Power and Mittag–Leffler laws for examining the dynamics of fractional unemployment model: A comparative analysis

Unemployment is a major problem worldwide and is one of the key factors determining a nation’s economic status. The issue of unemployment is made more difficult globally by the ongoing rise in labor force participation and the scarcity of job positions. In this work, we study the unemployment model with two distinct fractional-order derivatives: the Caputo operator and the Atangana–Baleanu operator in the sense of Caputo (ABC). These derivatives under consideration are the operators widely utilized in modeling real-world phenomena in fractional dynamics. The existence and uniqueness of the solutions to the fractional model under consideration are ascertained using the fixed-point theory. The Hyers-Ulam analysis is employed to determine stability. For the numerical results, we present an Adams-type predictor–corrector (PC) technique for Caputo derivative and an extended Adams Bashforth (ABM) method for Atangana–Baleanu derivative. The outcomes achieved with the Atangana–Baleanu–Caputo and Caputo derivatives are identical to those of the regular case when fractional order $\nu =1.00$. However, the results obtained change slightly as fractional order assumes values smaller than one, and this variation becomes most noticeable when the fractional order $\nu <0.72$. This is because of the fractional derivative definitions’ underlying kernel. It is shown that the Mittag–Leffler kernel derivative provides better results for smaller fractional orders.