Linearized Crank–Nicolson method for solving the nonlinear fractional diffusion equation with multi-delay

ABSTRACT

This paper is concerned with numerical solution of the nonlinear fractional diffusion equation with multi-delay. The studied model plays a significant role in population ecology. A linearized Crank–Nicolson method for such problem is proposed by combing the Crank–Nicolson approximation in time with the fractional centred difference formula in space. Using the discrete energy method, the suggested scheme is proved to be uniquely solvable, stable and convergent with second-order accuracy in both space and time for sufficiently small space and time increments. Several numerical experiments for solving the delay fractional Hutchinson equation and two real problems in population dynamics are provided to verify our theoretical results.

KEYWORDS:

• Delay fractional differential equation
• Crank–Nicolson method
• multi-delay system
• Hutchinson equation
• stability and convergence

2010 AMS SUBJECT CLASSIFICATIONS:

• 35K58
• 65M06
• 65M12

Acknowledgments

The author would like to thank the referees for the careful reading of the paper and the thoughtful comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.