Infinite Series Expansion Research Articles

Exponential series approximation of the SIR epidemiological model

IntroductionThe SIR (Susceptible-Infected-Recovered) model is one of the simplest and most widely used frameworks for understanding epidemic outbreaks.MethodsA second-order dynamical system for the R variable is formulated using an infinite exponential series expansion, and a recursion relation is established between the series coefficients. A numerical approximation scheme for the R variable is also developed.ResultsThe proposed numerical method is compared to a double exponential (DE) nonlinear approximate analytic solution, which reveals two coupled timescales: a relaxation timescale, determined by the ratio of the model’s time constants, and an excitation timescale, dictated by the population size. The DE solution is applied to estimate model parameters for a well-known epidemiological dataset—the boarding school flu outbreak.DiscussionFrom a theoretical standpoint, the primary contribution of this work is the derivation of an infinite exponential, Dirichlet, series for the model variables. Truncating the series yields a finite approximation, known as a Prony series, which can be interpreted as a sequence of coupled exponential relaxation processes, each with a distinct timescale. This apparent complexity can be approximated well by the DE solution, which appears to be of main practical interest.

Analysis of scalar fields with series convolution

Wave equations for some curved spacetimes may involve functions that prevent a solution in a closed form. In some cases, these functions can be eliminated by transformations and the solutions can be found analytically. In the cases where such transformations are not available, the infinite series expansions of these functions can be convoluted with the power series solution ansatz. We study such an example where the solution is based on a special function.

Emergent family of Tsallis entropies from the q-deformed combinatorics

We revisit the derivation of a formula for the q-generalised multinomial coefficient rooted in the q-deformed algebra, a foundational framework in the study of nonextensive statistics. Previous approximate expressions in the literature diverge as q approaches 2 (or 0, depending on convention). In contrast, our derived formula provides an exact, smooth function for all real values of q, expressed as an infinite series expansion involving Tsallis entropies with sequential entropic indices, coupled with Bernoulli numbers. This formulation is achieved through the analytic continuation of the Riemann zeta function, stemming from the q-deformed factorials. Our formula thus offers a distinctive characterisation of Tsallis entropy within the q-deformed combinatorics. Throughout this exploration, we also highlight a symmetry within the q-deformed theory that links different values of the entropic parameter. Furthermore, we discuss extending our results to encompass more general, affinity-sensitive cases, building on the previously established framework of affinity-based extended entropy.

Sparse Ramanujan Sequences Transform based OFDM for PAPR Reduction

Sparse Ramanujan Sequences (SRS) are periodic series derived from Ramanujan Sums (RS). RS is an exponential summation used to derive infinite series expansions for arithmetic functions, whose application pervades different fields of study from classical mathematics to digital signal processing. SRS has properties similar to RS such as periodicity and orthogonality. SRS transforms allow the sparse representation of signals, hence they have many potential applications, such as denoising and compression. The application of SRS transform to reduce the Peak to Average Power Ratio (PAPR) in Orthogonal Frequency Division Multiplexing (OFDM) is proposed in this paper. The Complementary Cumulative Distribution Function (CCDF) and Bit Error Rate (BER) are used to assess the PAPR reduction performance of the proposed system. The results show that the proposed SRS transform-based OFDM system significantly reduces PAPR and BER with reduced computational complexity in comparison with the existing systems.

Adaptive event-triggered optimal [formula omitted] tracking control for uncertain nonlinear systems: Comparative analysis applied to autonomic tractor-trailer system

In this paper, an event-triggered tracking optimization control method is proposed for nonlinear uncertain systems with H∞ discounted cost. First, the linearization method using the infinite series expansion theorem is given. And the tracking control design problem is solved by appropriate state augment. Secondly, the zero sum differential game strategy is introduced to address the control design problem of uncertain systems with event-triggered inputs. The actor-critic-disturbance networks framework is used to approximate the optimization of event- triggered controller strategy, optimization costs, and the disturbance strategy, respectively. Due to the consideration of discrete time and continuous time dynamics, we put the impulsive system method to use and choose appropriate Lyapunov functions to prove stability and boundedness, at the same time, there will be no infinite triggering situation. Finally, the effectiveness of the proposed scheme is demonstrated by an example of a practical tractor system with n trailers through simulation compared with another typical design method.

SOLUTION OF FUZZY VOLTERRA INTEGRAL EQUATIONS USING COMPOSITE METHOD

With the rapid development of of fuzzy Integral equations, the problem of finding its solution becomes increasingly important for knowledge propagation and putting researchers wisdom to work. A recent development trend of fuzzy integral equations is modeling many problems in the field of science and technology. If the unknown function in the considered equation has a solution in terms of infinite series expansion, this proposed method becomes more accurate to find the exact solution. According on the parametric form of a fuzzy number, a fuzzy integral equation converts to two systems of integral equations in the crisp case, and then proposed method is achieved to obtain the exact fuzzy solutions of fuzzy Volterra integral equations. To validate the effectiveness of proposed method, some examples of considered equations are solved and the results show that it achieved the best performance.

