Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties

We presented an analysis of evolutions equations generated by three fractional derivatives namely the Riemann-Liouville, Caputo-Fabrizio and the Atangana-Baleanu fractional derivatives. For each evolution equation, we presented the exact solution for time variable and studied the semigroup principle. The Riemann-Liouville fractional operator verifies the semigroup principle but the associate evolution equation does not. The Caputo-Fabrizio fractional derivative does not satisfy the semigroup principle but surprisingly, the exact solution satisfies very well all the principle of semigroup. However, the Atangana-Baleanu for small time is the stretched exponential derivative, which does not satisfy the semigroup as operators. For a large time the Atangana-Baleanu derivative is the same with Riemann-Liouville fractional derivative, thus satisfies semigroup principle as an operator. The solution of the associated evolution equation does not satisfy the semigroup principle as Riemann-Liouville. With the connection between semigroup theory and the Markovian processes, we found out that the Atangana-Baleanu fractional derivative has at the same time Markovian and non-Markovian processes. We concluded that, the fractional differential operator does not need to satisfy the semigroup properties as they portray the memory effects, which are not always Markovian. We presented the exact solutions of some evolutions equation using the Laplace transform. In addition to this, we presented the numerical solution of a nonlinear equation and show that, the model with the Atangana-Baleanu fractional derivative has random walk for small time. We also observed that, the Mittag-Leffler function is a good filter than the exponential and power law functions, which makes the Atangana-Baleanu fractional derivatives powerful mathematical tools to model complex real world problems.