A numerical method for solving Fredholm and Volterra integral equations of the second kind is presented. The method is based on the use of  the NEWTON–COTES quadrature rule and Lagrange interpolation polynomials. By the proposed method, the main problem is reduced to solve some nonlinear algebraic equations that can be solved by NEWTON’s method. Also, we prove some statements about the convergence of the method. It is shown that the approximated solution is uniformly convergent to the exact solution. In addition, to demonstrate the efficiency and applicability of the proposed method, several numerical examples are included, which confirms the convergence results.

Fuzzy NEWTON-COTES method for integration of fuzzy functions that was proposed by Ahmady in [1]. In this paper we construct error estimate of fuzzy NEWTON-COTES method such as fuzzy Trapezoidal rule and fuzzy Simpson rule by using Taylor's series. The corresponding error terms are proven by two theorems. We prove that the fuzzy Trapezoidal rule is accurate for fuzzy polynomial of degree one and fuzzy Simpson rule is accurate for polynomial of degree three. The accuracy of fuzzy Trapezoidal rule and fuzzy Simpson rule for integration of fuzzy functions are illustrated by two examples.

This paper presents an efficient numerical analysis method called the NEWTON-COTES-Hermite-Four-Point (NCH-4P) method for the time history analysis of structural systems with one and multiple degrees of freedom under earthquake effect. The method combines the numerical integration FORMULA of the NEWTON-COTES four-point method with the Hermite interpolation FORMULAs to form a new algorithm for solving the vibration equation. The method can handle both linear and nonlinear systems, as well as different types of loading, such as external forces and earthquake excitation. The method is developed for the first time for the analysis of linear and nonlinear damped and undamped structural systems with one and multiple degrees of freedom. The method shows remarkable performance superiority in terms of accuracy, speed, convergence and simplicity compared to the pseudo-analytical Newmark-beta method and the semi-analytical Duhamel integral method. The method modifies the differential equation of motion to have a suitable form for numerical integration. Unlike the nonlinear Newmark-beta method, the method does not require an independent process such as NEWTON’s iteration to account for nonlinear effects; instead, a series of simple repeated calculations leads to the convergence of the response in nonlinear behavior. The numerical results demonstrate the efficiency of the method in estimating the response of systems under the famous EL-Centro record.

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.

A fast and efficient numerical scheme is presented for time-history analysis of single-degree-of-freedom (SDOF) structural systems undergoing seismic excitation (Chopra, 2003). The new method is called NEWTON-COTES-4P-θ Method. It uses the most known 4-point NEWTON-COTES quadrature in its body to solve the vibration equation. Nonlinear analysis is covered as well as linear analysis. Any arbitrary external loadings of type force or seismic signals are welcome. The significant advantages of the new FORMULAtion are its great simplicity, running speed, and appropriate precision level compared with its counterparts such as Duhamel integral and Newmark-β methods. The accuracy level of the NEWTON-COTES-4P-θ is close to the semi-analytical method of Duhamel integration and its speed is similar to the Newmark-β algorithm. Notably, against the nonlinear Newmark-β method, the new method does not require a standalone procedure to handle nonlinear analysis; instead, it simply triggers iteration of the same computation used in its first processing round. Moreover, the Newmark-β method loses its performance dealing with stiff and near-conservative () systems; however, the NEWTON-COTES-4P-θ method does not loos its accuracy and keeps its well-performed analysis in this case. Numerical results reveal the superiority of the NEWTON-COTES- 4P-θ method against its counterparts such as the Duhamel integral, Newmark-β, and Wilson-θ methods (Babaei et al., 2021; Babaei et al., 2022; Babaei et al., 2023).

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.