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The simulation of the predissociation spectrum of the Na 2 2 3 Π g ∼ 3 3 Π g ← b 3 Π u (v = 14 , J = 14) transition, observed earlier with the Perturbation Facilitated Optical–Optical Double Resonance (PFOODR) experimental method, is done within the Optimizer project with our Split-operator computational package using Padé approximations for all functions involved in the model. This simulation reproduces the experimental spectrum satisfactory well. The parameters of the model functions involved in the simulations are determined and reported. A brief description of our computational methods and programs is presented. [Display omitted] • Simulation of the predissociation spectrum Na 2 2 3 Π g ∼ 3 3 Π g ← b 3 Π u (v = 14 , J = 14) is done. • New version of the multichannel Split-operator computational package is presented. • New version of the Padé-approximant computational package is presented. • All the programs are adapted to the Optimizer project and used for the modeling. • Quantitative parameters of all the functions involved in the model are determined. [ABSTRACT FROM AUTHOR]

The performed numerical analysis reveals that Wynn's identity for the compass 1 / (N − C) + 1 / (S − C) = 1 / (W − C) + 1 / (E − C) = 1 / η (here C stands for centre, the other letters correspond to the four directions of the compass) gives the long sought criterion, the minimal | η | or Image 1 -criterion, for the choice of the optimal Padé approximant. The work of this method is illustrated by calculation of multipoint Padé approximation by a new formula for calculation of this best rational approximation. The calculation of the optimal Padé approximant by this criterion is demonstrated in calculation of series summation – frequently encountered problem in theoretical physics. This study originates from a magneto-hydrodynamic problem of heating of solar corona by Alfvén waves, where the present method is used for a predictor in solution of differential equations. In such a way, an efficient and valuable control mechanism for N -point Padé approximant calculation is proposed. We believe that the suggested method and criterion can be useful for many applied problems in numerous areas not only in physics but in any scientific application where differential equations are solved. The obtained new solution of the Cauchy-Jacobi problem is illustrated by a Fortran program. The algorithm is generalized for the case of the first K -derivatives at N -nodal points. The numerical analysis of probability distribution function of the errors will review, which formula for extrapolation of functions deserves to be implemented in the applied mathematics software like Mathematica and Maple. [ABSTRACT FROM AUTHOR]

This paper presents an interval approach for the Holomorphic Embedding Load Flow Method (HELM) considering load and generation uncertainties. In the proposed method, power flow equations are considered as holomorphic complex functions, being their solutions determined recursively based on the expansion in Maclaurin series and analytical continuation using Padé approximants. The concepts of the interval arithmetic are applied into the HELM equations assuming active and reactive powers as uncertain data to obtain interval values associated with the voltage phasors for all the system buses. The procedures to calculate interval results are described along the paper including the use of interval arithmetic concepts applied into the HELM equations. IEEE test systems such as 6, 14, 30 and 57-bus are used for computational simulations, being the results compared with Monte Carlo (MC) approach, Krawczyk method, Taylor series, affine arithmetic and the traditional interval power flow. As important contribution, a novel interval power flow technique is presented allowing to compute interval values closer to the ones determined by MC simulations than other alternative methods. Results are obtained in a computational time approximately 500 times faster than the MC approach considering 20,000 simulations and the interval sensitivities are, at least, twice lower than other interval methods. • The Interval Holomorphic Embedding Load Flow Method (IHELM) is presented. • Load and generation uncertainties are considered. • Results are obtained in an advantageous computational time. • Voltage intervals are determined by a direct method. [ABSTRACT FROM AUTHOR]

Highlights • Historically, obtaining closed-form approximations required trial and error. • We show such approximations can be derived without iterative trial and error. • After impedance approximations we successfully test our method on Green's functions. • The method can be used on any series or integral that has no closed form solution. Abstract Engineering approximations of physical systems sometimes produce models in which real-valued model-based physical-distances are added to complex-valued distances and/or (for electrical systems), real-valued current/charge image intensities are replaced with complex-valued quantities. These models are arrived at often using ad hoc approximations that allow infinite integrals or series to be approximated in closed form. Arriving at accurate ad hoc approximations in a compatible analytic form is often the difficult step in the derivation of these approximations. In this paper, we show that this difficult ad hoc step can be replaced for many classes of functions with the use of analytic continuation via Padé approximants, along with some reasonable engineering judgement. We apply our approach to several existing approximations in the electrical engineering field (overhead transmission line impedance, underground cable impedance and Green's functions used in ground potential rise calculations) and show that these approximations can be derived elegantly, without the need for grand leaps of insight, and provide a basis for both distance parameters and current/charge intensities that are complex-valued. [ABSTRACT FROM AUTHOR]

