On the Approximation Solutions of linear and nonlinear Volterra Integral Equation of First and Second kinds by Using B-spline Tight Framelets Generated by Unitary Extension Principle and Oblique Extension Principle

Framelets methods are a very useful tool in the approximation for the piecewise smooth functions, and have fast decomposition and reconstruction algorithms associated with them. In this article, we present a numerical method for solving linear and nonlinear Volterra integral equations of the first and second kinds. Our method is based on the use of quasi-affine tight framelets systems generated by the unitary extension principle and oblique extension principle. Many different examples of framelets systems and their graphs are provided. The solution of the integral equation is based on converting the integral equation to a system of linear equations. We prove the convergence theorem for the numerical solution of linear Volterra integral equation of the second kind. Finally, we present numerical examples of solving linear and nonlinear Volterra integral equations to ensure the validity of our method. Comparisons of the results with other methods are included in the examples.