The classical fractional derivatives, such as Caputo, Riemann–Liouville and Grünwal, are based on singular kernels. On the other hand, recently, some new definitions of fractional derivative were derived based on nonsingular kernels. These derivatives include the Caputo–Fabrizio [1
] and Atangana–Baleanu fractional derivatives [2
]. These new definitions have been applied in many areas, including the groundwater flow within a confined aquifer [3
], the magnetohydrodynamic electroosmotic flow of Maxwell fluids [4
], the modeling of a financial system [5
], the modeling of various types of diseases or epidemics [7
] and many other fields [10
On the other hand, solving a fractional differential equation which is defined as an Atangana–Baleanu fractional derivative always is not an easy task. Furthermore, an analytical solution may not always possible, especially when involving variable coefficients. Hence, we need some reliable numerical methods. To overcome this problem, in [12
], the authors used fractional Adams–Bashforth methods to solve the fractional differential equation in a Atangana–Baleanu sense. The Crank–Nicholson difference method and reproducing kernel function had been applied for solving third order fractional differential equations in the sense of Atangana–Baleanu Caputo derivative [13
]. An implicit, linear and unconditionally stable finite difference method was derived for solving fractional differential equations in the Atangana–Baleanu sense [14
]. Other methods for solving special kinds of problems defined in the Atangana–Baleanu fractional derivative include the quasi wavelet approach of non-linear reaction diffusion and integro reaction-diffusion equation [15
], the modified homotopy analysis transform method for solving gas dynamics equations of arbitrary order [16
] and reproducing kernel functions [17
In this research direction, operational matrix related methods are getting considerable attention for solving fractional calculus problems; among them is the Legendre operational matrix for solving fractional differential equations in the Caputo–Fabrizio sense [18
]. Apart from that, for solving fractional differential equations defined in the Atangana–Baleanu fractional derivative, fifth-kind Chebyshev polynomials have been applied to derive operational matrices for multi-variable orders differential equations with non-singular kernels [19
]. Besides that, the shifted Legendre cardinal functions operational matrix method was developed for solving the nonlinear time fractional Schrödinger equation with variable-order defined in the Atangana–Baleanu derivative [20
]. Differently than the previous published results, here, for the first time, poly-Bernoulli polynomials which can be considered as a family of Bernoulli polynomials have been employed to derive the operational matrix. The poly-Bernoulli polynomials are derived using a generating function involving a polylogarithm function. Bernoulli polynomials have been used in solving various type fractional calculus problems, such as in [21
]. These kinds of semi-orthogonal polynomials, including Genocchi polynomials and Euler polynomials, are widely used in solving differential equations up to arbitrary orders [23
]. More specifically, we derive the new operational matrix based on poly-Bernoulli polynomials for solving the fractional differential equation in an Atangana–Baleanu sense. We also derive the delay operational matrix based on poly-Bernoulli polynomials to tackle the fractional delay differential equation in Atangana–Baleanu sense. In short, our proposed approach is able to overcome the difficulty of finding the solutions for variable coefficients of the fractional delay differential equation in an Atangana–Baleanu derivative via transforming the problem into solving a system of algebraic equations, which greatly reduces the complexity of the problem.
This paper is organized as follows. We briefly explain some preliminarie concepts, including the Atangana–Baleanu fractional derivative and our main tool used in this work—poly-Bernoulli polynomials—in Section 2
. Section 3
discusses the derivation of new operational matrix based on poly-Bernoulli polynomials for the Atangana–Baleanu fractional derivative. In Section 4
, we explain the new scheme and give some examples for solving fractional delay differential equations defined in the Atangana–Baleanu fractional derivative using our propose method via the new operational matrix based on poly-Bernoulli polynomials. Conclusions and some recommendations are highlighted in Section 5
In this work, we achieved the following results:
A new operational matrix based on poly-Bernoulli polynomials for ABC-derivative.
A new delay operational matrix based on poly-Bernoulli polynomials.
A collocation scheme via operational matrix and delay operational matrix based on poly-Bernoulli polynomials for fractional differential equations in an ABC-sense.
The poly-Bernoulli operational matrix is the general case for any integer k, while when , it reduces to a Bernoulli operational matrix. The numerical examples show that the proposed method is a powerful tool for obtaining the numerical solutions for variable coefficients of fractional delay differential equations in an ABC-sense. Hence, we suggest that the following points may be able to apply:
In this work, our delay operational matrix was limited to constant delay. Hence, for future work, it would also be interesting to study a similar problem but with proportional delay or variable delay.