An efficient cubic B-spline and bicubic B-spline collocation method for numerical solutions of multidimensional nonlinear stochastic quadratic integral equations

This paper aims to develop a novel numerical approach on the basis of B-spline collocation method to approximate the solution of one-dimensional and two-dimensional nonlinear stochastic quadratic integral equations. The proposed approach is based on the hybrid of collocation method, cubic B-spline, and bi-cubic B-spline interpolation and Itô approximation. Using this method, the problem solving turns into a nonlinear system solution of equations that is solved by a suitable numerical method. Also, the convergence analysis of this numerical approach has been discussed. In the end, examples are given to test the accuracy and the implementation of the method. The results are compared with the results obtained by other methods to verify that this method is accurate and efficient.