Study of composite fractional relaxation differential equation using fractional operators with and without singular kernels and special functions

Our aim in this article is to solve the composite fractional relaxation differential equation by using different definitions of the non-integer order derivative operator Dt , more specifically we employ the definitions of Caputo, Caputo–Fabrizio and Atangana–Baleanu of non-integer order derivative operators. We apply the Laplace transform method to solve the problem and express our solutions in terms of Lorenzo and Hartley’s generalised G function. Furthermore, the effects of the parameters involved in the model are graphically highlighted.

sciences. During the past few years, the main properties of these operators have been explored and their advantages have been extensively investigated for various practical cases [13][14][15][16][17][18].
The motion of the sphere dipped in an in-compressible viscous fluid poses a classical problem, which has many practical implications in flows of geophysical and engineering interest [19,20]. In order to linearise the Navier-Stokes equations characterising the motion of fluid, the low Reynolds number limit or slow motion assumption is considered. The problem of a sphere under the influence of gravity was first discussed independently by Boussinesq [19] in 1885 and by Basset [20] in 1888, who further introduced a special hydrodynamic force, regarding the history of the relative acceleration of the sphere, later known as Basset force. Practically, for the case when a body is immersed in a fluid, the acceleration of that body with respect to the fluid gives rise to an unsteady force that can be divided into two parts, namely the virtual mass effect and the Basset force. The Basset force deals with the viscous effects and explains the temporal-delay-in-boundary-layerdevelopment as the relative velocity changes with time [21]. It is also acknowledged as the history term.
The fractional differential equation along with the initial conditions u(0 + ) = u 0 is referred to as composite fractional relaxation equation (CFRE).
Here, D α t is an NOID operator of order α ∈ (0, 1). Moreover, in the equation a is a positive constant and a = ( 9ρ f 2ρ p +ρ f ) α , where ρ p and ρ f denote the densities of particle and fluid respectively, u(t) is the field variable, whereas the given function f (t) is presumed to be continuous.
Considering the NIOD operator in the sense of Caputo, this problem was discussed by Basset and was first solved by Boggio [25] for α = 1/2 in terms of Gauss and Fresnel integrals. Further, the solutions of the Basset problem are found in [26,27]. Moreover, Mainardi [22] solved the problem by using the Laplace transform method and expressed the solution in terms of Mittag-Leffler functions. Later on, in 2014 Anjara and Solofoniaina [28] solved the Basset problem by Adomian's method.
In solving non-integer order differential equations (DEs) using the Laplace transform method, the inverse Laplace transform is not trivial. In this regard we have to introduce some special functions. For example, Mittag-Leffler function, Robotnov and Hartley's function, Lorenzo and Hartley's generalised R function, generalised G functions etc. Such functions produce a direct solution and give important interpretations for the fundamental linear non-integer order DEs and corresponding IVPs. These functions are helpful in the solutions of the FC problems and more notably in the solution of fractional differential equations. Our aim in this article is to solve the composite fractional relaxation equation by using different definitions of NIOD operators, more specifically we employ the definitions of Caputo, Caputo-Fabrizio and Atangana-Baleanu of NIOD operators. We apply the Laplace transform method to solve the problem and express our solutions in terms of Lorenzo and Hartley's generalised G function. Moreover, to get some understanding regarding the influence of the two parameters α and a on the generalised as well as classical Basset problem, we present some diagrams for the particle's velocity, analogous to the solution of (1). For the sake of clarity, we assume a diminishing initial velocity condition and f (t) = H(t) and sin(ωt).
The manuscript is structured in six sections. Following this short introductory section, Sect. 2 describes some mathematical preliminaries. Section 3 discusses the solution of the problem employing various definitions of NIOD operators, while some special cases derived from the obtained results are reported in Sect. 4. Section 5 is devoted to the parametric analysis, and finally the useful conclusions are recorded in Sect. 6.

Mathematical preliminaries
In this section we present some basic definitions and properties of some important special functions and Caputo, CF and AB-time fractional derivative operators [29][30][31].

Special functions
As already mentioned in the introduction, for solving the NIOD DEs using the Laplace transform method, the inverse Laplace transform is not trivial. In this regard we have to introduce some special functions [32]. For example, Mittag-Leffler function [33], Robotnov and Hartley's function [34] and Lorenzo and Hartley's functions [35]. Such functions contribute a direct solution and critical insight for the fundamental linear fractional-order DEs and associated IVPs. Moreover, they are appropriate in the solutions of the problems related to FC and more importantly in the solution of NOIDEs.
In the following, we present some special functions together with their definitions, Laplace transforms and some examples.
1. Mittag-Leffler function. It is a significant function and has many applications in the field of FC. Analogous to the exponential function that arises in a natural way from the solutions of differential equations of integer order, the Mittag-Leffler function plays a similar role in the solution of non-integer order differential equations [36,37]. As a matter of fact, the exponential function is a special form of it. The Mittag-Leffler function is defined as [33] It is not difficult to observe that, for α = 1, we get Moreover, ; α > 0.

