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A second-order Crank-Nicolson finite difference method is designed to solve a 1D nonlocal Schrödinger equation on the whole real axis. We employ an asymptotically compatible scheme to discretize the spatially nonlocal operator, and apply the Crank-Nicolson scheme in time to achieve a fully discrete infinite system. An iterative technique for the second-order matrix difference equation is then developed to obtain Dirichlet-to-Dirichlet (DtD)-type artificial boundary conditions (ABCs) with the application of z z -transform for the resulting fully discrete system. After that, with the aid of discrete nonlocal Green’s first identity, we derive Dirichlet-to-Neumann (DtN)-type ABCs from DtD-type ABCs. The resulting DtN-type ABCs are available to reduce the infinite discrete system to a finite discrete system on a truncated computational domain, and make it possible to perform stability and convergence analysis for the reduced problem. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed approach.

In this paper, we will deal with the problem of calculating the average velocity of a celestial object revolving around another celestial object in an elliptical orbit. After proving our main theorem to this effect, we will give some alternate forms of the formula for the average velocity, and show that this average value is in fact, attained at certain points of the orbit. We will conclude the paper by providing an intuitively natural and straightforward amendment of this formula.

In our paper (Prohens and Torregrosa, 2013 [1] ) we are concerned with the problem of shape and period of isolated periodic solutions of perturbed analytic radial Hamiltonian vector fields in the plane. Actually, there is a mistake in the formula of the first order approximation of the period given in Corollary 4 . Here we give its proper drafting.

A radiotherapy treatment margin formula has been analytically derived when a standard deviation (SD) of systematic positioning errors Ʃ is relatively small compared to an SD of random positioning errors σ. The margin formula for 0 ≤ Ʃ ≤ σ was calculated by linearly interpolating two boundaries at Ʃ = 0 and Ʃ = σ, assuming that the van Herk margin approximation of k1Ʃ + k2σ is valid at Ʃ = σ. It was shown that a margin formula for 0 ≤ Ʃ ≤ σ may be approximated by k1σ + k2Ʃ, leading to a more general form of k1 max(Ʃ,σ) + k2 min(Ʃ,σ) which is a piecewise linear approximation for any values of Ʃ and σ.

For the Riemannian space Vn, the invariantly associated with Vn space V˜n2 is constructed, which implements a second order approximation for Vn. We studied conformal transformations when the original space is a space of non-zero constant curvature.For the Riemannian space Vn, the invariantly associated with Vn space V˜n2 is constructed, which implements a second order approximation for Vn. We studied conformal transformations when the original space is a space of non-zero constant curvature.

A second order scheme is constructed for the scalar conservation laws with flux function allowed to be discontinuous in the space variable. The corresponding numerical solutionis shown to converge to the (A,B) entropysolution. Numerical results are presented to validate the convergence analysis. In addition, it is inferred that these numerical solutions are comparable with those obtained using the standard minmod limiter.

We consider the initial boundary value problem for the time-fractional diffusion equation with a homogeneous Dirichlet boundary condition and an inhomogeneous initial data a ( x ) ∈ L 2 ( D ) in a bounded domain D ⊂ R d with a sufficiently smooth boundary. We analyze the homogenized solution under the assumption that the diffusion coefficient κ ϵ ( x ) is smooth and periodic with the period ϵ > 0 being sufficiently small. We derive that its first order approximation measured by both pointwise-in-time in L 2 ( D ) and L p ( ( θ , T ) ; H 1 ( D ) ) for p ∈ [ 1 , ∞ ) and θ ∈ ( 0 , T ) has a convergence rate of O ( ϵ 1 / 2 ) when the dimension d ≤ 2 and O ( ϵ 1 / 6 ) when d = 3 . Several numerical tests are presented to demonstrate the performance of the first order approximation.

This work concerns the analysis of wave propagation in random media. Our medium of interest is sea ice, which is a composite of a pure ice background and randomly located inclusions of brine and air. From a pulse emitted by a source above the sea ice layer, the main objective of this work is to derive a model for the backscattered signal measured at the source/detector location. The problem is difficult in that, in the practical configuration we consider, the wave impinges on the layer with a non-normal incidence. Since the sea ice is seen by the pulse as an effective (homogenized) medium, the energy is specularly reflected and the backscattered signal vanishes in a first order approximation. What is measured at the detector consists therefore of corrections to leading order terms, and we focus in this work on the homogenization corrector. We describe the propagation by a random Helmholtz equation, and derive an expression of the corrector in this layered framework. We moreover obtain a transport model for quadratic quantities in the random wavefield in a high frequency limit., 30 pages

The form of Maxwell's equations when expressed in material coordinates has been debated for decades. The most commonly accepted formulation is known to be inconsistent, even to first order approximation. It has been shown that self-consistent formulations are possible when convected coordinates are used. In this presentation, we review these formulations. We then demonstrate their use in solving for the coupled EM/Structural response of a simple parallel plate capacitor bent into an arc.

