An Operational Matrix Method Based on Poly-Bernoulli Polynomials for Solving Fractional Delay Differential Equations

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.

A Specific Method for Solving Fractional Delay Differential Equation via Fraction Taylor’s Series

“…According to [15], the exact solution of ( 7) is expanded by u(t) � 􏽐 ∞ j�1 f(t j ζ j (t)). e approximate 2 Mathematical Problems in Engineering solution of ( 7) is expanded by u N (t) � 􏽐 N j�1 f(t j )ζ j (t).…”

“…In [14], Abdulnasir adopts a new simple and effective new algorithm based on one of the Appell polynomials, namely, Genocchi polynomials, to solve the generalized Pantograph equations, the FDDE with neutral terms, and delay differential system with constant and variable coefficients. In [15], Phang uses an operational matrix method for solving the FDDEs. In [16], Salem gives the existence and uniqueness of the coupled system of nonlinear fractional Langevin equations (FLE) with multipoint and nonlocal integral boundary conditions (NIBCs).…”

An Efficient Numerical Scheme for Solving Multiorder Tempered Fractional Differential Equations via Operational Matrix

“…To date, various numerical or analytical methods were derived to find the solution for different fractional calculus problems, such as [11][12][13]. On top of that, the operational matrix method via different types of the polynomial is one of the common numerical schemes which had been widely used in solving various types of fractional calculus problems, such as the poly-Bernoulli operational matrix for solving fractional delay differential equation [14], poly-Genocchi operational matrix for solving fractional differential equation [15], Jacobi wavelet operational matrix of fractional integration for solving fractional integro-differential equation [16], and Fibonacci wavelet operational matrix of integration for solving of nonlinear Stratonovich Volterra integral equations [17]. Recently, the operational matrix method had been successfully extended to solve other fractional operator problems, such as solving Prabhakar fractional differential equation [18].…”

Application of a Novel Collocation Approach for Simulating a Class of Nonlinear Third-Order Lane–Emden Model

“…ese extensive works on the solutions of PD models open the door to exploit the wider applications of similar models of the reallife phenomenon. For more details regarding the applications and techniques for solving the PD models, one may refer to [20][21][22][23] and the references therein.…”

“…Proof. If we replace each term of (8) with its corresponding approximation given by equations ( 20)- (23) and by substituting x � x k collocation points defined previously, (33) can be written in matrix form as…”