The objective of present study is to investigate the two-dimensional magnetohydrodynamic (MHD) flow of a viscous fluid over a constant wedge immersed in a porous medium with velocity slip condition. The flow is induced by suction/injection and also by the mainstream flow that is assumed to vary in a power-law manner with co-ordinate distance along the boundary. Similarity transformations are used to convert the governing nonlinear boundary layer equations into a third order Falkner–Skan equation. This equation is solved analytically by using a novel analytical method called DTM-Padé technique which is a combination of the differential transformation method and the Padé approximation. This method is applied to give solutions of equation with boundary condition at infinity. Graphical results are presented to investigate the effects of the velocity slip parameter, Hartmann number, permeability, suction/injection parameter and nonlinear pressure gradient on the flow-field. Further, the results of the present analysis have been compared with the corresponding results available in literature. Our results have been found in excellent agreement. [ABSTRACT FROM AUTHOR]

A regular perturbation method in which the small parameter is taken as the characteristic frequency of variation of a two-dimensional slit wall height is applied to the creeping flow of a viscoelastic fluid. The Upper-Convected Maxwell model has been chosen as the constitutive equation. The equations of motion are expressed in terms of the stream function as the only unknown. The stream function is expressed then as a regular perturbation expansion up to the fourth order in the small parameter. When these expansions are written for slits with slowly varying walls of any shape, curvature and higher-order derivatives appear in the expansion solution. From the stream function solution, we distinguish two different types of viscoelastic higher-order effects: one is symmetric and the other is a loss of fore-and-aft flow symmetry due to the convected derivatives appearing in the constitutive equation. The expansion is formally validated with a ratio between two consecutive higher-order terms. It shows that the accuracy of the perturbation will depend on four factors: the small parameter ϵ, the Weissenberg number W e 0 , the local depth h and the shape of the slit. In some instances, Padé approximants can restore a diverging perturbation expansion and extend the range of viscoelastic levels for which the expansion is usable. Results for diverging and for converging flows in a wedge, as well as flows in corrugated slits, are checked against Finite Elements Method calculations. At the same level of viscoelasticity, the stress profiles are completely different in a diverging flow from a converging flow. In particular, the shear stress component of the stress tensor becomes non-monotonic in divergent flows. [ABSTRACT FROM AUTHOR]

We investigate the accuracy and efficiency of linear and non-linear transformations on the truncated series solutions obtained by asymptotic methods for simple shear and uniaxial elongational viscoelastic flows. The Upper Convected Maxwell, Oldroyd-B, exponential and linear Phan-Thien and Tanner, Giesekus, and FENE-P differential constitutive models are used. The major dimensionless quantity of interest is the Weissenberg number ( Wi ) in terms of which the unknown dependent flow variables are expanded suitably in series form. The standard regular perturbation method is performed at the Newtonian limit (zero Weissenberg number), while regular or singular perturbation techniques are required to perform the analysis at the purely elastic limit (infinite Weissenberg number). It is shown that the asymptotic approximations at low Wi converge very slowly and have limited radius of convergence, while at high Wi the asymptotic approximations are very accurate even for Wi close to unity. Further improved convergence is achieved by applying a variety of techniques on the original asymptotic solutions, such as Euler's linear transform, Shanks’ non-linear transform, and Padé-type approximants. It is found that the accuracy and the radius of convergence of the transformed solutions increase substantially compared to the original ones. By combining the transformed solutions from both limits, excellent agreement with the exact solutions is demonstrated over the entire range of the Weissenberg number, from zero to infinity, except for a very narrow region close to Wi = 1/2 under uniaxial elongational flow. The results show that, contrary to intuition, viscoelastic phenomena at Wi = O(1) can be predicted accurately using non-linear analysis. [ABSTRACT FROM AUTHOR]

We propose a new ARCH-type model that uses a rational function to capture the asymmetric response of volatility to returns, known as the “leverage effect”. Using 10 individual stocks on the Tokyo Stock Exchange and two stock indices, we compare the new model with several other asymmetric ARCH-type models. We find that according to the deviance information criterion, the new model ranks first for several stocks. Results show that the proposed new model can be used as an alternative asymmetric ARCH-type model in empirical applications. [ABSTRACT FROM AUTHOR]

