Pade Technique Research Articles

Nonlinear free vibration of conservative oscillators with inertia and static type cubic nonlinearities using homotopy analysis method

In this study, an accurate analytical solution for the nonlinear free vibration of a conservative oscillator with inertia and static type cubic nonlinearities is derived. This solution has been obtained using homotopy analysis method (HAM). Then, homotopy Pade technique is applied to accelerate the convergence rate of the series solution. This study shows that the HAM leads to an accurate analytical solution, which is valid for a wide range of considered system parameters. Unlike the other analytical methods, HAM can control and adjust the convergence region and rate of the approximation series solution. The excellent accuracy of the current results is demonstrated by comparing with the available analytical and numerical results.

Series solutions of boundary-layer flows in porous media with lateral mass flux

Approximate analytical solutions for free convection boundary layers on a heated vertical plate with lateral mass flux embedded in a saturated porous medium are presented using the modified Adomian decomposition method and Pade technique. Several values of the wall temperature exponent for illustrating the effects of suction/injection parameter on the flow and heat transfer are considered. This study also includes the influence of the exponent on an impermeable surface. The results obtained are comparable to the exact analytical solutions and elucidate reliability and efficiency of the technique.

The solution of linear and nonlinear systems of Volterra functional equations using Adomian–Pade technique

The purpose of this study is to implement Adomian–Pade (Modified Adomian–Pade) technique, which is a combination of Adomian decomposition method (Modified Adomian decomposition method) and Pade approximation, for solving linear and nonlinear systems of Volterra functional equations. The results obtained by using Adomian–Pade (Modified Adomian–Pade) technique, are compared to those obtained by using Adomian decomposition method (Modified Adomian decomposition method) alone. The numerical results, demonstrate that ADM–PADE (MADM–PADE) technique, gives the approximate solution with faster convergence rate and higher accuracy than using the standard ADM (MADM).

The solution of coupled Burgers’ equations using Adomian–Pade technique

The purpose of this study is to implement ADM–PADE (MADM–PADE) technique, which is a combination of Adomian decomposition method (modified Adomian decomposition method) and Pade approximation, for solving homogeneous (inhomogeneous) two dimensional parabolic equation. Also, we apply ADM–PADE technique to solve coupled Burgers’ equations. The results obtained by ADM–PADE (MADM–PADE) techniques are compared to those obtained by using ADM (MADM) alone. The numerical results demonstrate that ADM–PADE (MADM–PADE) technique gives the approximate solution with faster convergence rate and higher accuracy than using ADM (MADM).

The solution of Burgers’ and good Boussinesq equations using ADM–Padé technique

Abstract ADM–Pade technique is a combination of Adomian decomposition method (ADM) and Pade approximants. It is an approximate method, which can be adapted to solve nonlinear partial differential equations. In this paper, we solve Burgers’ and Boussinesq equation using ADM–Pade technique which gives the approximate solution with faster convergence rate and higher accuracy than using ADM alone.

Symbolic analytical solution of N‐dimensional radially symmetric polytropes

AbstractIn this paper, accelerated power series solutions are developed for N‐dimensional symmetric radially polytropes. The solutions are valid for any geometric and polytropic index. The implementation of Padé technique and changing of the independent variable give us identical polytrope solutions to the numerical one. Physical parameters are presented for Ndimensional radially polytropes. In case of the isothermal sphere, the density profile is accurate to 10–5 per cent whatever the value of ξ is. All the computations are performed using Mathematica software. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Efficient computation of resonant frequencies and quality factors of cavities via a combination of the finite-difference time-domain technique and the Pade approximation

An efficient method for analyzing cavity structures by using the fast Fourier transform (FFT)/Pade technique, in combination with the finite-difference time-domain (FDTD) method, is presented. Without sacrificing the accuracy of the results, this new method significantly reduces the computational time compared to that needed where the conventional FFT algorithm is used. The usefulness of this approach is demonstrated by modeling a lossy cavity and computing its resonant frequencies as well as Q.

Bound-state method with elementary-product wavefunctions

We propose a new version of the multipoint Pade technique to solve Schrodinger equations in their first-order Riccati form. Our compact product representation of wavefunctions interpolates or 'matches' available approximants at several points. The mechanism of 'matching' determines the energies and eliminates spurious solutions. On V(x)= mu 2x2+ lambda x2/(1+gx2), its efficiency is illustrated numerically.

Simplification of z-transfer function via Padé approximation of tangent phase function

Simplification of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</tex> -transfer function is performed in the bilinear-transformed domain via approximating tangent phase function by a Padé technique. A modification has been given to the transformed transfer function so that the reduced <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</tex> -transfer function has the same zero initial responses as the original system. The simplified <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</tex> -transfer function is always stable and minimum phase.