This paper is concerned with the use of semilocal smoothing splines (or S-splines) for constructing cubature formulas. Such a spline is a piecewise-polynomial function. The first coefficients of each of the polynomials are determined by the smooth joint conditions, and the remaining ones, by the least-squares method. Previous studies were concerned with splines of degree 3 and 5. In the present paper, we consider S-splines of degree n (n = 9, 10). Of special importance for calculation of integrals are the S-splines of class C 0 (the continuous ones). Such splines are employed in building quadrature and cubature formulas of high order of approximation for calculation of one-, two-, and three-dimensional integrals in a simply connected domain to 10th and 11th orders of approximation. The integrable function is assumed to lie in the class C (n+1) (n = 9, 10) in a somewhat larger domain than the original one (in which the integration takes place). It is also assumed that the boundary of the domain is given parametrically. This makes it possible to take into account, with high order of accuracy, the boundary of the domain. The corresponding convergence rates are estimates. A similar approach is also capable of building formulas for integration of smooth functions in multidimensional domains.

In the present work we propose an analytical method for the algebraic calculation of the low frequency expansion of the solution of a scalar scattering problem upon a smooth starshaped scatterer. The method is based on the far field expansion theorem, introduced by Atkinson and Wilcox in the middle of the last century, applied in the low frequency realm. In this view, the need for solving elliptical boundary value problems to obtain the low frequency approximations is replaced by an analytical procedure, easily encoded in an algorithm including only algebraic operators. As a demonstration example, the method is applied to the acoustic scattering problem with plane wave excitation upon three different non-penetrable spherical scatterers. The first low frequency approximations are deduced, yielding the validity of the proposed method by both recovering already known results and accurately deriving higher orders of approximation.

In this paper, we establish some theorems on approximation and Voronovskaja type results for complex modified Sz?sz-Mirakjan-Stancu operators attached to analytic functions having exponential growth on compact disks. Also, we estimate the rate of convergence and the exact order of approximation.

International audience; This paper investigates a first-order and a second-order approximation technique for the shallow water equation with topography using continuous finite elements. Both methods are explicit in time and are shown to be well-balanced. The first-order method is invariant domain preserving and satisfies local entropy inequalities when the bottom is flat. Both methods are positivity preserving. Both techniques are parameter free, work well in the presence of dry states, and can be made high order in time by using strong stability preserving time stepping algorithms. 1. Introduction. The objective of this paper is to develop an invariant domain preserving well-balanced approximation of the shallow water equation with bathymetry using continuous finite elements. There are many finite volume and dis-continuous Galerkin (DG) techniques available in the literature that can solve this problem efficiently up to second and higher order in space. Examples of schemes that are well balanced at rest and robust in the presence of dry states can be found, for example, in Audusse and Bristeau [1], Audusse et al. [2], Bollermann, Noelle, and Lukáčová-Medvidová [6], Gallardo, Parés, and Castro [14], Kurganov and Petrova [23], Perthame and Simeoni [27], Ricchiuto and Bollermann [28]. We refer the reader to the book of Bouchut [7] for a review on this topic, to the paper of Xing and Shu [32] for a survey on finite volume and DG methods, and to the paper [23] for a survey of central-upwind schemes. However, to the best of our knowledge, these types of approximations are not developed in the context of continuous finite elements. Or we should say that no robust continuous finite element technique is yet available in the literature that guarantees second-order accuracy, works properly in every regime (subcritical, transcritical, transcritical with hydraulic jumps, wet, and dry regions) and is well-balanced at rest. We propose such a method in the present paper. Two variants of the method are discussed: one variant is first-order accurate in space, positivity preserving, and preserves every convex invariant domain of the system in the absence of bathymetry; the other variant is second-order accurate in space and positivity preserving. Both variants are explicit in time and use continuous finite elements on unstructured meshes.