The present study investigates three different kinds of nonlocal plate theories for capturing the small length scale effect of microstructured plates in elasticity. One kind is the classical stress gradient Eringen’s theory as applied to the Kirchhoff–Love plate model, another kind is based on the continualization of the discrete lattice model and the last kind is a combination of Eringen’s model with some additional gradient curvature terms. From the three associated governing equations, analytical vibration and buckling solutions of these equivalent continuous systems are obtained for simply supported rectangular plates and their accuracies are assessed by comparison to the reference exact lattice model. The exact lattice solution is derived for the discrete microstructured plate model via the resolution of a linear difference boundary value problem. It is found that the continualized model (nonlocal continuum model constructed from the lattice equations) provides more accurate results than the traditional Eringen’s theory. This model introduces some nonlocal terms both in the constitutive law and in the balance equations. It appears that this new nonlocal continuum perfectly fits the lattice plate model with a length scale factor that only depends on the element size of the microstructured plate, independently on the kind of analysis, static versus dynamics. However, when considering the Eringen stress gradient model, the small length scale coefficient determined by comparing the solutions from the continuous models and the microstructured beam–grid model, is shown to be dependent on the buckling mode and the geometry of plate. Ultimately, the models are evaluated based on their ability to capture microstructured scale effects. [ABSTRACT FROM AUTHOR]

In this paper, we consider a transient vertical one-dimensional flow problem of water in unsaturated porous media, modelled by the non-linear Richards equation. Constitutive relations of Van Genuchten will be employed and Padé approximants are used to represent the hydraulic capacity and conduction in a simplified fashion. From the proposed methodology a construction of a functional solution is presented with the objective to define an initialisation of a recursive scheme in the spirit of Adomian decomposition. This solution is optimised and evaluated using the governing equation for a self-consistency test. The results are presented for some soil types and its related soil parameters, that are reported in the literature. [ABSTRACT FROM AUTHOR]

Padé approximants are used to model physicochemical, transport, and thermodynamic properties of seawater for industrial applications. Sophisticated mathematical models to calculate saline solution properties are presented in the literature for applications in which very high accuracy is needed, such as oceanographic studies. Such accuracy is not necessary in the design and optimization of processes in which seawater is used, mainly in the mining industry (mineral concentration, hydrometallurgy, and refining processes). The properties studied are density, specific heat, osmotic coefficient, viscosity, thermal conductivity, surface tension, enthalpy, entropy, vapor pressure, latent heat of vaporization, and boiling temperature elevation. Padé approximants represent a useful tool in applied mathematics and have not been used to model saline solutions properties in the way presented here. Padé results are compared with Taylor series and simple polynomial expansions. The general models obtained are of rational type and give deviations similar to those provided by more sophisticated models presented in the literature. [ABSTRACT FROM PUBLISHER]

The objective of present paper is to revisit the problem pertaining to the boundary layer equations governing the flow of viscous fluid due to a stretching surface. The series solution method has been effectively implemented to a related integro-differential equation. The condition at infinity is applied to a related Pade approximants of the obtained series solution. The features of the flow characteristic have been analyzed and discussed. Comparison of the obtained results for some particular cases of the present study has been done with the earlier results and they have been found in good agreement. [ABSTRACT FROM AUTHOR]

Efficient solvers for the unsteady Navier–Stokes equations are presented. A classic time-stepping scheme is combined with high-order nonlinear solvers coupling homotopy and a perturbation technique. Polynomial and rational representations are used to approximate the unknowns of the problem. A pseudo-residual criterion is proposed to improve the efficiency of the solvers. The numerical example considered in this paper is the time-periodic two-dimensional flow around a circular cylinder. Comparisons with the classical first order Newton–Raphson solver are performed. Numerical results reveal that a lower number of matrix factorization is needed with the proposed methods, decreasing the computational effort. [ABSTRACT FROM AUTHOR]

