New Method For Numerical Integration Research Articles

Quadrature Rules for the Fm-Transform Polynomial Components

The purpose of this paper is to reduce the complexity of computing the components of the integral Fm-transform, m≥0, whose analytic expressions include definite integrals. We propose to use nontrivial quadrature rules with nonuniformly distributed integration points instead of the widely used Newton–Cotes formulas. As the weight function that determines orthogonality, we choose the generating function of the fuzzy partition associated with the Fm-transform. Taking into account this fact and the fact of exact integration of orthogonal polynomials, we obtain exact analytic expressions for the denominators of the components of the Fm-transformation and their approximate analytic expressions, which include only elementary arithmetic operations. This allows us to effectively estimate the components of the Fm-transformation for 0≤m≤3. As a side result, we obtain a new method of numerical integration, which can be recommended not only for continuous functions, but also for strongly oscillating functions. The advantage of the proposed calculation method is shown by examples.

A New Method for Numerical Integration of Higher-Order Ordinary Differential Equations Without Losing the Periodic Responses

A new numerical method is presented for the solution of initial value problems described by systems of N linear ordinary differential equations (ODEs). Using the state-space representation, a differential equation of order n > 1 is transformed into a system of L = n×N first-order equations, thus the numerical method developed recently by Katsikadelis for first-order parabolic differential equations can be applied. The stability condition of the numerical scheme is derived and is investigated using several well-corroborated examples, which demonstrate also its convergence and accuracy. The method is simply implemented. It is accurate and has no numerical damping. The stability does not require symmetrical and positive definite coefficient matrices. This advantage is important because the scheme can find the solution of differential equations resulting from methods in which the space discretization does not result in symmetrical matrices, for example, the boundary element method. It captures the periodic behavior of the solution, where many of the standard numerical methods may fail or are highly inaccurate. The present method also solves equations having variable coefficients as well as non-linear ones. It performs well when motions of long duration are considered, and it can be employed for the integration of stiff differential equations as well as equations exhibiting softening where widely used methods may not be effective. The presented examples demonstrate the efficiency and accuracy of the method.

An Algorithm for the Numerical Integration of Perturbed and Damped Second-Order ODE Systems

A new method of numerical integration for a perturbed and damped systems of linear second-order differential equations is presented. This new method, under certain conditions, integrates, without truncation error, the IVPs (initial value problems) of the type: x″(t)+Ax′(t)+Cx(t)=εF(x(t),t), x(0)=x0, x′(0)=x0′, t∈[a,b]=I, which appear in structural dynamics, astrodynamics, and other fields of physics and engineering. In this article, a succession of real functions is constructed with values in the algebra of m×m matrices. Their properties are studied and we express the solution of the proposed IVP through a serial expansion of the same, whose coefficients are calculated by means of recurrences involving the perturbation function. This expression of the solution is used for the construction of the new numerical method. Three problems are solved by means of the new series method; we contrast the results obtained with the exact solution of the problem and with its first integral. In the first problem, a quasi-periodic orbit is integrated; in the second, a problem of structural dynamics associated with an earthquake is studied; in the third, an equatorial satellite problem when the perturbation comes from zonal harmonics J2 is solved. The good behavior of the series method is shown by comparing the results obtained against other integrators.

Application of non-local transformations for numerical integration of singularly perturbed boundary-value problems with a small parameter

Singularly perturbed boundary-value problems for second-order ODEs of the form εyxx′′=F(x,y,yx′) with ε→0 are considered. We present a new method of numerical integration of such problems, based on introducing a new non-local independent variable ξ, which is related to the original variables x andy by the equation ξx′=g(x,y,yx′,ξ). With a suitable choice of the regularizing function g, this method leads to more appropriate problems that allow the application of standard numerical methods with fixed stepsize of ξ (in the whole range of variation of the independent variable x, including both the boundary-layer region and the outer region). It is shown that methods based on piecewise-uniform grids are a particular (degenerate) case of the method of non-local transformations with a piecewise-smooth regularizing function of special form. A number of linear and non-linear test problems with a small parameter (including convective heat and mass transfer type problems) that have exact or asymptotic solutions (both monotonic and non-monotonic), expressed in elementary functions, are presented. Comparison of numerical, exact, and asymptotic solutions showed the high efficiency of the method of non-local transformations for solving singularly perturbed problems with boundary layers. In addition to non-local transformations, examples of the use of point (local) transformations for numerical integration of singularly perturbed boundary-value problems are also given.

