Neural minimization methods (NMM) for solving variable order fractional delay differential equations (FDDEs) with...

Area of Science:

• Numerical Analysis
• Computational Mathematics
• Differential Equations

Background:

• Delay differential equations (DDEs) are crucial for modeling real-world phenomena with time delays.
• Fractional delay differential equations (FDDEs) offer a more precise description of complex dynamics.
• Existing numerical methods may lack accuracy or efficiency for solving FDDEs.

Purpose of the Study:

• To introduce novel neural minimization (NM) techniques for solving FDDEs.
• To apply Chebyshev simulated annealing neural network (ChSANN) and Legendre simulated annealing neural network (LSANN) for numerical solutions.
• To enhance accuracy by reducing mean square error (MSE) using Chebyshev and Legendre polynomials with simulated annealing (SA).

Main Methods:

• Development and application of ChSANN and LSANN for solving DDEs and FDDEs.
• Integration of Chebyshev and Legendre polynomials with simulated annealing (SA) to optimize solutions.
• Utilizing neural minimization (NM) principles to achieve accurate numerical approximations.

Main Results:

• The proposed ChSANN and LSANN schemes provide accurate numerical solutions for FDDEs.
• Computational experiments demonstrate significant reduction in mean square error (MSE).
• Graphical and numerical analyses confirm the high accuracy and efficiency of the developed methods.

Conclusions:

• ChSANN and LSANN are effective and efficient techniques for solving fractional delay differential equations.
• The methods offer superior numerical approximations compared to existing approaches.
• The proposed schemes are readily implementable in software like Mathematica or MATLAB.