Approximants to the inverse Langevin and Brillouin functions appear in diverse contexts such as polymer science, molecular dynamics simulations, turbulence modeling, magnetism, theory of rubber. The exact inverses have no analytic representations, and are typically not implemented in software distributions. Various approximants for the inverse Langevin function L - 1 had been proposed in the literature. After proving asymptotic features of the inverse functions, that had apparently been overlooked in the past, we use these properties to revisit this field of ongoing research. It turns out that only a subset of existing approximations obeys the relationships. Here we are able to derive improved (or ‘corrected’) approximations analytically. We disqualify the classical Padé solution approach that is typically used to obtain coefficients for approximate forms, and recommend a simple rational function L FENE - 1 ( y ) / ( 1 + y 2 / 2 ) that has a maximum relative error of 1–2 orders of magnitude smaller compared with the usually employed approximations L FENE - 1 ( y ) = 3 y / ( 1 - y 2 ) (50% maximal relative error) or also L Cohen - 1 ( y ) = L FENE - 1 ( y ) ( 1 - y 2 / 3 ) (4.94%), while it is exactly as efficiently implemented and convenient for analytic (both force and energy) calculations. A number of applications is worked out. Moreover is the strategy proposed in this manuscript general and can be equally applied to fitting problems when asymptotic features are known. [ABSTRACT FROM AUTHOR]

The scaled boundary finite element method is employed to obtain highly accurate solutions of the stress intensity factors (SIFs) and T-stress for standard homogeneous and bimaterial fracture specimens. The solutions are assembled into a practical database (up to 6 significant figures) corresponding to various crack lengths. Using a curve fitting technique, precise Padé approximants of the SIFs and T-stress are also developed, which retain a maximum error of <0.28% for any specimen. The findings of this paper are useful for engineering analysis and design, as well as verification of current and future numerical schemes for computing SIFs and T-stress. [ABSTRACT FROM AUTHOR]

Sequence transformations are valuable numerical tools that have been used with considerable success for the acceleration of convergence and the summation of diverging series. However, our understanding of their theoretical properties is far from satisfactory. The Euler series E ( z ) ∼ ∑ n = 0 ∞ ( − 1 ) n n ! z n is a very important model for the ubiquitous factorially divergent perturbation expansions in theoretical physics and for the divergent asymptotic expansions for special functions. In this article, we analyze the summation of the Euler series by Padé approximants and by the delta transformation, which is a powerful nonlinear Levin-type transformation that works very well in the case of strictly alternating convergent or divergent series. Our analysis is based on a very recent factorial series representation of the truncation error of the Euler series. We derive explicit expressions for the transformation errors of Padé approximants and of the delta transformation. A subsequent asymptotic analysis proves rigorously the convergence of both Padé and delta. Our asymptotic estimates clearly show the superiority of the delta transformation over Padé. This is in agreement with previous numerical results. [ABSTRACT FROM AUTHOR]

In this article, we investigate Padé approximants of hyper-exponential to base on the first moments and the matrix-exponential representation of Padé approximated function. An explicit formula is given for the Laplace transform in the time to calculate finite time ruin probabilities of classical Cramer–Lundberg model. This formula generalizes the ultimate ruin probabilities formula of Asmussen and Rolski (1991) [12] . To illustrate this formula, several numerical examples with different values u are given. [ABSTRACT FROM AUTHOR]

Abstract: Master equations for the evolution of complex networks with positive (birth) and negative (death) transition probabilities per unit time are analyzed. Explicit equations for the time evolution of the total number of nodes and for the relative node frequencies are given. It is shown that, in the continuous limit, the master equation reduces to a Fokker–Planck equation (FPE). The basic dynamical function for its stationary solution is the ratio between its drift and diffusion coefficients. When this ratio is approximated by partial fractions (Padé’s approximants), a hierarchy of stationary solutions of the FPE is obtained analytically, which are expressed as an exponential times the product of powers of monomials and binomials. It is also shown that if the difference between birth and death transition probabilities goes asymptotically to zero, the exponential factor in the solution is absent. Fits to real complex network probability distribution functions are shown. Comparison with rank-ordered data shows that, in general, the value of this exponential factor is close to unity, evidencing crossovers among power-law scale invariant regimes which might be associated to an underlying criticality and are related to a generalization of the beta distribution. The time dependent solution is also obtained analytically in terms of hyper-geometric functions. It is also shown that the FPE has similarity solutions. The limitations of the approach here presented are also discussed. [Copyright &y& Elsevier]

