Solving nonlinear stochastic differential equations via fourth-degree hat functions

Abstract

The article focuses on introducing novel fourth-degree hat functions (FDHFs) for constructing new operations matrices used to find numerical solutions for nonlinear stochastic differential equations (NSDEs). The technique’s effort is summarized by utilizing FDHFs to construct operations matrices that transform the given problem into nonlinear algebraic equations. The advantage of this technique lies in its simplicity for calculating the unknown coefficients of the function’s approximation without requiring any integration. As a result, the proposed approach incurs low computational expenses. Additionally, an error analysis of this approach was conducted, demonstrating a convergence rate of O(h5). Several examples were implemented to support and illustrate the effectiveness and capability of the proposed technique.

Keywords

• Fourth-degree hat basis functions
• Itô integral
• Brownian motion process
• Nonlinear stochastic differential equations
• Operational matrix
• Error analysis