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In this study, we develop a modified version of the two-dimensional differential transform (TDDT) method for solving proportional delay partial differential equations (PDPDEs) that frequently arise in engineering and scientific models. This modification is achieved by integrating the TDDT method with the Laplace transform and the Padé approximant, thereby leveraging the strengths of each technique to improve overall performance. Theorems are provided in a general manner to cover various types of PDEs, with constant or variable coefficients. To validate the approach, we apply it to three test problems, demonstrating its effectiveness in extending the convergence domain of the traditional TDDT approach, reducing computational complexity, and yielding analytic solutions with fewer computational steps. Results indicate that the method is a viable alternative for addressing PDPDEs, especially in scenarios where traditional analytic solutions are challenging to obtain. This combination opens new avenues for efficiently solving complex delayed systems in engineering and science, potentially outperforming existing numerical and analytical techniques in both speed and reliability.

This work aims to present a reliable algorithm that can effectively generate accurate piecewise approximate analytical solutions for third- and fourth-order reaction-diffusion singular perturbation problems. These problems involve a discontinuous source term and exhibit both interior and boundary layers. The original problem was transformed into a system of coupled differential equations that are weakly interconnected. A zero-order asymptotic approximate solution was then provided, with known asymptotic analytical solutions for the boundary and interior layers, while the outer region solution was obtained analytically using an enhanced residual power series approach. This approach combined the standard residual power series method with the Padé approximation to yield a piecewise approximate analytical solution. It satisfies the continuity and smoothness conditions and offers higher accuracy than the standard residual power series method and other numerical methods like finite difference, finite element, hybrid difference scheme, and Schwarz method. The algorithm also provides error estimates, and numerical examples are included to demonstrate the high accuracy, low computational cost, and effectiveness of the method within a new asymptotic semi-analytical numerical framewor.

Abstract In this work, our main focus is to establish asymptotic expansions for the triple gamma function in terms of the triple Bernoulli polynomials. As application, an asymptotic expansion for hyperfactorial function is also obtained. Furthermore, using these asymptotic expansions, Padé approximants related to the triple gamma function are derived as a consequence. The results obtained are new, and their importance is demonstrated by deducing several interesting remarks and corollaries.

This work presents a reliable algorithm to obtain approximate analytical solutions for a strongly coupled system of singularly perturbed convection–diffusion problems, which exhibit a boundary layer at one end. The proposed method involves constructing a zero-order asymptotic approximate solution for the original system. This approximation results in the formation of two systems: a boundary layer system with a known analytical solution and a reduced terminal value system, which is solved analytically using an improved residual power series approach. This approach combines the residual power series method with Padé approximation and Laplace transformation, resulting in an approximate analytical solution with higher accuracy compared to the conventional residual power series method. In addition, error estimates are extracted, and illustrative examples are provided to demonstrate the accuracy and effectiveness of the method.

Abstract In this paper, we present Padé approximations of some functions involving complete elliptic integrals of the first kind K ( x ) $K(x)$ , and motivated by these approximations we also present the following double inequality: 1 − x 2 1 − x 2 + x 4 62 < 2 e 2 π K ( x ) − 1 ( 1 + 1 1 − x 2 ) < 1 − 96 100 x 2 1 − 96 100 x 2 + x 4 64 , x ∈ ( 0 , 1 ) . $$ \frac{1-x^{2}}{1-x^{2}+\frac{x^{4}}{62}}< \frac{2 e^{\frac{2}{\pi }K(x)-1}}{ (1+\frac{1}{\sqrt{1-x^{2}}} )}< \frac{1-\frac{96}{100}x^{2}}{1-\frac{96}{100}x^{2}+\frac{x^{4}}{64}},\quad x\in ( 0,1 ). $$ Our results have superiority over some new recent results.