We obtain highly accurate solutions to the Thomas–Fermi equations for atoms and atoms in very strong magnetic fields. We apply the Padé–Hankel method, numerical integration, power series with Padé and Hermite–Padé approximants and Chebyshev polynomials. Both the slope at origin and the location of the right boundary in the magnetic-field case are given with unprecedented accuracy. [ABSTRACT FROM AUTHOR]

A study of the heat conduction through a miniature composite slab showing large differences in scale is presented. The system consists of a highly conductive thin sensing film, subjected to self-heating and encapsulated by insulating layers. The formalism of the Laplace transfer functions is used to relate the operating temperature of the coated sensing film to the unknown outermost temperatures and to self-heating. The use of Padé approximants provides approximations of these transfer functions by rational functions that are much more accurate than most of the concurrent approaches. Then the real-time evolution of the operating temperature of the sensing film is established without resorting to the Laplace inversion, but using recursive digital filters constructed from the Padé approximants. Finally, the inverse analysis that is proposed allows to regain the evolution of the outermost temperatures from the knowledge of the operating temperature, even in the presence of a significant self-heating and a nonlinear regime. The present systemic approach could be applied to optimize and accurately describe the real-time operation of many miniature resistive sensors used for example in biomedical instrumentation. [ABSTRACT FROM AUTHOR]

Abstract: The numerical simulation of colliding solitary waves with compact support arising from the Rosenau–Hyman equation requires the addition of artificial dissipation for stability in the majority of methods. The price to pay is the appearance of trailing tails, amplitude damping, and delays as the solution evolves. These undesirable effects can be corrected by properly counterbalancing two sources of artificial dissipation; this procedure is designed by using the slow time evolution of the parameters of the solitary waves under the presence of the dissipation determined by means of adiabatic perturbation methods. The validity of the tail removal methodology is demonstrated on a Padé numerical scheme. The tails are completely removed leaving only a small compact ripple at the original position of their front, and the numerical stability of the scheme under compacton collisions is preserved, as shown by extensive numerical experiments for several values of n. [Copyright &y& Elsevier]

Abstract: We derive explicit expressions for the eigenvalues (spectrum) of the discontinuous Galerkin spatial discretization applied to the linear advection equation. We show that the eigenvalues are related to the subdiagonal Padé approximation of when pth degree basis functions are used. We derive an upper bound on the eigenvalue with the largest magnitude as . We demonstrate that this bound is not tight and prove that the asymptotic growth rate of the spectral radius is slower than quadratic in p. We also analyze the behavior of the spectrum near the imaginary axis to demonstrate that the spectral curves approach the imaginary axis although there are no purely imaginary eigenvalues. [Copyright &y& Elsevier]

Abstract: A nonlinear third order singular ordinary differential equation is a model for a viscous fluid which drains over a wet surface as a thin film down a vertical wall. Here, we discuss the physical processes of a thin liquid film evolution. The classical thin film equation is being derived, and the series solution is obtained by employing the Adomian decomposition method (ADM). In this paper, a new way for the phase planes analysis and traveling wave solution by ADM is presented. The proposed algorithm is implemented using mathematical software Mathematica 6.0. [Copyright &y& Elsevier]

Abstract: The exact expression for the maximum tension of a pendulum string is used to obtain a closed-form approximate expression for the solution of a simple pendulum in terms of elementary functions. This approximate solution has a rational harmonic expression and depends on an unknown function, which must be determined. This unknown function is expanded using the Padé approximant and two new parameters are introduced which are determined by means of a term-by-term comparison of the power series expansion for the approximate maximum tension with the corresponding series for the exact one. Using this approach, accurate approximate analytical expressions for the periodic solution are obtained. We also compared the Fourier series expansions of the approximate solutions and the exact ones. This allowed us to compare the coefficients for the different harmonics in these solutions. We also compared the approximate and exact solutions as a function of time for several oscillation amplitudes. Finally, in this procedure we used some of the approximate expressions for the simple pendulum frequency which can be found in the bibliography; however, the procedure can be applied using other approximate frequencies. [Copyright &y& Elsevier]

