For the finite elements analyses, the stresses and strains and the elements of the deformation-displacement matrices in solid mechanics, the analysts in general are needing actually to obtain the results at the levels of nods of elements, such that the extrapolation **functions** and especially at the edge nods, don’t allow to offer good precisions. Moreover, the proposed **integration** formulas for polynomials, which addressed especially to the finite elements developers, such as they could be more favorable for finite elements analysts, and also for the **integration** of polynomials in general. In addition, such as the same obtained **numerical** results using the developed formulas and would be otherwise, obtained using Gauss **quadrature** and exact direct **integration** ones, thereof the intervention of the first derivatives ordinates herein, mean just that the complete liberty of the choosing points positions, and their contributions in this effect to get

Show more
2) The element type: In general, a higher **polynomial** degree of the shape **functions** requires a higher **integration** order to ensure exact **numerical** **integration** regarding the Jacobian determinant. Therefore, the computing effort for the Gaussian **quadrature** increases. At the same time, the computation effort of the factorization-based method also grows, since a finite element of higher order has a larger number of nodes n. These competing factors must be considered in terms of computational effort and time.

The Meshless Local Petrov-Galerkin (MLPG) method is a **numerical** framework for solving partial differential equations. This method is unique in that it uses the governing equations in the local symmetric weak form and does not rely on a prescribed node connectivity. MLPG does not specify the **numerical** tools used for accomplishing the required tasks. In this work, Gaussian **quadrature** was used for **integration** over one-dimensional and two-dimensional domains. The Moving Least Squares (MLS) method was used for approximation of data and spatial derivatives of the data. The primitive variable form of the Navier-Stokes equations was used. MLPG results for benchmark fluids applications as well as the Rayleigh Taylor Instability (RTI) are presented. The Shape Function Interpolation Method (SFIM), a novel treatment for gradient type boundary conditions, is introduced. The SFIM method uses the MLS shape **functions** to identify the nodal boundary value that satisfies the prescribed gradient condition. A capability for computing a inward facing normal vector for a boundary node, based upon the other boundary nodes in the identified region, was also created. Using the standard method for the treatment of Neumann boundary conditions, the **quadrature** and test function require modification for new geometry types. This feature eliminates the need for modified **quadrature** and allows the weight function and MLS approximation to use their native Cartesian representations. These two additions minimize the phenomena or geometry specific coding required to apply MLPG to wide range of applications on complex domains.

Show more
124 Read more

In what follows, we establish some new error estimates which are sometimes exact and convenient for evaluation of the **quadrature** error for the Lipschitz con- tinuous **functions** using a new approach to **numerical** **integration**. The result can be generalized for continuous **functions** and some types of cubature formulae.

11 Read more

In this chapter we shall state and prove a result from which we may obtain an expression, in terms of a contour integral, for the remainder term of a general quadrature formula of the fo[r]

249 Read more

There are a lot of methods that evaluate a deﬁnite integral, numerically. Some of these methods use the end points (closed rules) of integral and some do not (open rules). Some of them are based on using interpolating **polynomial**. The most popular of such methods are Trapezoidal, Simpson, and mid-point methods which are special cases of Newton-Cots method. There are some other methods that are based on the exact **integration** of poly- nomials of increasing degree; in which no subdivision of the **integration** interval are used. Basic properties of these methods can be found in many textbooks such as [1, 18]. The ordinary Taylor’s series has been generalized by many authors. Hardy [3] introduced a new version of the generalized Taylor’s series that uses Reimann-Liouville fractional integral and Trujillo et al. [19] obtained a new formula that is based on Reimann-Liouville frac- tional derivatives. For the concept of fractional derivative Odibat [14] adopted Caputo deﬁnition which is a modiﬁcation of the Reimann-Liouville deﬁnition and introduced a generalized Taylor’s series. Zaid Odibat introduced a generalized method for solving lin- ear partial diﬀerential equations of fractional order [12, 15] and introduced a novel method for nonlinear fractional partial diﬀerential equations [13]. Hashemiparast et al. [6] intro- duced a method using derivations of function for **numerical** **integration**. There are some good textbooks in this area [11, 16], and some new works have been done on **numerical**

Show more
11 Read more

12 Read more

22 Read more

The finite element method has become a powerful tool for the **numerical** solution of a wide range of engineering problem, particularly when analytical solutions are not available or very difficult to arriving the results. **Numerical** methods for **integration** approximate a definite integral of a given function by a weighted sum of function values at specified points. There are many **quadrature** methods available for approximating integrals. Surface integrals are used in multiple areas of physics and engineering. In particular, they are used for Problems involving calculations of mass of a shell, center of mass and moments of inertia of a shell, fluid flow and mass flow across a surface, electric charge distributed over a surface, plate bending, plane strain, heat conduction over a plate, and similar problems in other areas of engineering which are very difficult to analyse using analytical techniques, These problems can be solved using the finite element method. The method proposed here is termed as Generalized Gaussian rules, since the Generalized Gaussian **quadrature** nodes and weights for products of **polynomial** and logarithmic function given in [9] by Ma et al. are used in this paper

Show more
While testing **functions** are deﬁned analytically, they are often used numerically by employing a number of **integration** points on testing domains. Hence, at the implementation level, testing **functions** are reduced to a set of testing vectors with proper weights that are combinations of **integration** weights and the values of the used testing **functions**. Considering that the accuracy of MFIE heavily depends on the choice of the testing **functions**, it may be possible to select (and even optimize) the testing vectors numerically such that the accuracy of MFIE is improved, without deriving any analytical expression for the testing **functions**. In this study, we show that this approach is feasible and such **numerical** testing **functions** can be designed and constructed based on the compatibility of MFIE systems with the corresponding EFIE solutions. The designed testing **functions** provide more **accurate** results for both MFIE and the combined-ﬁeld integral equation (CFIE) that is more popular for closed conductors. The proposed **numerical** approach and its advantages for multi-frequency applications are demonstrated on two diﬀerent scattering problems.