We describe three-dimensional and planar models of hyperelastic fiber reinforced materials characterized by statistical distribution of the fiber orientation. Our models are based on a second order approximation of the strain energy density in terms of the fourth pseudo-invariant I ¯ 4 , typically employed in the description of fiber reinforced materials. For a particular choice of the strain energy density associated to the fiber reinforcement, it is possible to derive the explicit expression of the material and spatial stress tensors and of the stress covariance tensors. The mechanical behavior of the models is assessed through uniaxial, biaxial and shear tests.

In the present paper, we consider nonclassical problems for multidimensional elliptic equations. A finite difference method for solving these nonlocal boundary value problems is presented. Stability, almost coercive stability and coercive stability for the solutions of first and second orders of approximation are obtained. The theoretical statements for the solutions of these difference schemes are supported by numerical examples for the two-dimensional elliptic equations. Copyright © 2014 John Wiley & Sons, Ltd.

In this study, we investigate approximation properties and obtain Voronovskaja type results for complex modified Szász-Mirakjan operators. Also, we estimate the exact orders of approximation in compact disks and prove that the complex modified Szász-Mirakjan operators attached to an analytic function preserve the univalence, starlikeness, convexity and spirallikeness in the unit disk.

To obtain the approximately analytical solution of double-walled carbon nanotubes (DWNTs) nonlinear vibration. In this study, homotopy perturbation method (HPM) was used to solve nonlinear vibration equation of DWNTs. Novel and accurate analytical solutions for the frequency and displacement are derived. Comparison of the result obtained by the HPM with exact solutions reveals that only the first or second order approximation of the HPM leads to higher accurate solution.

In this paper we discuss the application of second-order adjoint based perturbation techniques to linear and nonlinear time dependent problems. These techniques are based on the property of the adjoint problem which allows calculating the set of first and second order coefficients by solving a number of adjoint systems. The derivation of a second order adjoint theory that can be used to calculate the second order Taylor components of a functional response is first presented for nonlinear problems, and then reformulated for linear systems. This implementation is based on the Adjoint Sensitivity Analysis Procedure, originally presented by Cacuci. The theoretical part is followed by the application of the technique to some classical reactor physics problems: a fuel depletion calculation, a linear and a nonlinear point-kinetic problem. The results obtained using the second order theory are compared with the values obtained using a first order approximation and a direct solution of the forward problem. Our first results show that the procedure provides good estimations in presence of higher order perturbation components, being able to reconstruct the responses of interest even in presence of large perturbations. The main differences between linear and nonlinear formulations and their computational implications are also discussed.

In this study, analytical solutions are obtained by homotopy perturbation method (HPM) for the nonlinear vibration equation of single-wall nanotube (SWNT). Novel and accurate analytical solutions for the frequency and displacement are derived. Comparison of the result obtained by the HPM with exact solutions reveals that only the first or second order approximation of the HPM leads to higher accurate solution.

In this paper, a quantitative measure of partial observability is de_ned for PDEs. The quantity is proved to be consistent in well-posed approximation schemes. A first order approximation of an unobservability index using empirical gramian is introduced. For linear systems with full state observability, the empirical gramian is equivalent to the observability gramian in control theory. The consistency theorem is exemplified using a Burgers' equation.

The influence of the quadratic and cubic terms on non-linear dynamic characteristics of the angle-ply composite laminated rectangular plate with parametric and external excitations is investigated. The method of multiple time scale perturbation is applied to solve the non-linear differential equations describing the system up to and including the second-order approximation. All possible resonance cases are extracted and investigated at this approximation order. Two cases of the sub-harmonic resonances cases (Ω2 ≅ 2ω1 and Ω2 ≅ 2ω2) in the presence of 1:2 internal resonance ω2 ≅ 2ω1 are considered. The stability of the system is investigated using both frequency response equations and phase-plane method. It is quite clear that some of the simultaneous resonance cases are undesirable in the design of such system as they represent some of the worst behavior of the system. Such cases should be avoided as working conditions for the system. Some recommendations regarding the different parameters of the system are reported. Comparison with the available published work is reported.

