Numerical solution of fractional delay differential equation by shifted Jacobi polynomials

ABSTRACT

In this paper, the fractional delay differential equation (FDDE) is considered for the purpose to develop an approximate scheme for its numerical solutions. The shifted Jacobi polynomial scheme is used to solve the results by deriving operational matrix for the fractional differentiation and integration in the Caputo and Riemann–Liouville sense, respectively. In addition to it, the Jacobi delay coefficient matrix is developed to solve the linear and nonlinear FDDE numerically. The error of the approximate solution of proposed method is discussed by applying the piecewise orthogonal technique. The applicability of this technique is shown by several examples like a mathematical model of houseflies and a model based on the effect of noise on light that reflected from laser to mirror. The obtained numerical results are tabulated and displayed graphically.

KEYWORDS:

• fractional differential equations
• delay differential equations
• operational matrix
• shifted Jacobi polynomials
• Newton's iterative method

2010 AMS SUBJECT CLASSIFICATIONS:

• 34A08
• 33C45

Acknowledgments

The authors would like to express their sincere thanks to the Editor in Chief, Associate Editor's and anonymous reviewers for helpful comments and suggestions to improve the quality of this manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work was supported by National Board for Higher Mathematics, Mumbai, India under the grant No: 2/48(5)/2013/NBHM (R.P.)/RD-II/688 dt 16.01.2014.