New probabilistic solutions of the generalized shallow water wave equation with dual random dispersion coefficients

Abstract

AbstractIn this paper, some exact solutions of the stochastic generalized nonlinear shallow water wave equation are investigated. This equation is important in fluid mechanics’ fields since it can model the propagation of disturbances in water and other incompressible fluids. Opposite to what is usually considered in the literature, the two dispersion coefficients of the nonlinear terms are considered dependent random quantities as a more realistic case. The modified extended-tanh function (METF) method is combined with the random variable transformation (RVT) technique to get full probabilistic solutions of the problem via computing the probability density functions (PDFs) of the solution processes. Based on the probability density function, any statistical moment of the solution can be evaluated. Through two different applications for the input random variables (dispersion coefficients), my findings are applied efficiently. Finally, numerical results are presented graphically along the spatial dimension at a certain wave speed and time. The obtained results ratify that the proposed technique is efficient and powerful for obtaining analytical probabilistic solutions for the problem.

Keywords

• Stochastic differential equations
• random coefficients
• random variable transformation
• modified extended tanh-function
• probability density function
• Marco Montemurro, Arts et Metiers ParisTech – Centre de Bordeaux-Talence, FRANCE