An Efficient Numerical Scheme for Solving a Fractional-Order System of Delay Differential Equations

Area of Science:

• Applied Mathematics
• Numerical Analysis
• Dynamical Systems

Background:

• Fractional-order delay differential equations model complex phenomena in population dynamics, neural networks, ecology, and physiology.
• Accurate numerical solutions are essential for understanding these systems.

Purpose of the Study:

• To introduce a novel implicit numerical scheme for solving fractional-order systems of delay differential equations.
• To provide a rigorous error analysis for the proposed method.

Main Methods:

• The study extends the L1 numerical scheme to develop a new implicit method.
• The method's error estimate is shown to be O(h).
• The scheme is applied to solve various non-trivial fractional-order delay differential equation systems.

Main Results:

• The proposed implicit numerical scheme achieves an error estimate of O(h).
• Numerical examples demonstrate superior accuracy compared to the fractional Adams method and the three-term new predictor-corrector method.
• The method exhibits convergence even for very small fractional orders.

Conclusions:

• The developed implicit numerical scheme offers a highly accurate and efficient approach for solving fractional-order delay differential equations.
• This method provides a valuable tool for researchers in applied mathematics and related scientific fields.
• The scheme's robustness and accuracy make it suitable for complex dynamical system analysis.