A novel approach for stochastic solutions of wick-type stochastic time-fractional Benjamin–Bona–Mahony equation for modeling long surface gravity waves of small amplitude

Abstract

Objectives: In the paper, two new reliable analytical methods have been devised for getting new exact analytical solutions of wick-type stochastic time-fractional Benjamin-Bona-Mahony (BBM) equation. Moreover, the Hermite transform andinverse Hermite transform have been utilized for converting fractional stochastic differential equation to deterministic fractional partial differential equation and vice versa respectively. Here for reducing fractional partial differential equations (FPDE) to the ordinary differential equation (ODE), fractional complex transform has been utilized.Methods: The authors have used a newly proposed method and Kudryshov method for getting the solutions for wick-type stochastic time-fractional Benjamin-Bona-Mahony (BBM) equation.Results: By using two reliable methods, here, the authors find the new exact solutions for the governing equations.Conclusion: Two new approaches to find solutions of the aforementioned equation have been established. Also, the new exact solutions have been obtained for stochastic differential equation by using two methods.