Numerical simulation of a fractional stochastic delay differential equations using spectral scheme: a comprehensive stability analysis

Abstract

The fractional stochastic delay differential equation (FSDDE) is a powerful mathematical tool for modeling complex systems that exhibit both fractional order dynamics and stochasticity with time delays. The purpose of this study is to explore the stability analysis of a system of FSDDEs. Our study emphasizes the interaction between fractional calculus, stochasticity, and time delays in understanding the stability of such systems. Analyzing the moments of the system’s solutions, we investigate stochasticity’s influence on FSDDS. The article provides practical insight into solving FSDDS efficiently using various numerical techniques. Additionally, this research focuses both on asymptotic as well as Lyapunov stability of FSDDS. The local stability conditions are clearly presented and also the effects of a fractional orders with delay on the stability properties are examine. Through a comprehensive test of a stability criteria, practical examples and numerical simulations we demonstrate the complexity and challenges concern with the analyzing FSDDEs.

Exploring the Exact Solution of the Space-Fractional Stochastic Regularized Long Wave Equation: A Bifurcation Approach

“…Likewise, the family of unbounded orbits in green will approach the unbounded orbit in pink when f approaches zero. Hence, for the later orbit, the solution in (27) becomes…”

“…Using Equation ( 8), we obtain α = 0.01041666667 and β = 0.07500000000. Since α > 0 and β > 0, Equation ( 33) has solutions given by ( 19), ( 21), ( 23), ( 25), (27), and (29).…”

“…Using both fractional and stochastic generalizations in one model offers a more flexible tool for modeling systems that exhibit both memory effects and randomness. Research on stochastic fractional differential equations (SFDEs) is a growing area in applied mathematics, e.g., [20][21][22][23][24][25][26][27].…”

“…Finding solutions for the SFRLW Equation ( 2) is equivalent to solving the dynamical system (9). Thus Lemma 1 can be used to obtain the solutions depicted in Figures 1b and 2a,b from ( 19), ( 21), ( 23), ( 25), (27), and (29).…”

“…By setting p = 1 and ρ = 0, the solutions ( 19), ( 21), ( 23), ( 25), (27), and ( 29) give new solutions for the deterministic Equation (2) (RLW) with integer-order derivatives.…”