Abstract: The exchange interactions (J BB and J AB are the intra and the inter-sublattice exchange interactions between neighbouring spins, respectively) are obtained by using the general expressions of canting angle and critical temperature obtained by mean field theory of Li0.5Fe2.5−2x Al x Cr x O4. The expression of magnetic energy of Li0.5Fe2.5−2x Al x Cr x O4 is obtained for different spin configurations and dilution x. The saturation magnetisation of Li0.5Fe2.5–2x Al x Cr x O4 is obtained with different values of dilution x. The magnetic phase diagram of Li0.5Fe2.5–2x Al x Cr x O4 materials is obtained by high temperature series expansions (HTSEs). The critical exponent associated with the magnetic susceptibility of Li0.5Fe2.5−2x Al x Cr x O4 is deduced. [Copyright &y& Elsevier]

Abstract: In this paper, Shooting Type Laplace–Adomian Decomposition Algorithm (STLADA), is applied to some boundary value problems with one of the boundary conditions at infinity. The analytic solution obtained by using this method converges rapidly, highly effective in terms of accuracy and very close to the exact solution. The solution is compared with the existing results. Shooting Type Laplace–Adomian Decomposition Algorithm is an efficient method for solving these types of problems occurring especially in fluid mechanics. [Copyright &y& Elsevier]

Abstract: As a continuation of the well known connection between the theory of orthogonal polynomials on the unit circle and the interval , in this paper properties concerning error and convergence of certain rational approximants associated with the measures and supported on and the unit circle respectively are deduced. Numerical illustrations are also given. [Copyright &y& Elsevier]

Abstract: The properties of fully three-dimensional gravity waves propagating on the interface between two finite layers of different densities are investigated. Solutions are calculated via a computer-generated perturbation expansion in wave steepness. Series solutions are analytically computed to third-order for general wave parameters, and numerically to 27th-order for five specific values of and 0.99, where is the ratio of the density of the upper fluid to that of the lower fluid. For near limiting waves, the series of frequency, kinetic energy and potential energy are summed using Padé approximants. For both two and three-dimensional cases, the present theory is found to coincide with previous theories such as two-dimensional interfacial standing waves, two-dimensional interfacial progressive waves and three-dimensional surface gravity waves respectively, showing the validity and general applicability of the solutions. The numerical results demonstrate the influence of the ratio density and thicknesses of the two fluids on the wave profile and wave frequency bifurcation. Particular attention is paid to the harmonic resonances where multiple solutions are possible. [Copyright &y& Elsevier]

Abstract: In this article, a variational iteration method (VIM) is performed to give approximate and analytical solutions of nonlinear ordinary differential equation systems such as a model for HIV infection of CD4+ T cells. A modified VIM (MVIM), based on the use of Padé approximants is proposed. Some plots are presented to show the reliability and simplicity of the methods. [Copyright &y& Elsevier]

Abstract: We show that a simple and straightforward rational approximation to the Thomas–Fermi equation provides the slope at origin with unprecedented accuracy and that Padé approximants of relatively low order are far more accurate than more elaborate approaches proposed recently by other authors. We consider both the Thomas–Fermi equation for isolated atoms and for atoms in strong magnetic fields. [Copyright &y& Elsevier]

Abstract: The Adomian decomposition method and the asymptotic decomposition method give the near-field approximate solution and far-field approximate solution, respectively, for linear and nonlinear differential equations. The Padé approximants give solution continuation of series solutions, but the continuation is usually effective only on some finite domain, and it can not always give the asymptotic behavior as the independent variables approach infinity. We investigate the global approximate solution by matching the near-field approximation derived from the Adomian decomposition method with the far-field approximation derived from the asymptotic decomposition method for linear and nonlinear differential equations. For several examples we find that there exists an overlap between the near-field approximation and the far-field approximation, so we can match them to obtain a global approximate solution. For other nonlinear examples where the series solution from the Adomian decomposition method has a finite convergent domain, we can match the Padé approximant of the near-field approximation with the far-field approximation to obtain a global approximate solution representing the true, entire solution over an infinite domain. [Copyright &y& Elsevier]

Abstract: Using Padé approximation to the exponential function, we obtain new identities involving values of the Rieman zeta function at integers. Applications to series associated with zeta numbers are proved. In particular, expansion of ζ(3) (resp. ζ(5)) in terms of ζ(4j +2) (resp. ζ(4j)) are proved. [Copyright &y& Elsevier]

