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The study of the linear methods of approximation, which are given by sequences of positive and linear operators, studied extremely, in relation to different subjects of analysis, such as numerical analysis. The principal objective of this manuscript is to develop a new and more comprehensive version of Gamma type operators and presented their approximation features. For this purpose, we benefit from two sequences of functions, which are αn(x) and βn(x), and from the function τ(x). To indicate how the function τ play a significant role in the construction of the operator, we reconstruct the mentioned operators which preserve exactly two test functions from the set {1, τ, τ2}. Then we established Voronovskaya type theorem and order of approximation properties of the newly defined operators utilizing weighted modulus of continuity to show that their approximation properties. At the end of this note, we present a series of numerical results to show that the new operators are an approximation technique.

The main feature of this paper is an application of the modified Bernstein operator in the approximate and numerical solution of integral equations. The stability of the algorithm is discussed in the context of errors resulting from the numerical approximation of Volterra integral equations. The mechanism of the proposed method is explained in detail, including its main features. Furthermore, a comparison with an alternative numerical technique is made, and the superiority of the proposed solution is shown. Numerical experiments are also performed to verify the validity of the proposed method and to assess its accuracy. Finally, several conclusions are drawn from the results of the numerical experiments. The proposed method provides a powerful and efficient tool for the approximate solution of Volterra integral equations, and its results are promising. The results obtained from this algorithm are useful in the numerical analysis of integral equations. [ABSTRACT FROM AUTHOR]

This research investigates the second Hankel determinant for a specific class of functions associated with the Daehee polynomial. To achieve this, we introduce new subclasses of starlike functions in the context of Daehee polynomials. In complex analysis, establishing precise bounds for coefficient estimates in bi-univalent functions is essential, as these coefficients define the fundamental properties of conformal mappings. In this study, we derive sharp bounds for coefficient estimates within new subclasses of starlike functions related to Daehee polynomials, with most of the obtained limits demonstrating high accuracy. This work aims to inspire further exploration of rigorous bounds for analytic functions associated with innovative mapping domains. [ABSTRACT FROM AUTHOR]

In this manuscript, linear and positive operators described on bounded and unbounded intervals that fix the function sets {1, φ} and {1, φ2} such that φ ∈ C[0, 1] are presented. Then we present different types of operators by choosing different functions and values. Finally, Voronovskaya type theorems are given for this newly defined sequences of linear and positive operators.

In this manuscript, we present and test a numerical scheme with an algorithm to solve Volterra and Abel's integral equations utilizing generalized sampling operators. Illustrative computational examples are included to indicate the validity and practicability of the proposed technique. All of the computational examples in this research have been computed on a personal computer implementing some programs coded in MATLAB. [ABSTRACT FROM AUTHOR]

In this study we introduced and tested retarded conformable fractional integral inequalities utilizing non-integer order derivatives and integrals. In line with this purpose, we used the Katugampola type conformable fractional calculus which has several practical properties. These inequalities generalize some famous integral inequalities which provide explicit bounds on unknown functions. The results provided here had been implemented to the global existence of solutions to the conformable fractional differential equations with time delay.

In this article, a new type of Gamma operators which preserves 1 and x2 has been presented and tested. In order to show the approximation properties of the newly defined operator, Voronovksya type theorems, weighted approximation, rate of convergence, and pointwise estimates have been presented. Additionally, numerical examples implemented by MATLAB have been provided to show its approximation properties by numerically. [ABSTRACT FROM AUTHOR]

Mathematical modelling and simulation in biophysics and its applications in terms of both theoretical and biological/physical/ecological point of view arise in a number of research problems ranging from physical and chemical processes to biomathematics and life science. As known, the modeling of a biophysical system requires the analysis of the different interactions occurring among the different components of the system. This editorial article deals with the topic of this special issue, which is devoted to the new developments in the modelling and simulation in biophysical applications with special attention to the interplay between different scholars. [ABSTRACT FROM AUTHOR]

The present paper deals with new modification of Gamma operators preserving polynomials in Bohman-Korovkin sense and study their approximation properties: Voronovskaya type theorems, weighted approximation and rate of convergence are captured. The effectiveness of the newly modified operators according to classical ones are presented in certain senses as well. Numerical examples are also presented, highlighting the performance of the new constructions of Gamma operators in the context of one dimensional approximation. [ABSTRACT FROM AUTHOR]