The main objective of this paper is to establish some new inequalities related to the open NEWTON-COTES FORMULAs in the setting of q-calculus. We establish a quantum integral identity first and then prove the desired inequalities for $q$-differentiable convex functions. These inequalities are useful for determining error bounds for the open NEWTON-COTES FORMULAs in both classical and $q$-calculus. This work distinguishes itself from existing studies by employing quantum operators, leading to sharper and more precise error estimates. These results extend the applicability of NEWTON-COTES methods to quantum calculus, offering a novel contribution to the numerical analysis of convex functions. Finally, we provide mathematical examples and computational analysis to validate the newly established inequalities.

In this paper, we investigate the connection between closed NEWTON-COTES FORMULAe, trigonometrically-fitted methods, symplectic integrators and efficient integration of the Schrodinger equation. The study of multistep symplectic integrators is very poor although in the last decades several one step symplectic integrators have been produced based on symplectic geometry (see the relevant literature and the references here). In this paper we study the closed NEWTON-COTES FORMULAe and we write them as symplectic multilayer structures. Based on the closed NEWTON-COTES FORMULAe, we also develop trigonometrically-fitted symplectic methods. An error analysis for the one-dimensional Schrodinger equation of the new developed methods and a comparison with previous developed methods is also given. We apply the new symplectic schemes to the well-known radial Schrodinger equation in order to investigate the efficiency of the proposed method to these type of problems.

A collocation procedure is developed for the linear and nonlinear Fredholm and Volterra integro-differential equations, using the globally defined B-spline and auxiliary basis functions. The solution is collocated by cubic B-spline and the integrand is approximated by the NEWTON-COTES FORMULA. The error analysis of proposed numerical method is studied theoretically. Numerical results are given to illustrate the efficiency of the proposed method which shows that our method can be applied for large values of N. The results are compared with the results obtained by other methods to illustrate the accuracy and the implementation of our method.

The objective of this paper is to examine integral inequalities related to multiplicatively differentiable functions. Initially, we establish a novel identity using the two-point NEWTON-COTES FORMULA for multiplicatively differentiable functions. Using this identity, we derive Companion of Ostrowski's inequalities for multiplicatively differentiable convex mappings. The work also provides the results' applications.

In this paper, we use parametric form of fuzzy number, then an iterative approach for obtaining approximate solution for a class of nonlinear fuzzy Fredholm integro-differential equation of the second kind is proposed.This paper presents a method based on NEWTON-COTES methods with positive coefficient. Then we obtain approximate solution of the nonlinear fuzzy integro differential equations by an iterative approach.

A numerical method for solving nonlinear mixed Hammerstein integral equations is presented in this paper. The method is based upon hybrid of rationalized Haar functions approximations. The properties of hybrid functions which are the combinations of block-pulse functions and rationalized Haar functions are first presented. The NEWTON-COTES nodes and NEWTON-COTES integration method are then utilized to reduce the nonlinear mixed Hammerstein integral equations to the solutions algebraic equations. The method is computationally attractive, and applications are demonstrated through illustrative examples.