Positivity Preserving Nonstandard Finite Difference Schemes Applied to Cancer Growth Model

When one solves differential equations, modeling biological or physical phenomena, it is of great importance to take physical constraints into account. More precisely, numerical schemes have to be designed such that discrete solutions satisfy the same constraints as exact solutions. In this work, we introduce explicit finite difference schemes based on the nonstandard discretization method to approximate solution of the cross-diffusion system from bioscience. The proposed schemes improve the accuracy and guarantee the positivity requirement, as is demanded for the solution of such system. We apply new methods for numerical integration of the cancer growth model for illustrating the performance of them.

A NEW CLASS OF L-STABLE HYBRID ONE-STEP METHODS FOR THE NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS

In this paper, a class of one-step hybrid methods for the numerical solution of ordinary differential equations (ODEs) are considered. The accuracy and stability properties of these methods are investigated. By judicious choice of the coefficients in these formulae a class of method is derived which is shown to be L-stable and so is appropriate for the solution of certain ordinary differential and stiff differential equations. We apply the new method for numerical integration of some famous stiff chemical problems such chemical Akzo-Nobel problem, ROBER problem (suggested by Robertson) and some others which are very popular in numerical studies.

Bandlimited implicit Runge–Kutta integration for Astrodynamics

We describe a new method for numerical integration, dubbed bandlimited collocation implicit Runge–Kutta (BLC-IRK), and compare its efficiency in propagating orbits to existing techniques commonly used in Astrodynamics. The BLC-IRK scheme uses generalized Gaussian quadratures for bandlimited functions. This new method allows us to use significantly fewer force function evaluations than explicit Runge–Kutta schemes. In particular, we use a low-fidelity force model for most of the iterations, thus minimizing the number of high-fidelity force model evaluations. We also investigate the dense output capability of the new scheme, quantifying its accuracy for Earth orbits. We demonstrate that this numerical integration technique is faster than explicit methods of Dormand and Prince 5(4) and 8(7), Runge–Kutta–Fehlberg 7(8), and approaches the efficiency of the 8th-order Gauss–Jackson multistep method. We anticipate a significant acceleration of the scheme in a multiprocessor environment.

A new method for numerical integration of singular functions on the plane

Region transformation methods have been used by many authors to integrate singular multivariate integrands, the most notable being the Duffy transformation that maps squares onto triangles and cubes onto pyramids. In this paper, a new method that generalizes and improves the existing ones is provided. We show that behind these variable transformation methods there is an underlying linear structure involving the shape functions of the integration domains. This structure can be modified with further composition of the original transformation with regular mappings on the unit square in order to improve the performance of the numerical integration. The proposed method is neutral in the sense that the transformation composition is suitable for different types of singularities, without making any particular assumption on the regular part of the integrand. Moreover, this is achieved without a significant increase in the computational cost and making use of standard Gaussian quadrature rules only. We illustrate the efficiency of the proposed methods with comparative numerical examples. This technique proves to be both efficient and practical and has a better convergence than those based on classical transformations such as power or trigonometric maps.