Attaching to a compact disk \({\overline{\mathbb{D}_{r}}}\) in the quaternion field \({\mathbb{H}}\) and to some analytic function in Weierstrass sense on \({\overline{\mathbb{D}_{r}}}\) the so-called q-Bernstein operators with q ≥ 1, Voronovskaja-type results with quantitative upper estimates are proved. As applications, the exact orders of approximation in \({\overline{\mathbb{D}_{r}}}\) for these operators, namely \({\frac{1}{n}}\) if q = 1 and \({\frac{1}{q^{n}}}\) if q > 1, are obtained. The results extend those in the case of approximation of analytic functions of a complex variable in disks by q-Bernstein operators of complex variable in Gal (Mediterr J Math 5(3):253–272, 2008) and complete the upper estimates obtained for q-Bernstein operators of quaternionic variable in Gal (Approximation by Complex Bernstein and Convolution-Type Operators, 2009; Adv Appl Clifford Alg, doi:10.1007/s00006-011-0310-8, 2011).

Efficient predictions of sound diffraction around objects is vital in room-acoustics simulations. Recent efforts using the Physical Theory of Diffraction (PTD) [Rozynova, A. and Xiang, N., J. Acoust. Soc. Am., 141, pp. 3785 (2017)] have been reported on some canonical objects of infinite size. This research has been further developed based on solving diffraction problems on finite, rectangular plates which are free from grazing singularities. Separate solutions are considered for diffraction from two pairs of the rigid plate edges, the edge waves around the plate corner are ignored due to the lower order of diffraction contributions in the far-field. This paper will discuss numerical implementations of the theoretical predictions to demonstrate the efficiency of the PTD in room-acoustics simulations for finite sized rectangular plates. The implemented numerical results will also be compared with some preliminary experimental results carried out with a goniometer.

We present an elementary method to obtain the equations of the shallow-water solitary waves in different orders of approximation. The first two of these equations are solved to get the shapes and propagation velocities of the corresponding solitary waves. The first-order equation is shown to be equivalent to the Korteweg-de Vries (KdV) equation, while the second-order equation is solved numerically. The propagation velocity found for the solitary waves of the second-order equation coincides with a known expression, but it is obtained in a simpler way. By measuring the propagation velocity of solitary waves in the laboratory, we demonstrate that the second-order theory gives a considerably improved fit to experimental results., Comment: 15 pages, 8 EPS figures, uses IOP class file for LaTeX2e, slightly revised version

A large number of papers are adopting the Beavers and Joseph (BJ) condition (Beavers and Joseph, J Fluid Mech 30(1):197–207, 1967) for describing the boundary condition between a saturated porous medium and a free fluid, in place of the adherence condition. The aim of the paper is to bring some insight into the domain of validity of the BJ condition. After a short review of some papers on the subject, we point out that the experimental conditions of BJ do not show a good separation of scales. That makes the BJ condition not transposable to different macroscopic situations. When the separation of scales is good, an intrinsic boundary condition is obtained by using the homogenization technique of multiple scale asymptotic expansions. As in the BJ condition and other theoretical works, e.g., Jager and Mikelic (Transp Porous media, 2009, to appear) we obtain the adherence condition of the free fluid at the first order approximation. However, the corrector to the adherence condition is \({\mathcal{O}(\varepsilon^2)}\) whereas it is \({\mathcal{O}(\varepsilon)}\) in the BJ condition, where e is the separation of scales parameter.

In this study, an accurate analytical solution for Duffing equations with cubic and quintic nonlinearities is obtained using the Homotopy Analysis Method (HAM) and Homotopy Pade technique. Novel and accurate analytical solutions for the frequency and displacement are derived. Comparison between the obtained results and numerical solutions shows that only the first order approximation of the Homotopy Pade technique leads to accurate solution with a maximum relative error less than 0.4%.

International audience; We investigate stochastic homogenization for some degenerate quasilinear pa rabolic PDEs. The underlying nonlinear operator degenerates along the space variable, uniformly in the nonlinear term: the degeneracy points correspond to the degeneracy points of a reference diffusion operator on the random medium. Assuming that this reference diffusion operator is ergodic, we can prove the homogenization property for the quasilinear PDEs, by means of the first order approximation method. The (nonlinear) limit operator needn't be nondegenerate. Concrete examples are provided.

This paper deals with the non-variational Galerkin- B -spline method recently developed by the authors. It is applied to circular thin flexible plates within the context of Von Karman equations. Our main result shows that the low orders of approximation have been sufficient to build the solutions from the basis functions in a high quality. Comparison of solutions obtained to variational approach with known ones is performed.

