Hybrid Operators for Approximating Nonsmooth Functions and Applications on Volterra Integral Equations with Weakly Singular Kernels

Abstract

This research concerns some theoretical and numerical aspects of hybrid positive linear operators for approximating continuous functions on $\left[0,1\right]$ that have unbounded derivatives at the initial point. These operators are defined by using Modified Bernstein–Kantorovich operators $K_{n,\alpha },$ where n is positive integer, $\alpha >0$ is a fixed constant and reduces to the classical Bernstein–Kantorivich operators when $\alpha =1.$ To show the importance and the applicability of the given hybrid operators we develop an algorithm which implements them for solving the second kind linear Volterra integral equations with weakly singular kernels. Furthermore, applications are also performed on first kind integral equations, by utilizing regularization. Eventually, it is shown that the numerical realization of the given algorithm is easy and computationally efficient and gives accurate approximations to nonsmooth solutions.

Keywords:

• Asymptotic rate of convergence
• hybrid operators
• regularization
• singular kernel
• Volterra integral equations

2020 Mathematics Subject Classification:

• 41A36
• 41A25
• 45D05
• 65D15

Disclosure statement

The authors declare that they have no competing interest.

Data availability

No data are used.

Additional information

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.