On approximation of definite integrals by compound quadrature formulas using derivatives

Рассмотрена задача вычисления определенного интеграла функции, для которой известны значения ее самой и набора производных до заданного порядка в точках отрезка интегрирования. Построены составные квадратурные формулы, которые используют значения функции и ее производных до m-го порядка включительно. Получено представление остаточного члена, выраженное через производную соответствующего порядка и число узловых точек. Приведены примеры интегрирования заданных функций с исследованием погрешности и ее оценки. Дано сравнение с известными численными методами и формулой Эйлера-Маклорена, которое показало повышенную точность и лучшую сходимость метода двухточечного интегрирования The problem of computing a definite integral of a function for which the values of itself and the set of derivatives up to a given order at the points of the interval of integration are known is considered. Composite quadrature formulas are constructed that use the values of the function and its derivatives up to the m-th order inclusive. A representation of the remainder is obtained, expressed in terms of the derivative of the corresponding order and the number of nodal points. Examples of integration of the given functions with the study of the error and its estimation are given. A comparison is made with the known numerical methods and the Euler-Maclaurin formula, which showed increased accuracy and better convergence of the two-point integration method.

Approximate energy spectra and statistical mechanical functions of some diatomic molecular hydrides

In this study, we have investigated the statistical mechanical properties of the Varshni potential model for some diatomic molecular hydrides via the Euler–Maclaurin formula. This was done using the approximate analytical energy eigenvalues, which were obtained by solving the radial Schrödinger equation with the Greene–Aldrich approximation and suitable coordinate transformation schemes. The effect of high temperatures and upper bound vibration quantum number on the vibrational partition function and other thermodynamic functions of the selected diatomic molecular hydrides were studied. We also show that these effects on the thermodynamic functions considered were similar for all the diatomic molecular hydrides selected.

Duffin–Kemmer–Petiau equation and thermodynamic quantities

We investigate the generalized form of Duffin–Kemmer–Petiau (DKP) equation in the presence of both a position-dependent electrical field and curved spacetime for the 2-dimensional anti-de Sitter spacetime. Moreover, we derive both the asymptotic wave function and construct energy quantization with the help of the properties of gamma function. All thermodynamic quantities of the system have been calculated with the help of the Euler–MacLaurin formula in the final state.

Contour integrals of analytic functions given on a grid in the complex plane

Ability to evaluate contour integrals is central to both the theory and the utilization of analytic functions. We present here a complex plane realization of the Euler–Maclaurin formula that includes weights also at some grid points adjacent to each end of a line segment (made up of equispaced grid points, along which we use the trapezoidal rule). For example, with a $5\times 5$ ‘correction stencil’ (with weights about two orders of magnitude smaller than those of the trapezoidal rule), the accuracy is increased from $2$nd to $26$th order.

Thermal properties of three dimensional Morse potential for some diatomic molecules via Euler-Maclaurin approximation

The purpose of this study is to develop a method of calculating the vibration partition function of diatomic molecules for the Morse potential energy. After a brief introduction about the eigensolutions obtained for the problem in question, Via the Euler-Maclaurin formula, we have determined the thermal properties for four diatomics such as \text{H}_{2}, HCl, LiH, and CO. Different situation has been exposed and explained by the appropriate curves of the thermal properties for these diatomics molecules in consideration. In addition, we have shown that our method exposed to calculate these thermal properties can be used to determine these thermodynamic quantities.

Finite Heisenberg Quantum Spin Chain

The Bethe ansatz genuinely considers a finite system. The extraction of finite-size results from the Bethe ansatz equations is of genuine interest, especially against the background of the results of finite-size scaling and conformal symmetry in finite geometries. The mathematical techniques introduced in chapter 19 permit a systematic treatment in this chapter of finite-size corrections as corrections to the thermodynamic limit of the system. The application of the Euler-Maclaurin formula transforming finite sums into integrals and finite-size corrections transforms the Bethe ansatz equations into Wiener–Hopf integral equations with inhomogeneities representing the finite-size corrections solvable using the Wiener–Hopf technique. The results can be compared to results for finite systems obtained from other approaches that are independent of the Bethe ansatz method. It briefly discusses higher-order corrections and offers a general assessment of the finite-size method.