Numerical analysis employing matrix spectral collocation step method for stochastic conformable differential problems stimulated by one-dimensional Brownian motion with constant discrete delays

Complex, powerful, and representative fractional SDDMs are used to model fractional-order systems of stochasticity and time delays. This study aims to construct an approximate spectral collocation scheme, numerically solving certain types of these models. Specifically, a suitable model of conformable fractional operator sense where the stochastic term is of the standard one-dimensional SBM type, and the time delay is discrete and constant, is established and solved. The proposed scheme is based on the spectral collocation method, with SLP basis functions and SLGL collocation points. To obtain the desired approximate solutions utilizing the present method, the domain under consideration is discretized into steps, and at each step, the solution is approximated using the spectral collocation technique. Simply, complicated problems are evolved into a system of algebraic equations where the unknowns are the Legendre coefficients. These equations are solved utilizing an appropriate numerical method executed by MATHEMATICA. For convenience, the convergence analysis under Lipschitz properties is presented. To validate the reality of the method, some applications of linear and nonlinear types of the suggested model are solved, and the errors are computed. Moreover, the Log-Log plots are sketched to confirm the efficiency of the presented method when more collocation points are used. The results formulated in tables, figures, and discussions illustrate the high accuracy and capability of the proposed methodology. Final remarks and future work are reviewed and debated, too.