A Numerical Method for Solving Systems of Fractional Stochastic Delay Differential Equations

Abstract

This paper presents a novel stepwise numerical method based on Fibonacci wavelets for solving fractional stochastic delay differential systems. Such systems arise in numerous applications where memory effects, randomness, and time delays coexist, and their accurate long‐time numerical treatment remains challenging. The proposed method combines the fractional derivative operational matrices associated with the Fibonacci wavelet basis with a stepwise discretization strategy, thereby integrating high spectral accuracy with stable computation over extended time intervals. First, the arbitrary‐order fractional derivative operational matrix for the Fibonacci wavelet basis functions is constructed via block pulse functions. Then, the original problem is solved on successive subintervals using a recursive stepwise procedure that efficiently incorporates the delay term. At each step, the solution is approximated by a Fibonacci wavelet expansion; applying collocation conditions together with the operational matrix reduces the problem to a system of algebraic equations. A rigorous convergence analysis and error bounds of the proposed method are provided. Finally, several numerical examples and comparisons with existing methods demonstrate the efficiency and robustness of the algorithm. The results confirm that the method achieves remarkable accuracy while maintaining computational efficiency for problems defined on long time domains.