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The two-parametric Mittag–Leffler function (MLF), E α , β , is fundamental to the study and simulation of fractional differential and integral equations. However, these functions are computationally expensive and their numerical implementations are challenging. In this paper, we present a unified framework for developing global rational approximants of E α , β ( - x ) , x > 0 , with { ( α , β ) : 0 < α ≤ 1 , β ≥ α , ( α , β ) ≠ ( 1 , 1 ) } . This framework is based on the series definition and the asymptotic expansion at infinity. In particular, we develop three types of fourth-order global rational approximations and discuss how they could be used to approximate the inverse function. Unlike existing approximations which are either limited to MLF of one parameter or of low accuracy for the two-parametric MLF, our rational approximants are of fourth order accuracy and have low percentage error globally. For efficient utilization, we study the partial fraction decomposition and use them to approximate the two-parametric MLF with a matrix argument which arise in the solutions of fractional evolution differential and integral equations.

Linear piezoelectric energy harvesters exhibit coupled vibration phenomena under external excitation. This study proposes a novel semi-analytical method for examining the vibration response of linear piezoelectric energy harvesters in vortex-induced vibration environments. In this study, the governing equations of a vortex-induced piezoelectric energy harvester, derived with the help of Euler–Lagrange equation, are solved using the multi-step differential transform-Padé approximation method. The accuracy of this method is validated against the fourth-order Runge–Kutta method. Furthermore, the influences of the bluff body’s diameter and the length of piezoelectric beam on the system’s output power are examined. Finally, optimization of the model is conducted using the non-dominated sorting genetic algorithm II with the objectives of maximizing output power and minimizing system mass, in which response surface methodology is employed to tackle with the time-consuming problem in computation process. The results indicate that the output power at the turning points calculated through idealized points is 13.37% greater than that of the initial design point, while the total system mass is reduced by 3.06%.

We show that the logarithmic (Hencky) strain and its derivatives can be approximated, in a straightforward manner and with a high accuracy, using Padé approximants of the tensor (matrix) logarithm. Accuracy and computational efficiency of the Padé approximants are favourably compared to an alternative approximation method employing the truncated Taylor series. As an application, Hencky-type hyperelasticity models are considered, in which the elastic strain energy is expressed in terms of the Hencky strain, and of our particular interest is the anisotropic energy quadratic in the Hencky strain. Finite-element computations are carried out to examine performance of the Padé approximants of tensor logarithm in Hencky-type hyperelasticity problems. A discussion is also provided on computation of the stress tensor conjugate to the Hencky strain in a general anisotropic case.

This paper tackles the persistent challenge of slow convergence and numerical instability in the fractional calculus when applied to power series–representable functions , limitations that compromise accuracy in scientific applications. A novel reformulation of fractional derivatives and integrals is achieved by applying Padé approximation to conventional power series solutions, replacing them with optimized rational functions. The modified operators demonstrate significantly improved accuracy and enhanced convergence properties compared to established methods. This approach enables more reliable fractional modeling in physics and engineering domains where traditional operators fail. The work constitutes the first systematic integration (differentiation) of Padé approximation into the foundational definition of fractional operators, overcoming convergence barriers inherent in prior series‐based techniques.

Padé approximation is the rational generalization of Hermite interpolating polynomial. On its own merits, it has earned a relevant place in the theory of constructive approximation. In this article, we will develop an exhaustive analysis of two-point Padé approximations to Herglotz-Riesz transforms. We study the convergence problem when the poles are partially preassigned. In this analysis, the Stieltjes polynomials on the unit circle naturally arise. Finally, some illustrative numerical examples are discussed.

Based on the blossoming theory, in this work we develop a new method for deriving Hermite-Padé approximants of certain hypergeometric series. Its general principle consists in building identities generalising the Hermite identity for exponentials, and in then applying their blossomed versions to appropriate tuples to simultaneously produce explicit expressions of the approximants and explicit integral representations of the corresponding remainders. For binomial series we use classical blossoms while for q -hypergeometric series we have to use q -blossoms.

Padé approximations are approximations of holomorphic functions by rational functions. The application of Padé approximations to Diophantine approximations has a long history dating back to Hermite. In this paper, we use the Maier–Chudnovsky construction of Padé-type approximation to study irrationality properties about values of functions with the form f(x)=∑k=0∞xkk!(bk+s)(bk+s+1)⋯(bk+t), where b,t,s are positive integers and obtain upper bounds for irrationality measures of their values at nonzero rational points. Important examples includes exponential integral, Gauss error function and Kummer’s confluent hypergeometric functions.

We present a completely explicit transcendence measure for e . This is a continuation and an improvement to the works of Borel, Mahler, and Hata on the topic. Furthermore, we also prove a transcendence measure for an arbitrary positive integer power of e . The results are based on Hermite–Padé approximations and on careful analysis of common factors in the footsteps of Hata.

We present a general class of derivative free iterative methods with optimal order of convergence for solving nonlinear equations. The methodology is based on quadratically convergent Traub–Steffensen scheme and inverse Padé approximation. Unlike that of existing higher order techniques the proposed technique is attractive since it leads to a simple implementation. Numerical examples are provided to confirm the theoretical results and to show the feasibility and efficacy of the new methods. The performance is compared with well established methods in literature. Computational results, including the elapsed CPU-time, confirm the accurate and efficient character of proposed techniques. Moreover, the stability of the methods is checked through complex geometry shown by drawing basins of attraction.

One-dimensional flows of an incompressible viscoelastic polymer fluid that are qualitatively similar to the solutions of Burgers’ equation are described on the basis of mesoscopic approach for the first time. The corresponding initial boundary-value problem is posed for the system of quasilinear differential equations. The numerical algorithm for solving it is designed and verified. The algorithm uses the explicit fifth-order scheme to approximate unknown functions with respect to time variable and the rational barycentric interpolations with respect to space variable. A method for localization of singular points of the solution in the complex plain and for adaptation of the spatial grid to them is implemented using the Chebyshev-Padé approximations. Two regimes of evolution of the solution to the problem are discovered and characterized while using the algorithm: regime 1—a smooth solution exists in a sufficiently large time interval (the singular point moves parallel to the real axis in the complex plane); regime 2—the smooth solution blows up at the beginning of evolution (the singular point reaches the segment of the real axis where the problem is posed). We study the influence of the rheological parameters of fluid on the realizability of these regimes and on the length of time interval where the smooth solution exists. The obtained results are important for the analysis of laminar-turbulent transitions in viscoelastic polymer continua.