Bivariate modified Bernstein–Kantorovich operators for the numerical solution of two‐dimensional fractional Volterra integral equations

Abstract

A class of two‐dimensional fractional Volterra integral equations (2D‐FVIEs) of the second kind is considered. The solution may have unbounded derivatives near the integral domain boundary. Therefore, smoothing transformations are employed to change the original 2D‐FVIEs into new transformed 2D‐FVIEs with better regularity. The novelty in this research concerns both the theoretical investigation of the bivariate modified Bernstein–Kantorovich (B‐MBK) operators and the numerical application of these operators for approximating the unknown solution of 2D‐FVIEs. In this regard, an algorithm is given utilizing the B‐MBK operators and discretization that approximates the solution of the transformed discretized equation. Further, an inverse transformation is applied to obtain the solution of the original equation. Additionally, we illustrate the applicability of the proposed method on examples from the literature.

Editorial of the special issue on modelling, analysis, and applications

“…The artificial Hamiltonian method, also known as partial Hamiltonian or nonstandard Hamiltonian system, is used to analyze biological population models, namely, the Easter Island, Verhulst, Lotka‐Volterra, and four‐dimensional SEIR and SIRD models, in [14]. The 2D‐fractional Volterra integral equations of second type are the focus of a research issue, in which Bernstein‐Kantorovivh operators are analyzed and numerically applied to approximate their solution [15]. The Adomian decomposition method is used to solve a class of nonlinear differential equations of fractional order, with modified Caputo derivatives and integral boundary conditions.…”