Orthonormal Bernoulli polynomials collocation approach for solving stochastic Itô‐Volterra integral equations of Abel type

In this paper, orthonormal Bernoulli collocation method has been developed to obtain the approximate solution of linear singular stochastic Itô‐Volterra integral equations. By applying this method, linear stochastic integral equation converts to linear system of algebraic equations. This system is achieved by approximating functions that appear in the stochastic integral equations by using orthonormal Bernoulli polynomials (OBPs) and then substituting these approximations into consideration equation. This linear system of algebraic equations can be solved via an appropriate numerical method and approximate solution of integral equation is obtained. A main advantage of this technique is that the condition number of the coefficient matrix of the system is small, which verify that THE proposed method is stable. Also, convergence and error analysis of the present method are discussed. Finally, two examples are given to show the pertinent properties, applicability, and accuracy of the present method.