Balancing Polynomial for Solution Nonlinear Stochastic Itô–Volterra Integral Equations

ABSTRACT This paper introduces an innovative method for solving nonlinear stochastic Itô–Volterra integral equations using balancing polynomials and their associated operational matrices. This approach effectively transforms these complex stochastic equations into a system of nonlinear algebraic equations that can be solved using the Newton method. Balancing polynomials are chosen for their ability to enhance stability and convergence, providing a more reliable and manageable framework for tackling challenging stochastic problems. The paper also includes a convergence analysis and error estimation for the proposed method. Additionally, the effectiveness of this approach is demonstrated through four numerical examples. The results obtained from this method are compared with the exact solution and those from other established techniques, including the block‐pulse function method, the shifted Jacobi operational matrix (SJOM) method, and the shifted Jacobi spectral Galerkin (SJSG) method. These comparisons highlight the superior performance and accuracy of the proposed method. All numerical computations were performed using MATLAB.