In this paper we study analytically a class of waves in the variant of the classical second-order approximation Boussinesq system given by ∂ t u − b ∂ x x t u + c ∂ x x x x t u = − ∂ x v − ∂ x ( u v ) − ( 1 3 − 2 b ) ∂ x x x v + b ∂ x x x ( u v ) + b ∂ x ( u ∂ x x v ) − a ∂ x x x x x v , ∂ t v − b ∂ x x t v + c ∂ x x x x t v = − ∂ x u + 1 2 ∂ x x x u − v ∂ x v − ∂ x ( u ∂ x x u ) + b ∂ x ( v ∂ x x v ) − d ∂ x x x x x u , where a , b , c , d are some real constants. This equation is ill-posed and most initial conditions do not lead to solutions. Nevertheless, we show that, for almost every a , b , c and d it admits solutions that are quasiperiodic in time.

We present a closed form solution for the upscaled diffusion coefficient and derive a first-order homogenized approximation to linear flow equations with periodic and rapidly oscillating coefficients. The coefficients are defined as step functions describing inclusions of various shapes in a main matrix. This constitutes the n-dimensional upscaled version of Darcy’s law for linear flow in such systems. We consider the two-scale asymptotic expansion of the solution of the flow equation, and develop a corrector to an analytical approximation for the solution of the periodic cell-problem. We demonstrate that the proposed analytical form for the effective coefficient satisfies the generalized Voigt–Reiss’ inequality and is in agreement with other known theoretical results, including the geometric average for the checkerboard geometry, and with some published numerical results. The zeroth-order approximation in H1(Ω) is readily obtained and the first-order approximation in L2(Ω) is derived from the proposed analytical approximation to the basis functions. The analytical basis functions are also used to define a correction function that incorporates the heterogeneous features into the zeroth-order approximation to the gradient and flux, which considerably improves the convergence results. We illustrate the procedure with coefficients describing square inclusions with contrast ratios between the inclusion and the matrix as 10:1, 100:1, 1000:1 and 1:10, respectively. We demonstrate numerically that the convergence properties of the proposed approximations agree with the classical theoretical results in homogenization theory.

In this paper, we consider the orders of approximation of functions on the whole real axis by operators of Fejer type in the Banach space with the so-called generalized Holder metric.

The solution of a wide-angle parabolic wave equation solution using shift-map and finite-difference techniques is presented. The so-called Claerbout equation allows propagation angles up to 450 from the paraxial direction. The shift-map technique can be incorporated into this scheme allowing a varying terrain to be considered. The Claerbout one-order solution with respect to the terrain slope performs well for slopes up to about 15" and discontinuous slope changes up to about 30" and corresponds to the well-known split-step solution. The Claerbout second-order approximation is able to give better results than the split-step solution. Besides, under the condition of the phase error is less than 0.002, the acceptable propagation angle limits of the Claerbout and Greene are 360 and 470 respectively. This paper derives Greene second-order solution. To a certain extent, the Greene approximation can certainly give better results.

In this paper, asymptotic behavior of convolution of distributions belonging to two subclasses of distributions with exponential tails are considered, respectively. The precise second-order tail asymptotics of the convolutions are derived under the condition of second-order regular variation., Comment: 22 pages

We design a general type of spherical mean operators and employ them to approximate $L_p$ class functions. We show that optimal orders of approximation are achieved via appropriately defined K-functionals of fractional orders.

This paper studies a general method for estimating the length of a parametric curve using only samples of points. We show that by making a special choice of points, namely the Gauss–Lobatto nodes, we get higher orders of approximation, similar to the behaviour of Gauss quadrature, and we derive some explicit examples.

In recent works [1], [4], it has been shown that the damping of a linear time invariant system relates to the so-called characteristic ratios (αk, k = 1, ..., n - 1) which are defined by coefficients of the denominator of the transfer function. However, the exact relations are not yet fully understood. For the purpose of exploring the issue, this paper presents the analysis of time response sensitivity to the characteristic ratio change. We begin with the sensitivity of output to the perturbations of coefficients of the system denominator and then the first order approximation of the αk perturbation effect is computed by an explicit transfer function. The results are extended to all-pole systems in order to investigate which characteristic ratios act dominantly on step response. The same analysis is also performed to a special class of systems whose denominator is composed of so called -polynomial. Finally, some illustrative examples are given.