To improve the efficiency for managing power systems with constant time delay, an improved frequency-weighted model reduction is proposed. First, the Padé approximant allows to represent the time delay in the state-space form and build a linear model of the closed-loop power system containing a wide-area damping controller. Then, a weight function is introduced to solve the equilibrium transformation matrix and balance the original system. Finally, the residual order reduction is applied to the model of the power system considering time delay. The proposed method is evaluated in the two-area four-machine test system and New England system, and the effectiveness of the proposed reduction method is verified.

In this paper, we derive and analyze a new one-parameter family of modified Cauchy method free from second derivative for obtaining simple roots of nonlinear equations by using Padé approximant. The convergence analysis of the family is also considered, and the methods have convergence order three. Based on the family of third-order method, in order to increase the order of the convergence, a new optimal fourth-order family of modified Cauchy methods is obtained by using weight function. We also perform some numerical tests and the comparison with existing optimal fourth-order methods to show the high computational efficiency of the proposed scheme, which confirm our theoretical results. The basins of attraction of this optimal fourth-order family and existing fourth-order methods are presented and compared to illustrate some elements of the proposed family have equal or better stable behavior in many aspects. Furthermore, from the fractal graphics, with the increase of the value m of the series in iterative methods, the chaotic behaviors of the methods become more and more complex, which also reflected in some existing fourth-order methods.

This paper presents an effective procedure for model order reduction of discrete time control system. The exact model derived from complex dynamic systems proves to be very complicated for analysis, control and design. This necessity brings about using a tool known as model order reduction technique or model simplification. A novel mixed method has been implemented in this paper for reducing the order of the large scale dynamic discrete system. Dominant pole based pole clustering method has been used to derive the coefficients of denominator polynomial while Padé approximation has been applied to obtain the coefficients of numerator polynomial of the reduced order model. The proposed method is quite simple and able to generate a stable reduced order model from high order stable discrete systems. The dominancy of poles has been decided by values of the ratio of residue to its pole. The pole is considered dominant which have larger ratio value. An illustrative example has been considered to show the various reduction steps. The result obtained confirms the effectiveness of the approach.

Abstract Inequalities are frequently used for solving practical engineering problem. There are two key issues of bounding inequalities; one is to find the bounds, and the other is to prove the bounds. This paper takes Wilker type inequalities as an example, presents a two-point-Padé-approximant-based method for finding the bounds, and it also provides a method to prove the bounds in a new way. It not only recovers the estimates in Mortici’s method, but it also provides new improvements of estimates obtained from prevailing methods. In principle, it can be applied for other inequalities.

Abstract A two-point Padé approximant method is presented for refining some remarkable trigonometric inequalities including the Jordan inequality, Kober inequality, Becker–Stark inequality, and Wu–Srivastava inequality. Simple proofs are provided. It shows to achieve better approximation results than those of prevailing methods.

This paper shows that using the Padé–Laplace (PL) method for deconvolution of multi-exponential functions (stress relaxation of polymers) can produce ill-conditioned systems of equations. Analysis of different sets of generated data points from known multi-exponential functions indicates that by increasing the level of Padé approximants, the condition number of a matrix whose entries are coefficients of a Taylor series in the Laplace space grows rapidly. When higher levels of Padé approximants need to be computed to achieve stable modes for separation of exponentials, the problem of generating matrices with large condition numbers becomes more pronounced. The analysis in this paper discusses the origin of ill-posedness of the PL method and it was shown that ill-posedness may be regularized by reconstructing the system of equations and using singular value decomposition (SVD) for computation of the Padé table. Moreover, it is shown that after regularization, the PL method can deconvolute the exponential decays even when the input parameter of the method is out of its optimal range.