Abstract: Numerical methods for solving Ordinary Differential Equations (ODEs) have received considerable attention in recent years. In this paper a piecewise-linearized algorithm based on Krylov subspaces for solving Initial Value Problems (IVPs) is proposed. MATLAB versions for autonomous and non-autonomous ODEs of this algorithm have been implemented. These implementations have been compared with other piecewise-linearized algorithms based on Padé approximants, recently developed by the authors of this paper, comparing both precisions and computational costs in equal conditions. Four case studies have been used in the tests that come from stiff biology and chemical kinetics problems. Experimental results show the advantages of the proposed algorithms, especially when the dimension is increased in stiff problems. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we apply the modified Laplace decomposition method (MLDM) to obtain series solutions of the boundary layer equation. The technique is based on the application of Laplace transform to boundary layers in fluid mechanics. The nonlinear terms can be easily handled by the use of Adomian polynomials. The obtained series solution is combined with the diagonal Padé approximants to handle the boundary condition at infinity. Comparison of the present solution is made with the existing solution and excellent agreement is noted. [ABSTRACT FROM AUTHOR]

Abstract: Using the Padé approximation of the exponential function, we obtain a general recurrence relation for values of the zeta function which contains, as particular cases, many of relations already proved. Applications to Bernoulli polynomials are given. At last, we derive some new recurrence relations with gap of length 4 for zeta numbers. [ABSTRACT FROM AUTHOR]

Abstract: Commonly used techniques to study non-perturbative aspects of the strong interactions have a deep connection with rational approximants, and in particular with Padé approximants to meromorphic functions. However, only recently this connection has been acknowledged and efforts at fully exploiting it are only starting. In this article I will briefly review the most prominent techniques used in non-perturbative strong interactions with special emphasis on its relation with Padé approximants. I will then concentrate on a set of open problems outside the scope of these conventional techniques where Padé approximants might be extremely useful. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we give some new algebraic properties of a sequence transformation due to Drummond for accelerating the convergence. In the particular case where the terms of the sequence to be transformed are the partial sums of a formal power series, this transformation leads to Padé-type approximants. Then, using various operators, some extensions of Drummond''s process are proposed, and their properties are given. A new expression for the E-transformation, using generalized divided differences, is obtained, and it is also generalized. The recurrence relation for these differences can be used for its implementation, and the Ford–Sidi algorithm is also related to them. Lubkin''s T and W-transformations, which could be related to Drummond''s, are discussed in Appendix A. [Copyright &y& Elsevier]

Abstract: This paper presents an exact solution of linear elasticity theory for bending of sandwich plate-like beams due to temperature difference at the plate faces. It is assumed that the heat flow is stationary. The exact solution yields the temperature profile, stress and displacement distribution across the plate thickness. The analytical results are complemented by an example of a simply-supported sandwich beam. [Copyright &y& Elsevier]

Abstract: The magnetic properties of diluted spinels Zn1−x M x Fe2O4 (M=Mg, Ni) systems have been studied by mean field theory, probability law and high-temperature series expansions method in the range 0≤ x ≤1. The exchange interactions are calculated by the first and second theory, respectively. The magnetic phase diagrams of Zn1−x M x Fe2O4 (M=Mg, Ni) are calculated by the high-temperature series expansions, combined with the Padé approximants method. The critical exponent associated with the magnetic susceptibility (γ) is deduced. [Copyright &y& Elsevier]

Abstract: Two widely used methods for computing matrix exponentials and matrix logarithms are, respectively, the scaling and squaring and the inverse scaling and squaring. Both methods become effective when combined with Padé approximation. This paper deals with the computation of exponentials of skew-symmetric matrices and logarithms of orthogonal matrices. Our main goal is to improve these two methods by exploiting the special structure of skew-symmetric and orthogonal matrices. Geometric features of the matrix exponential and logarithm and extensions to the special Euclidean group of rigid motions are also addressed. [Copyright &y& Elsevier]

Abstract: Using the Padé approximation of the exponential function, we obtain recurrence relations between Apostol–Bernoulli and between Apostol–Euler polynomials. As applications, we derive some new lacunary recurrence relations for Bernoulli and Euler polynomials with gap of length 4 and lacunary relations for Bernoulli and Euler numbers with gap of length 6. [Copyright &y& Elsevier]

Abstract: The vacuum polarization of a quark, when considered in terms of the external momentum , is a function of the Stieltjes type. Consequently, the mathematical theory of Padé approximants assures that the full function, at any finite value of away from the physical cut, can be reconstructed from its low-energy power expansion around . We illustrate this point by applying this theory to the vacuum polarization of a heavy quark and obtain the value of the constant governing the threshold expansion at order . [Copyright &y& Elsevier]

