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

Highlights

• A derivative with random walk and non-random walk properties.

• A derivative with Markovian and non-Markovian properties.

• A derivative with Gaussian and Non-Gaussian properties.

• Semigroup principle failures to capture more natural phenomena.

• A non-validity of the index-law in fractional calculus.

Abstract

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.

Introduction

The concept of fractional differential and integration is one of the most used in the field of applied mathematics and mathematics as they are able to include memory effect and filter effect into partial or ordinary differential equations describing real world problems [[1], [2], [3], [4], [5], [6], [7]]. One of the law that was posed by some Platonian mathematicians [8] is that these non-local operators will be considered as fractional derivative if they satisfy the semigroup principle as it is well known that they play very important rule for evolution equations. In fact we shall recall that the semi-group involves in this problem is the strongly continuous on-parameter-semigroup, which is the generalization of the exponential function [[9], [10]]. We shall mentioned that, this concept while working with classical differential operator, it can be used to get the solution of a large class but not all problems commonly known as evolutions equations. This class of differential equations appear in many field of science, we can name physics, biology, chemistry, engineering and to cite few, economics [[9], [10]]. They generally portray by initial value problems, within the context of semi-group, the evolution is broken down into transitional steps, thus the system will evolve from step 1 to step 2, and then from step 2 to step 3 and so on. Thus while dealing with probability theory, it is nowadays well established that the semigroup are directly link to the well-known Markov process [[11], [12]]. We shall recall that, a Markov process is a stochastic model portraying a structure of conceivable actions in which the probability of each action depends only on the step attained in previous action. Keeping in mind that, the process is also referred to as a memoryless is also in a layman terms a process during which one can make predictions for the future even using only the present state thus one can know the full history [[11], [12]]. However, we can agree that, many physical problems that occur in a non-control environment rather follow a non-Markovian process than Markovian. Thus, we will observe that the nature produces more non-Markovian processes than Markovian. Let us present some examples of non-Markovian processes: If for instance, car failures depend upon the whole lifetime of a car, taking into account facts like wear out, instead of just the present condition of the car, the process is nothing more than a non-Markovian process. Another daily example for those who travel via plane is the journey from one airport to another, when we have the ticket, we see the duration of the journey however, if there is a sudden bad weather, the journey can be longer than predicted that is a non-Markovian process. Another common example is that, if tomorrow technology level of warehouse relies on long-term periodic conditions and not today technology, thus the process is non-Markovian. Let us check within the field of fractional calculus, one will notice that, the fractional Brownian motion having Hurst index deviating from $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$, is nothing more than a continuous-time procedure, which performs locally as a Brownian motion, nevertheless, portray a long-range dependence. One will find a well-established application of this in biology, climatology, finance, geohydrology, finance, among many others [[13], [14], [15], [16], [17], [18], [19], [20]]. Swimming within a river can be a good example of a non-Markovian process, when swimmer stops; he will obviously feel a mass of water hitting him from behind after a short time, noting that he induces the movement of mass of water while swimming, this wave is thus a non-Markovian property. We now start asking question about some statements made within the field of fractional differentiation and integration. If these operators must satisfy semi-group properties that are directly linked to the Markovian processes, how it is that in several papers published by the same authors, they speak about the fractional differential operators able to portray the long-range dependence, which is directly linked to the non-Markovian processes? Must a fractional differential operator satisfy semi-group principle while they are non-Markovian? In this paper, we will check with care and answer some of those questions. We must first realize that, evolution equations generated with local operator of differentiation are exponentially stable, which is not the case for those evolution equations generated by the well-known and used Riemann–Liouville fractional derivative. To express this in mathematical terms, let us consider a simple partial differential equation, the heat equation.

$$\left\{\begin{array}{c}{\partial }_{t}u\left(x,t\right)={\partial }_x^{2}u\left(x,t\right),x\in \left(0,1\right),t>0\hfill \\ u\left(x,t\right)=0,x=1or0,t>0\hfill \\ u\left(x,t\right)=u\left(x,0\right),x\in \left(0,1\right),t=0\hfill \\ X=L^2\left(\left(0,1\right);R\right).\hfill \end{array}\right.$$

We let the second order partial derivative operator to be A with a domain

$$D\left(A\right)=\left\{u\in H^2\left(\left(0,1\right);R\right)|u\left(0\right)=u\left(1\right)=0\right\}.$$

With of course $H^2$ being the Hardy, space. With this, the above equation is converted to

$$\left\{\begin{array}{c}\hfill \dot{u}\left(t\right)=Au\left(t\right)\hfill \\ \hfill u\left(0\right)=u_0.\hfill \end{array}\right.$$

The exact solution ought to be

$$u\left(t\right)=u_{0}exp\left(tA\right).$$

To be rigorous a meaning must be given to the exponential of $At$. Therefore as a function of $t$, $exp\left(At\right)$ is of course a semigroup of operators from the set $X$ to $X$, which clearly takes the initial state $u_0$ at the time $t=0$ to the state $u_{0}exp\left(tA\right)$ at time $t$. The operator $A$ is said to be the infinitesimal generator of the semigroup. In the next section, we verify evolution equations with Riemann–Liouville in Caputo sense, Caputo–Fabrizio and Atangana–Baleanu derivatives.