A new adaptive integration method for the peridynamic theory

Peridynamics is a new formulation of solid mechanics based on direct interactions between material points in a continuum separated by a finite distance. Integration of interactions between material points plays a crucial role in the formulation of peridynamics. To overcome the deficiencies in numerical integration methods for peridynamics in the literature, the work presented here focuses on a new method of numerical integration for the peridynamic theory. In this method, the continuum is discretized into cubic cells, and different geometric configurations over the cell and the horizon of interaction are classified in detail. Integration of the peridynamic force over different intersection volumes are calculated accurately using an adaptive trapezoidal integration scheme with a combined relative-absolute error control. Numerical test examples are provided to demonstrate the accuracy of this new method. The bond-based peridynamic constitutive model is used in the calculation but this new method is also applicable to state-based peridynamics.

Deterministic computation of functional integrals

A new method of numerical integration in functional spaces is described. This method is based on the rigorous definition of a functional integral in complete separable metric space and on the use of approximation formulas which we constructed for this kind of integral. No preliminary discretization of space and time is required in this method, as well as no simplifying assumptions like semiclassical approximation, collective excitations, introduction of “short-time” propagators, etc., are necessary in our approach. The comparison with results of other authors shows that our method gives significant economy of computer time and memory versus other known methods while providing the results with the same or better accuracy. The method is proved to have advantages also in a case of systems with many degrees of freedom. Examples of application of the method to the numerical study of some physical models are presented.

New method for numerical integration of the radial electronic Schrödinger equation

We present a new method for numerical integration of the radial electronic Schrödinger equation with these characteristics: (i) it uses a quantity directly related to the logarithmic derivative of the wave function, thereby facilitating the matching of solutions obtained for different radial regions; (ii) it avoids difficulty from the singularity of the logarithmic derivative at the nodes of the wave function; and (iii) it takes appropriate cognizance of the asymptotic form of the wave function at infinite radius. Examples are presented showing that eigenvalues can be obtained by the new method by outward integration alone, but that a combination of inward and outward integration leads to efficiencies which compare favorably with those achievable by the most popular previously existent method, that of Numerov.

A method for real time numerical integration

Some important problems connected with real time numerical integration are discussed. Some popular algorithms are mentioned. Several possibilities for required memory and computing interval reduction are noted. A new method for numerical integration using a microprocessor technique with a drastic reduction in the computing interval is described. Practical realization with the MC6840 Programmable Timer in high resolution spectroscopy experiment is demonstrated.

A new approach to numerical integration by means of steady-state flux in a slab

The definite integral is generally interpreted geometrically as an “area”. An alternate interpretation as a steady-state “flux” through a unit slab is derived, which leads to a new method of numerical integration. The usual sum of a large number of approximate areas is replaced by the flux through a “single” increment. The method involves the solution of a system of linear finite difference equations. The coefficient matrix is tri-diagonal and is solved efficiently by the Thomas algorithm. During the iterative process the coefficients are determined by simple quadrature schemes applied to each increment. Error analysis revealed that an expression could be derived for the roundoff error associated with the final Thomas iteration. It is shown that the roundoff error is smallest when the matrix coefficients a k\\S>1. The method is shown to be superior to the classical methods due to its simplicity and tolerance for variable increment size. In addition, a new function is determined which is useful in diffusion studies. Numerical data are presented confirming these results.

A new method for numerical integration of airplane equations of motion

I has been found in Ref. 1 that application of linear transformation methods to the integration of the airplane equations of motion can result in significant savings of computer time over more conventional methods. One of these methods, named the approximate inverse Laplace method, is applied below to the integration of rigid airplane equations of motion. These equations form a set of six simultaneous nonlinear differential equations. By expressing these equations consistently in terms of the dependent variables £7, a, 0, /3, <£, and \(/, it is possible to write these equations in the following form: Longitudinal (Body Axes)

A New Method of Numerical Integration of Differential Equations

A New Method of Numerical Integration of Differential Equations

A new method of numerical integration of differential equations

(1) y= f(x, y), y(xo) = yo is Heun's method. However, this method requires two substitutions into the differential equation (1) for each step forward. A method described by Lotkin [1] achieves comparable accuracy but requires only one substitution in (1). This can be quite a saving in time when (1) is a complicated function. This article illustrates another method which also requires only one substitution in (1).