Abstract: We apply high-order many-body perturbation theory for the calculation of ground-state energies of closed-shell nuclei using realistic nuclear interactions. Using a simple recursive formulation, we compute the perturbative energy contributions up to 30th order and compare to exact no-core shell model calculations for the same model space and Hamiltonian. Generally, finite partial sums of this perturbation series do not show convergence with increasing order, but tend to diverge exponentially. Nevertheless, through a simple resummation via Padé approximants it is possible to extract rapidly converging and highly accurate results for the ground-state energy once perturbative contributions beyond 5th order are included. [Copyright &y& Elsevier]

Abstract: The magnetic properties of the spinels systems Zn1−x Cd x Cr2Se4 and Hg1−x Cd x Cr2S4 have been studied by mean-field theory and high-temperature series expansions in the range 0≤ x ≤1. The nearest neighbour and the next-nearest-neighbour super-exchange interaction J 1(x) and J 2(x) respectively, are calculated for the two systems, using the first theory in the range 0≤ x ≤1. The intra-planar and the inter-planar interactions are deduced. The corresponding classical exchange energy for the magnetic structures is obtained. The second theory has been applied to the spinel systems Zn1−x Cd x Cr2Se4 and Hg1−x Cd x Cr2S4, combined with the Padé approximants method, to determine the magnetic phase diagrams (T C versus dilution x) in the range 0≤ x ≤1. The obtained theoretical results are in agreement with the experimental ones obtained by magnetic measurements. The critical exponents associated with the magnetic susceptibility (γ) and with the correlation lengths () are deduced for the two systems in the ordered phase. [Copyright &y& Elsevier]

Abstract: This work deals with damped nonlinear forced vibrations of thin elastic rectangular plates subjected to harmonic excitation by an asymptotic numerical method. Using the harmonic balance method and Hamilton’s principle, the governing equation is converted into a static formulation. A mixed formulation is used to transform the problem from cubic nonlinearity to quadratic one sequence. Displacement, stress and frequency are represented by power series with respect to a path parameter. Equating the like powers of this parameter, the nonlinear governing equation is transformed into a sequence of linear problems with the same stiffness matrix. Through a single matrix inversion, a considerable number of terms of the perturbation series can easily be computed with a limited computation time. The starting point, corresponding to a regular solution, is obtained by the Newton–Raphson method. In order to increase the step length, Padé approximants are used. Numerical tests are presented and compared with numerical and analytical results in the literature, for different boundary conditions, excitations and damping coefficients. [Copyright &y& Elsevier]

Abstract: In this paper we propose a reliable algorithm for the solution of nonlinear oscillators. Our algorithm is based upon the homotopy perturbation method (HPM), Laplace transforms, and Padé approximants. This modified homotopy perturbation method (MHPM) utilizes an alternative framework to capture the periodic behavior of the solution, which is characteristic of oscillator equations, and to give a good approximation to the true solution in a very large region. The current results are compared with those derived from the established Runge–Kutta method in order to verify the accuracy of the MHPM. It is shown that there is excellent agreement between the two sets of results. Results also show that the numerical scheme is very effective and convenient for solving strongly nonlinear oscillators. [Copyright &y& Elsevier]

Abstract: In this paper, we apply the variational iteration method using He’s polynomials for finding the analytical solution of unsteady flow of gas through a porous medium. The proposed method is an elegant combination of He’s variational iteration and the homotopy perturbation methods. The suggested algorithm is quite efficient and is practically well suited for use in such problems. The proposed iterative scheme finds the solution without any discretization, linearization or restrictive assumptions. The diagonal Pade approximants are effectively used in the analysis to capture the essential behavior of the solution and to determine the initial slope. A clear advantage of this technique over the decomposition method is that no calculation of Adomian’s polynomials is needed. [Copyright &y& Elsevier]

Abstract: ADM-Padé technique is a combination of Adomian decomposition method (ADM) and Padé approximants. We solve two nonlinear lattice equations using the technique which gives the approximate solution with higher accuracy and faster convergence rate than using ADM alone. Bell-shaped solitary solution of Belov–Chaltikian (BC) lattice and kink-shaped solitary solution of the nonlinear self-dual network equations (SDNEs) are presented. Comparisons are made between approximate solutions and exact solutions to illustrate the validity and the great potential of the technique. [Copyright &y& Elsevier]