Erdelyi's function. It is the generalisation of Mittag-Leffler function and is defined as
; α, β > 0.
For α = 1 and β = 2, we have Similarly, we have and where erfc denotes the complementary error function and is defined as [32] [34] and was studied by Robotnov for application to solid mechanics. It is defined as [39]. It is defined as
Similarly, setting a = 1, β = αν yields Moreover, 6. Generalised G-function. This function was also introduced by Lorenzo and Hartley [35], it is the generalisation of the R-function. It is defined as follows: t). γ (a, s).
t p is the kernel of the derivative, "*" denotes the convolution, and H 1 (a, b), a < b, p ∈ [0, 1), then the CF-fractional derivative is given by [10,29]

Definition 2.3 Let h ∈
where N(p) is a normalisation function fulfilling the condition N(0) = N(1) = 1, k ABC (p, t) is the kernel of the derivative and E p (·) is a one-parametric form of the Mittag-Leffler function [12]. Also Furthermore, the normalisation function can be any function fulfilling the condition N(0) = N(1) = 1. For example, it could be chosen as N(p) = 1p + p (p) [40]. In the present work we choose N(p) to be identically one. Again Taking the Laplace transform of (8), we get also where δ(t) is Dirac's delta function. Moreover, and ABC a Equation (12) and Eq. (13) represent the relation between the ABC-fractional derivative and the classical derivative.

Solution of the problem
In this part, we solve Eq. (1) by customising the operator D α t (·) according to Caputo, Caputo-Fabrizio and Atangana-Baleanu.

Solution of the problem using Caputo NIOD operator
Replace the operator D α t (·) in Eq. (1) by using Definition 2.1, Eq. (9). Basset equation in the sense of Caputo derivative operator becomes Employing Laplace transform [41] to Eq. (14) and making use of the initial conditions lead to or Taking an inverse Laplace transform, we get where G is the Lorenzo-Hartley generalised G function [35].

Solution of the problem using CF-NIOD operator
Replacing the operator D α t (·) in Eq. (1) by using Definition 2.2, Eq. (11), we get Employing LT [41] to Eq. (18) and making use of the initial conditions, we get where Employing the inverse LT, after long but straightforward computation, leads to

Solution of the problem using ABC-NIOD operator
Replacing the operator D α t (·) in Eq. (1) by using Definition 2.3, Eq. (13), we get Application of LT [41] to Eq. (23) and making use of the initial conditions lead to where γ = α 1-α . Finally, employing the inverse LT, we get

Special cases
In this section, from our general solutions, we discuss two cases, namely when excitation function is H(t) and sin(ωt).

Analysis of the influence of parameters on the Basset problem
With the aim to have proper understanding associated with the influence of the parameters α and a on the classical as well generalised Basset problem, we prepare some diagrams for the velocity of the particle, analogous to the solution of (1). Further, for the sake of clarity, we assume u(0) = 0, i.e. vanishing initial velocity and f (t) = H(t). We will discuss three scenarios for α, namely α = 1/2 (the classical Basset problem) and α = 1/4, 3/4 (the generalised Basset problem . For the case when the Caputo-Fabrizio derivative is used in the fractional relaxation equation (see Fig. 2), the same trend is observed as in the case when the Caputo derivative operator is employed. Additionally, it is reported that all velocities converge to the same constant value showing that the influence of the fractional order parameter diminishes with time. From Fig. 3, it is noticed that when the Atangana-Baleanu definition of NIOD is used for increasing values of α, velocity increases, and the range of velocity is larger as compared to the case when Caputo or Caputo-Fabrizio derivative operators are employed. In Fig. 4, the comparison of velocities with three definitions of fractional derivatives is presented. It is noted that, for α = 1/4 and α = 3/4, the behaviour of velocity when Caputo and Caputo-Fabrizio derivative operators are employed is the same. On the

Conclusions
In this article, we have solved the composite fractional relaxation equation by using different definitions of NIOD operators. More specifically, we have employed the definitions NIOD operators proposed by Caputo, Caputo-Fabrizio and Atangana-Baleanu. The solution of the non-integer order differential equation is obtained by applying the Laplace transform method. Moreover, the solutions of the problem are expressed in terms of Lorenzo and Hartley's generalised G-function that is the generalisation of many special functions that arise in the solution of non-integer order differential equations. Furthermore, the effects of the parameters involved in the model of generalised and classical Basset problem are shown, and the comparison of these model in terms of different definitions of NIOD operators is done by graphical analysis. The useful conclusions are as follows: 1. When Caputo, Caputo-Fabrizio as well as Atangana-Baleanu fractional order derivative operators are used, velocities increase for increasing values of α, and after some time they attain a constant value for large times. Moreover, time to achieve that constant value is getting small for increasing values of the ratio