This paper applies the technique of successive approximation to the nonlinear theory of elastic materials with voids, as developed by Nunziato and Cowin [2]. The complete set of equations for the first and second order approximation is derived. The torsion problem within the second-order theory is studied. It is shown that the solution can be expressed in terms of solutions of some two-dimensional problems. The theory is applied to study the torsion of a circular cylinder. It is shown that an infinitesimal twist produces a second-order volume fraction field which is proportional to the square of the amount of torsion.

Two versions of multiple scales and Krylov–Bogoliubov–Mitropolsky (KBM) methods are extended to obtain the asymptotic solutions of a class of strongly nonlinear oscillators with slowly varying parameters. A comparison of the two methods is made to show that the multiple scales method is equivalent to the KBM method for the first order approximation. An example of pendulum with slowly varying length is given to illustrate the comparison of the two methods.

We establish multi-valued and infinite-dimensional versions of stability theorems in the first-order approximation. The differential inclusions treated as first-order approximations can be nonautonomous and, in several cases under study, nonhomogeneous with respect to the phase variable. We outline applications in stability theory of solutions to parabolic inclusions.

A systematic exposition of single-point pdf methods for reacting flows based on cumulant expansion techniques is carried out. A second order approximation for the pdf equation is established. It is shown that transport in velocity-scalar space is governed by drift and diffusion formally consistent with the Fokker–Planck equation. The main result is the explicit representation of the diffusivity as a time integral over Lagrangian two-time correlations. This diffusivity is not positive definite, hence the width of the pdf may be reduced or increased depending on the evolution of the diffusivity. The relation of the present approach to the standard method is discussed for the example of the single scalar pdf.

The analysis using the Carnahan Muller shape factor approach for the original chain-SAFT equation of state, which is based on the Carnahan-Starling hard-sphere reference equation of state, has been performed. The performance of the second order approximation for shape factor of every proposed equation of state of hard convex fluid has been found to be of comparable accuracy when compared to their respective original equation of state.

A control strategy is proposed for variable damping elements (VDEs) used together with auxiliary stiffness elements (ASEs) that compose a time-varying non-linear Maxwell (NMW) element, considering near-future excitation influence. The strategy first composes a state equation for the structural dynamics and the mechanical balance in the NMW elements. Next, it establishes a cost function for estimating future responses by the weighted quadratic norms of the state vector, the controlled force and the VDEs' damping coefficients. Then, the Euler equations for the optimum values are introduced, and also approximated by the first-order terms under the autoregressive (AR) model of excitation information. Thus, at each moment t k , the strategy conducts the following steps: (1) identify the obtained seismic excitation information to an AR model, and convert it to a state equation; and (2) determine VDEs' damping coefficients under the initial conditions at t k and the final state at t k+L , using the first-order approximation of the Euler equations. The control effects are examined by numerical experiments.

In his application of the homotopy perturbation (HP) method to the problem of the three-dimensional flow past a stretching sheet, P.D. Ariel [The three-dimensional flow past a stretching sheet and the homotopy perturbation method, Comput. Math. Appl. 54 (2007) 920–925] encountered secular terms in the second order approximation, which he could not avoid. It is shown here that these secular terms can be removed by coordinate straining. Moreover, the form of an HP solution, which is free from secular terms at all levels of approximation, and which can be determined recursively, is indicated.

In this note we relate two methods of convergence acceleration for ordinary continued fractions, the first one is due to Lorentzen and Waadeland (Jacobsen and Waadeland, 1980, 1998), the second one to Waadeland (1986, 1987, 1987, 1987, 1988). (C) 1999 Elsevier Science B.V. All rights reserved. AMS classification: 40A15; 65B99; 65Q05. ispartof: Journal of computational and applied mathematics vol:101 issue:1 pages:167-175 status: published

In this paper we present a generalized theory of higher order vector finite elements. This theory aims at generating tctrahedral Whitney elements with tangential continuity, having better orders of approximation compared to the well-known first order edge elements. The proposed methodology deals with some fundamental theoretical issues about Whitney elements, such as the proper definition of degrees of freedom and the correct modeling of the nullspace of the curl operator, corresponding to the class of irrotational fields. The whole set of properties is enforced to the shape functions, in order to produce their explicit expressions. The new second and third order vector finite elements are implemented in a formulation for 3-D electromagnetic scattering. A second order absorbing boundary condition for rectangular boundaries is used to truncate the infinite space. The numerical results show the improved accuracy of the approximation, obtained by the use of higher order Whitney elements, thus render...