Numerical Solution of the Fredholm and Volterra Integral Equations by Using Modified Bernstein–Kantorovich Operators

Abstract

The main aim of this paper is to numerically solve the first kind linear Fredholm and Volterra integral equations by using Modified Bernstein–Kantorovich operators. The unknown function in the first kind integral equation is approximated by using the Modified Bernstein–Kantorovich operators. Hence, by using discretization, the obtained linear equations are transformed into systems of algebraic linear equations. Due to the sensitivity of the solutions on the input data, significant difficulties may be encountered, leading to instabilities in the results during actualization. Consequently, to improve on the stability of the solutions which imply the accuracy of the desired results, regularization features are built into the proposed numerical approach. More stable approximations to the solutions of the Fredholm and Volterra integral equations are obtained especially when high order approximations are used by the Modified Bernstein–Kantorovich operators. Test problems are constructed to show the computational efficiency, applicability and the accuracy of the method. Furthermore, the method is also applied to second kind Volterra integral equations.

Theoretical and Numerical Study for Volterra−Fredholm Fractional Integro-Differential Equations Based on Chebyshev Polynomials of the Third Kind

“…Tere has been a lot of interest in solving FI-DEs [26][27][28][29], specially using collocation methods. In [30], the authors used a wavelet collocation method to solve Fredholm FI-DEs.…”

Theoretical and numerical aspects of a non-stationary preconditioned iterative method for linear discrete ill-posed problems

“…The iterated Tikhonov algorithm can be seen as a preconditioned Landweber iteration, where the preconditioner is chosen to approximate a regularized version of the pseudo-inverse of A. Other possible regularizing preconditioner are described in [4,5,6].…”

Uniqueness of the Fredholm-Stiltjes Linear Integral Equations Solutions of the Third Kind

“…In [4], a theory is presented and numerical methods are used for solving non-to an increasing function was introduced, using linear Fredholm-Stiltjes integral equations of the first kind in [16] [17] [18]. Numerical solution of the Fredholm and Volterra Integral equations by using modified Bernstein-Kantorovich operators [19], second kind [20], and third kind of nonlinear Volterra-Stieltjes integral equations with the solution by Lavrentyev regularizing operator were also described [21].…”