In this work we show how by using a Padé type analytical continuation scheme, based on the Schlessinger point method, it is possible to find higher production thresholds in hadron physical problems. Recently, an extension of this numerical approach to the complex energy plane enabled the calculations of auto-ionization decay resonance poles in atomic and molecular systems. Here we use this so-called Resonances via Padé (RVP) method, to show its convergence beyond the singular point in hadron physical problems. In order to demonstrate the capabilities of the RVP method, two illustrations for the ability to identify singularities and branch points are given. In addition, two applications for hadron physical problems are given. In the first one, we identify the decay thresholds from a numerically calculated spectral function. In the second one, we use experimental data. First, we calculate the resonance pole of the f0(500) or σ meson using the S0 partial wave amplitude for ππ scattering in very good agreement with the literature. Second, we use data on the cross section ratio R(s) for e+e− collisions and discuss the prediction of decay thresholds which proves to be difficult if the data is noisy.

Abstract Based on the Padé approximation method, in this paper we determine the coefficients a j $a_{j}$ and b j $b_{j}$ ( 1 ≤ j ≤ k $1\leq j \leq k$ ) such that 1 e ( 1 + 1 x ) x = x k + a 1 x k − 1 + ⋯ + a k x k + b 1 x k − 1 + ⋯ + b k + O ( 1 x 2 k + 1 ) , x → ∞ , $$ \frac{1}{e} \biggl( 1+\frac{1}{x} \biggr) ^{x}= \frac{x^{k}+a_{1}x^{k-1}+ \cdots +a_{k}}{x^{k}+b_{1}x^{k-1}+\cdots +b_{k}}+O \biggl( \frac{1}{x ^{2k+1}} \biggr) , \quad x\to \infty , $$ where k ≥ 1 $k\geq 1$ is any given integer. Based on the obtained result, we establish new upper bounds for ( 1 + 1 / x ) x $( 1+1/x ) ^{x}$ . As an application, we give a generalized Carleman-type inequality.

Abstract The Padé approximation is a useful method for creating new inequalities and improving certain inequalities. In this paper we use the Padé approximant to give the refinements of some remarkable inequalities involving inverse trigonometric functions, it is shown that the new inequalities presented in this paper are more refined than that obtained in earlier papers.

Abstract In this paper we, respectively, give simple proofs of some remarkable trigonometric inequalities, based on the Padé approximation method. We also obtain rational refinements of these inequalities. We are convinced that the Padé approximation method offers a general framework for solving many other similar inequalities.

The work addressed in this paper is a comparative study between convergence of the acceleration techniques, diagonal pad'{e} approximants and shanks transforms, on Homotopy analysis method and Adomian decomposition method for solving differential equations of integer and fractional orders.

We present examples of Padé approximations of the α -effect and eddy viscosity/diffusivity tensors in various flows. Expressions for the tensors derived in the framework of the standard multiscale formalism are employed. Algebraically, the simplest case is that of a two-dimensional parity-invariant six-fold rotation-symmetric flow where eddy viscosity is negative, indicating intervals of large-scale instability of the flow. Turning to the kinematic dynamo problem for three-dimensional flows of an incompressible fluid, we explore the application of Padé approximants for the computation of tensors of magnetic α -effect and, for parity-invariant flows, of magnetic eddy diffusivity. We construct Padé approximants of the tensors expanded in power series in the inverse molecular diffusivity 1 / η around 1 / η = 0 . This yields the values of the dominant growth rate to satisfactory accuracy for η , several dozen times smaller than the threshold, above which the power series is convergent. We do computations in Fortran in the standard “double” (real*8) and extended “quadruple” (real*16) precision, and perform symbolic calculations in Mathematica.

The modified differential transform method (MDTM), Laplace transform and Padé approximants are used to investigate a semi-analytic form of solutions of nonlinear oscillators in a large time domain. Forced Duffing and forced van der Pol oscillators under damping effect are studied to investigate semi-analytic forms of solutions. Moreover, solutions of the suggested nonlinear oscillators are obtained using the fourth-order Runge-Kutta numerical solution method. A comparison of the result by the numerical Runge-Kutta fourth-order accuracy method is compared with the result by the MDTM and plotted in a long time domain.