On the recurrent computation of fractional operator with Mittag-Leffler kernel
Introduction
The usage of fractional-order derivatives in differential models was proven to be effective while modeling various phenomena, particularly, in hydrology, economics, and medicine [17], [8], [10], [28], [18]. Such derivatives are integro-differential operators that generalize the concept of derivative and can have different forms when various integral kernels are used. Usually, two types of kernels are distinguished: singular ones (such as of the classical Caputo derivative [19], [26]) and non-singular ones (such as of the Caputo-Fabrizio derivative [9]). One of the latest developments in the field of fractional derivatives with non-singular kernels is the Atangana-Baleanu derivative [2]. Despite the usage of fractional derivatives with singular kernels in differential models poses some restrictions on problems' initial conditions [32], their effective applications are also reported (e.g. [18], [5], [15], [31]).
The main methods used to numerically solve time-fractional differential equations involving the Atangana-Baleanu derivative are the finite-difference method (e.g. [33]) and spectral techniques [22], [21], [20] that are mostly applied to one-dimensional problems. As fractional derivatives are, first of all, integral operators, their discretizations in many cases have the form of sums with the number of terms proportional to grid size (e.g. [34]). When time-fractional differential equations are solved, on one time step it leads to linear computational complexity with respect to time step number. As for traditional differential equations such complexity has constant order, simulation using their fractional counterparts, allowing describing broader classes of processes, however, is significantly slower.
Two approaches that allows decreasing the computational complexity to the close-to-constant order while calculating the values of fractional derivatives can be singled out. The first one consists in the neglecting of contribution of some previously obtained solutions [14], [13] while the second one is based on the approximation of integral kernels [1], [4], [6], [7].
Common methods for computing the values of the Mittag-Leffler function, which is involved in an integral kernel of the Atangana-Baleanu derivative, use the combination of its series and integral representations' approximations (e.g. [30]). Direct implementation of such methods while solving time-fractional differential equations still yields linear order of computational complexity when iteration number increases.
To lower computational complexity we propose a new algorithm based on a variables separation technique [6], [7] that was proven to be effective for fractional derivatives with singular kernels. The novelty here lies in the application of variables separation to the series and the integral representations of the Mittag-Leffler function with their subsequent incorporation into L1-finite-difference approximation of the Atangana-Baleanu derivative. The main advantage of this scheme compared with straightforward finite-difference approximations or spectral techniques lies in the close-to-constant computational complexity of determining a single derivative's value in the process of sequential computations when solving fractional differential equations. This allows speeding up simulations on large time interval.
The paper is organized as follows. In Section 2 we present two computational schemes for the Atangana-Baleanu derivative based on variables separation in the series and the integral representations of the Mittag-Leffler function. In Sections 3, 4 the estimates of accuracy and performance for these schemes are derived. Section 5 presents an algorithm for incorporation of the computational procedure into the finite-difference scheme for the time-fractional convection-diffusion equation. Results of computational experiments aimed at the testing of accuracy and performance estimates given in Sections 3, 4 are presented in Section 6. Results on testing the overall accuracy of the Atangana-Baleanu derivative values computation are given in Section 7. Testing results while solving the convection-diffusion equation are given in Section 8.
Section snippets
Approximation of the Atangana-Baleanu derivative
We consider the Atangana-Baleanu derivative [2] of the continuously differentiable function f of the order α that has the following form: where is the Mittag-Leffler function [24].
Approximating in (1) with the first order of accuracy we obtain for (τ is the time step length, is the time step number) the following L1-finite-difference representation
Accuracy and performance estimates for truncated series expansion
The estimate of the error of series (4) truncation can be obtained the following way.
Let us denote where is the falling factorial.
As the series upon k in (4) are alternating for , we can state that
As increases with the increase of k, the lower bound will have the form
Accuracy and performance estimates for integral representation
We estimate an error of the approximation (7) of the integral (6) the following way.
Let us denote
As the first derivative of has singularity at , we will use the accuracy estimate for the midpoint quadrature in the form [12]. As is decreasing for large enough v, we assume that the integrated function has
Incorporation into finite-difference scheme for fractional convection-diffusion equation
Let us consider a convection-diffusion equation in the form (e.g. [33]) with the initial and boundary conditions
The necessity of posing zero initial conditions for the equation (17) comes from the properties of the Atangana-Baleanu derivative [34]. However, even with this restriction, the equation (17) is able to describe such mass-transfer processes as soil contamination after some accident in which substance suddenly
Testing of accuracy and performance estimates for computation
To test the obtained error estimates (9), (14) we performed a series of computational experiments evaluating using the truncated series (4) and the integral representation's approximation (7). We compare in such a way obtained values of with the values calculated by numerical integration on the base of recursive subdivision algorithm [27]. For this purpose, the values of varied in the range for and , , . Then, for and the
Testing of accuracy while calculating values of the Atangana-Baleanu derivative
To test the accuracy of the overall procedure for the Atangana-Baleanu derivative's values computation, we consider computing the values of for . As, from the best of our knowledge, an analytical form of the Atangana-Baleanu derivative of power function is obtained only for integer values of β, for the sake of comparison, we calculate them on the base of recursive subdivision quadrature [27]. The obtained approximations of the derivative's values for ,
Testing of accuracy and performance while solving time-fractional convection-diffusion equation
For testing purposes we consider three cases of the function in the equation (17) studied previously in [33] for which the equation has the following analytic solutions :
Following [33], we consider an error estimate of the numerical solution with respect to the grid parameter M in the form
Conclusions
Incorporation of argument's series expansion with subsequent variables separation into a computational scheme for the Mittag-Leffler function allowed us to construct an efficient algorithm for computing the values of the Atangana-Baleanu derivative. The obtained theoretical and experimental estimates of its accuracy and performance show that it allows reducing computational complexity while solving time-fractional convection-diffusion equation without significant loss in accuracy. This is due
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