Chebyshev cardinal wavelets for nonlinear stochastic differential equations driven with variable-order fractional Brownian motion

Abstract

Abstract This paper is concerned with a computational approach based on the Chebyshev cardinal wavelets for a novel class of nonlinear stochastic differential equations characterized by the presence of variable-order fractional Brownian motion. More precisely, in the proposed approach, the solution of a nonlinear stochastic differential equation is approximated by the Chebyshev cardinal wavelets and subsequently the intended problem is transformed to a system of nonlinear algebraic equations. In this way, the nonlinear terms are significantly reduced, due to the cardinal property of the basis functions used. The convergence analysis of the expressed method is theoretically investigated. Moreover, the reliability and applicability of the approach are experimentally examined through the numerical examples. In addition, the presented method is implemented for some famous stochastic models, such as stochastic logistic problem, stochastic population growth model, stochastic Lotka–Volterra problem, stochastic Brusselator problem, stochastic Duffing-Van der Pol oscillator problem and stochastic pendulum model. As another new finding, a procedure is established for constructing the variable-order fractional Brownian motion. Indeed, the standard Brownian motion together with the block pulse functions and the hat functions are utilized for generating the variable-order fractional Brownian motion.