Perturbation series for the mass gap of the (1+1)-dimensional O(2)-model

The strong coupling mass gap series to 10th order in the dimensionless coupling is presented for the (1+1)-dimensional O(2)-model. The second logarithmic derivative of the Euler transformed series is examined using Pade techniques to yield the critical parameter estimates lambda c approximately=0.89+0.02 and sigma approximately=0.50+or-0.08. These results are consistent with the model exhibiting a Kosterlitz-Thouless transition.

Some considerations on non-minimality, ill-conditioning, and instability of Padé approximants

Critical conditions of model simplification are pointed out and investigated. In particular, the problem of setting up an rth order model for an nth order system (with r<n) using the Pade technique is considered. The different situations of model uniqueness, non-existence, and non-minimality are classified according to the ranks of Hankel matrices formed from the original system Markov parameters (for example, the MacLaurin expansion coefficients of its transfer function). The case is then analysed where the reduced-order model parameters are given by an ill-conditioned set of equations so that small changes in the original system cause large variations of the model. Finally, the problem of reduced-order model instability is investigated and its relation to the ill-conditioned situation is pointed out.

Stability equation method to reduce the order of fast oscillating systems

A method for linear order reduction is proposed which, for fast oscillating systems, is superior to a number of other techniques (stability equation method, classic Pade technique etc.). For low-frequency systems the proposed method and the stability equation method yield identical results.

Alternative to Padé technique for iterative solution of integral equations

It is well known that when the kernel of Fredholm integral equations of the second kind has the absolute magnitude of one or more eigenvalues greater than unity the Neumann series of such equations diverge. Pade technique is a practical way for constructing the correct solution from the divergent Neumann series. Here we propose an alternative method for the iterative solution of such equations, which relies on writing auxiliary integral equations with “weaker” kernels so that the auxiliary equations have a convergent Neumann series. Then the solution of the original equation is expressed in terms of the (convergent Neumann series) solution of the auxiliary equations. The kernel of the auxiliary equations can systematically be made weaker and the rate of convergence of the Neumann series of the auxiliary equations improved in order to make the method more useful. The method is illustrated numerically in certain cases.

Singular-perturbation—strong-coupling field theory and the moments problem

Motivated by recent work of Bender, Cooper, Guralnik, Mjolsness, Rose, and Sharp, a new technique is presented for solving field equations in terms of singular-perturbation---strong-coupling expansions. Two traditional mathematical tools are combined into one effective procedure. Firstly, high-temperature lattice expansions are obtained for the corresponding power of the field solution. The approximate continuum-limit power are subsequently obtained through the application of Pad\'e techniques. Secondly, in order to reconstruct the corresponding approximate global field solution, one must use function-moments reconstruction techniques. The latter involves reconsidering the traditional moments problem of interest to pure and applied mathematicians. The above marriage between lattice methods and reconstruction procedures for functions yields good results for the ${\ensuremath{\varphi}}^{4}$ field-theory kink, and the sine-Gordon kink solutions. It is argued that the power are the most efficient dynamical variables for the generation of strong-coupling expansions. Indeed, a momentum-space formulation is being advocated in which the long-range behavior of the space-dependent fields are determined by the small-momentum, infrared, domain.

Three-Body Calculation of d- Elastic Scattering

S-, P-, D-waves and some higher wave phase shifts of d-a elastic scattering are calculated by solving the Amado-Lovelace equations in order to discuss the mechanism of this scattering and breakup reaction. Equations are solved by applying Pade technique. It is shown that this technique is especially useful to calculate elastic scattering. An interesting example of the convergence of the Born series is given. Remarkable improvements on Shanley's results are obtained in all partial waves. Effects of 'Li resonances are discussed and it is shown that the 1 + resonance play,; an important role to understand this scattering. It is suggested that the recently observed peak of the imaginary eigenphase by Griiebler et a!. does not suggest a new resonance of 'Li, but can be understood by this well-k~own resonance. In order to investigate the role of the Pauli principle, sensitivity to the s-wave a-nucleon interac­ tion is discussed.

Novel two-variable approximants for studying magnetic multicritical behavior

Abstract Series expansions in two variables, e.g. χ (T, H) = ∑ jk a jk ( J k B T ) j ( mH k B T k ) , are important in studying magnetism but established extrapolation methods (e.g. ratio, Pade, and Canterbury techniques) fail near multicritical points. A novel method of series extrapolation is explained which can exactly represent the expected two-variable, two-exponent (γ, φ), asymptotic scaling behavior.