffler function (MLF). Furthermore, we use the Schauder and Banach fixed point theorems (FPTs) to prove the existence and uniqueness (E&U) of the solution of the model. Moreover, we discover the equilibrium point and identify the nullcline points of the suggested model. Then we use the Lyapunov function to investigate the global stability of the discovered equilibrium point at certain criteria, leading to the discovery of a globally stable solution. To obtain numerical results, we use the fractional Adams-Bashforth technique of order 3. We also analyze the residual error to evaluate the correctness of the proposed method. After that, we perform simulations with different parameter values and fractional orders to show the applicability of the method in different contexts. Additionally, our results can be applied to the fractional generalized Atangana-Baleanu-Caputo (GABC), Atangana-Baleanu-Caputo (ABC), and Caputo–Fabrizo-Caputo (CFC) derivatives as special cases at certain parameters. The results confirm that the technique can produce accurate answers in many settings.