Show more
The **numerical** **integration** of signals given in tabular form is usually conducted using **quadrature** formulas, and experimental errors are not taken into consideration. In fact, **quadrature** formulas yield unpredictable results for various reasons. The first reason lies in the impossibility of establishing a priori smoothness of input data, and the second is that there is no way to evaluate the result of **integration**. Theoretical approximations of the error of **quadrature** formulas (3) may not be useful for actual values due to the inability to accurately calculate the residual term. The traditional approach consists of applying various smoothing filters. In this case, algorithms are used that do not depend on the task of integrating itself, which leads to excessive smoothing. The authors propose a new method for solving the problem of numerically integrating inaccurate signals that minimizes the residual term of the **quadrature** formula (4) for the set of unknown values of the signal by using ill-posed problem algorithms [1].

Show more
Abstract. It is well-known that the trapezoidal rule, while being only second-order **accurate** in general, improves to spectral accuracy if applied to the **integration** of a smooth periodic function over an entire period on a uniform grid. More precisely, for the function that has a square integrable derivative of order r the convergence rate is o N −(r−1/2)

C o cal **integration** based on interpolating poly- nstruction of **quadrature** rules for numeri- nomials is done by many authors that these poly- nomials are used to find weights corresponding to nodes. To see some **quadrature** rules based on polynomials, one can refer to [4]-[11] and the references therein. As we know, wavelets anal- ysis plays an important role in different areas of mathematics [12]-[16]. So, many authors ap- plied wavelets to approximate the solution of in- tegral equations, ordinary differential equations and partial differential equations. Recently, in [2], the authors applied Haar wavelets and hybrid **functions** to find **numerical** solution of definite in- tegrals with constant limits. In [1], the authors extended the scope of applicability of the method

Show more
The method of moments (MoM) is an efficient way to analyze the electromagnetic scattering problems formulated in terms of integral equations [1–3]. The conventional straightforward application of MoM involves low-order basis **functions** with the mesh cells’ size in the order of one tenth of a wavelength, leading to a dense system of linear equations. With N being the number of unknowns, the memory requirement is O(N 2 ), and the solution complexity is O(N 3 ) for a direct solver and O(N 2 ) for an iterative one. Several techniques have been proposed to reduce the memory demands as well as the solution complexity of the conventional MoM. Fast integral equation solvers, such as the multilevel fast multiple method (MLFMM) [4, 5], adaptive integral method (AIM) [6, 7] and its close counterpart, the precorrected FFT (PC-FFT) [8, 9], reach the solution complexity and memory

Show more
11 Read more

differential Equation (1) by certain finite difference. Extrapolation principles in [7] are applied to improve the performance of the method presented in [17]. Kia Dithelm in [18] studied that a fast algorithm for the **numerical** solution of initial value problems of the form (1) in the sense of Caputo identifies and discusses potential prob- lems in the development of generally applicable schemes. More recently, Lagrange multiplier method and the homotopy perturbation method are used to solve numerically multi-order fractional differential equation see [19]. Micul [20] considered the problem

Show more
for such services have not been fully examined. This study begins to address this gap by exploring the patients’ pref- erences of South African chronic disease for this service. Finding[r]

mentioned in [6] and [7] of perturbed trapezoid type for k.k ∞ - norm. The inequality is then functional for a Divider of the interval [ρ, [] to acquire many composite **quadrature** rules. The inequality is also functional to special means by Appropriately choosing selecting the function involved to get Certain direct association between different means.

22 Read more

Stieltjes integral, Functions of bounded variation, Lipschitzian func- tions, Monotonic functions, Quadrature rules.. 1..[r]

12 Read more

Runge-Kutta methods [5]-[10], are used for resolving this type of differential equations linear systems. These methods don’t need the calculus of partial de- rivatives of the perturbation function, but they have the problem of a lot of in- termediate calculations with a computational cost increase. It is preferred that the **numerical** methods used to resolve this type of systems verify the property of integrating exactly the homogeneous problem. The methods based on the Schei- fele - **functions** show this good property [1] [2] [11] [12] [13]. Runge-Kutta methods also have this property but it is necessary to do a transformation in the differential equations. This process is complicated because we need to change the **integration** order, furthermore the characteristic equation roots have to be known. The - function series methods are based on a refinement of Taylor ex- pansion [14] [15] and they are applied to problems where the solution has a near sinusoidal behaviour which is not possible to be effectively resolved by Fourier series and Taylor’s expansions.

Show more
13 Read more

A critical point in the construction of the **quadrature** matrices is the inversion of special Vandermonde matrices. In the present implementation, only the selection of non-uniform Chebyshev abscissas was performed with excellent results. In practice, using Chebyshev **polynomial** bases, it can be expected that quadratures of much higher order could be used to solve problems demanding such sophistication.