Direct Operational Vector Scheme for First-Kind Nonlinear Volterra Integral Equations and Its Convergence Analysis

Abstract

In the current paper, an efficient direct numerical scheme is developed to approximate the solution of nonlinear Volterra integral equations of the first kind. This computational method is based upon operational matrices and vectors which eventually lead to the sparsity of the coefficients matrix of the obtained system. For this purpose, the operational vector for hybrid block pulse functions and Chebyshev polynomials is constructed. Hybrid functions are powerful tools to approximate functions locally, and capable to adjust the orders of block-pulse functions and Chebyshev polynomials to achieve highly accurate numerical solutions. Its simple structure to implement, low computational cost and perfect approximate solutions are the major points of the presented method. The convergence analysis of the numerical approach under the $$L^2$$ -norm is completely studied. Finally, numerical experiments indicate that the proposed approach is effective and powerful to deal with smooth and non-smooth solutions and verify the obtained theoretical results.