Abstract
In this work, we derive the operational matrix using poly-Bernoulli polynomials. These polynomials generalize the Bernoulli polynomials using a generating function involving a polylogarithm function. We first show some new properties for these poly-Bernoulli polynomials; then we derive new operational matrix based on poly-Bernoulli polynomials for the Atangana–Baleanu derivative. A delay operational matrix based on poly-Bernoulli polynomials is derived. The error bound of this new method is shown. We applied this poly-Bernoulli operational matrix for solving fractional delay differential equations with variable coefficients. The numerical examples show that this method is easy to use and yet able to give accurate results.
Highlights
The classical fractional derivatives, such as Caputo, Riemann–Liouville and Grünwal, are based on singular kernels
A new operational matrix based on poly-Bernoulli polynomials for ABC-derivative
A new delay operational matrix based on poly-Bernoulli polynomials
Summary
The classical fractional derivatives, such as Caputo, Riemann–Liouville and Grünwal, are based on singular kernels. 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]. We derive the new operational matrix based on poly-Bernoulli polynomials for solving the fractional differential equation in an Atangana–Baleanu sense. We derive the delay operational matrix based on poly-Bernoulli polynomials to tackle the fractional delay differential equation in Atangana–Baleanu sense.
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