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This article focuses on an efficient and highly accurate approximate solver for a class of generalized singular boundary value problems (SBVPs) having nonlinearity and with two‐term fractional derivatives. The involved fractional derivative operators are given in the form of Liouville–Caputo. The developed algorithm for solving the generalized SBVPs consists of two main stages. The first stage is devoted to an iterative quasilinearization method (QLM) to conquer the (strong) nonlinearity of the governing SBVPs. Secondly, we employ the generalized Genocchi polynomials (GGPs) to treat the resulting sequence of linearized SBVPs numerically. An upper error estimate for the Genocchi series solution in the L2$$ {L}^2 $$ norm is obtained via a rigorous error analysis. The main benefit of the presented QLM‐GGPs procedure is that the required number of iteration in the first stage is within a few steps, and an accurate polynomial solution is obtained through computer implementations in the second stage. Three widely applicable test cases are investigated to observe the efficacy as well as the high‐order accuracy of the QLM‐GGPs algorithm. The comparable accuracy and robustness of the presented algorithm are validated by doing comparisons with the results of some well‐established available computational methods. It is apparently shown that the QLM‐GGPs algorithm provides a promising tool to solve strongly nonlinear SBVPs with two‐term fractional derivatives accurately and efficiently. [ABSTRACT FROM AUTHOR]

The present paper introduces a new modification of Gamma operators that protects polynomials in the sense of the Bohman–Korovkin theorem. In order to examine their approximation behaviours, the approximation properties of the newly introduced operators such as Voronovskaya-type theorems, rate of convergence, weighted approximation, and pointwise estimates are presented. Finally, we present some numerical examples to verify that the newly constructed operators are an approximation procedure. [ABSTRACT FROM AUTHOR]

In the present article, we give a Bézier variant of Păltăanea operators, which involves Gould-Hopper polynomials. First, we investigate the rate of convergence by using Ditzian-Totik modulus of smoothness, weighted modulus of continuity and also for a class of Lipschitz function. Furthermore, we obtain the quantitative Voronovskaja type theorem in terms of Ditzian-Totik modulus of smoothness. In the last section, we study the rate of point-wise convergence for the functions having a derivative of bounded variation. [ABSTRACT FROM AUTHOR]

In this article, we give a generalization of integral operators which involves Sheffer polynomials introduced by Sucu and Büyükyazici. We obtain approximation properties of our operators with the help of the universal Korovkin's theorem and study convergence properties by using modulus of continuity, the second order modulus of smoothness and Peetre's K-functional. We have also established Voronovskaja type asymptotic formula. Furthermore, we study the convergence of these operators in weighted spaces of functions on the positive semi-axis and estimate the approximation by using weighted modulus of continuity. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to give a Chlodowsky type generalization of Szász operators defined by means of the Brenke type polynomials. We obtain convergence properties of our operators with the help of universal Korovkin-type property and also establish the order of convergence by using a classical approach, the second order modulus of continuity and Peetre’s K -functional. We also give a Voronoskaja type theorem. Furthermore, we study the convergence of these operators in a weighted space of functions on a positive semi-axis. Some graphical examples for the convergence of our operators and error estimation are also given. [ABSTRACT FROM AUTHOR]