Comparative Study of Guided Electromagnetic Wave Propagation for Two Models of an Open Tape Helix

Rigorous methods to solve the homogeneous boundary value problems and the techniques to resolve illposed mathematical issues arising in the problem formulation of electromagnetic wave propagation of an Open tape helix slow wave structure with anisotropically conducting and completely conducting model is presented. The dispersion equation of both the models require convergent coeefficients of the infinte linear homogeneous equations whose determinant of the coefficent matrix is zero. The coefficient matrix of the anisotropically conducting model is rapidly convergent, while in the case of completely conducting model, the matrix entries which are infinite summation series representations turns out to be divergent, the not so well-posed boundary value problem is regularised with the method of mollification functions. The dispersion characteristics are plotted after truncating the infinite series expansions to adequate number of terms and summing the terms of converging series expansions(after regularization in the case of the completely conducting tape-helix model) in dispersion equations. The tape-current distribution for both models are estimated from the null-space vector of the truncated coefficient matrix corresponding to a particular (β <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> – <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> ) root of the dispersion equation. A comparison of the numerical results for both models show that the neglect of the perpendicular tape-current density component entails a substantial modification of the dispersion characteristics of guided waves supported by an open tape helix.

Potentials from the Polynomial Solutions of the Confluent Heun Equation

Polynomial solutions of the confluent Heun differential equation (CHE) are derived by identifying conditions under which the infinite power series expansions around the z=0 singular point can be terminated. Assuming a specific structure of the expansion coefficients, these conditions lead to four non-trivial polynomials that can be expressed as special cases of the confluent Heun function Hc(p,β,γ,δ,σ;z). One of these recovers the generalized Laguerre polynomials LN(α), and another one the rationally extended X1 type Laguerre polynomials L^N(α). The two remaining solutions represent previously unknown polynomials that do not form an orthogonal set and exhibit features characteristic of semi-classical orthogonal polynomials. A standard method of generating exactly solvable potentials in the one-dimensional Schrödinger equation is applied to the CHE, and all known potentials with solutions expressed in terms of the generalized Laguerre polynomials within, or outside the Natanzon confluent potential class, are recovered. It is also found that the potentials generated from the two new polynomial systems necessarily depend on the N quantum number. General considerations on the application of the Heun type differential differential equations within the present framework are also discussed.

Magnetic neutron scattering from spherical nanoparticles with Néel surface anisotropy: analytical treatment.

The magnetization profile and the related magnetic small-angle neutron scattering cross section of a single spherical nanoparticle with Néel surface anisotropy are analytically investigated. A Hamiltonian is employed that comprises the isotropic exchange interaction, an external magnetic field, a uniaxial magnetocrystalline anisotropy in the core of the particle and the Néel anisotropy at the surface. Using a perturbation approach, the determination of the magnetization profile can be reduced to a Helmholtz equation with Neumann boundary condition, whose solution is represented by an infinite series in terms of spherical harmonics and spherical Bessel functions. From the resulting infinite series expansion, the Fourier transform, which is algebraically related to the magnetic small-angle neutron scattering cross section, is analytically calculated. The approximate analytical solution for the spin structure is compared with the numerical solution using the Landau-Lifshitz equation, which accounts for the full nonlinearity of the problem. The signature of the Néel surface anisotropy can be identified in the magnetic neutron scattering observables, but its effect is relatively small, even for large values of the surface anisotropy constant.

Multipole infinite series expansion of the Coulomb repulsion term using analytical spherical Bessel-like functions

Special functions are particular mathematical functions which arise in the solutions of various classical problems in physics. The functions are quite useful in mathematical analysis. Mostly, they find applications in applied mathematics, theoretical physics, and engineering. Special functions are usually uniquely described by a generating function, a Rodrigues’ formula, and a recurrence relation. In this work, we derive analytical spherical Bessel-like functions corresponding to the infinite degree power series expansion of the functions obtained in the multipole series expansion of the Coulomb repulsion term. The convergence of the truncated power series expansion of the functions with the corresponding derived analytical functions, as well as the comparison of analytical functions of the first and second kinds, is analyzed. Rodrigues’ formula and a recurrence relation for obtaining these analytical functions are also presented. It is shown that the higher order even and odd spherical Bessel-like functions of the first kind are derivable from the lowest order functions of the first and second kinds, respectively.

Some new infinite series expansions for the first passage time densities in a jump diffusion model with phase-type jumps

We propose for the first time some explicit closed-form Laguerre series expansion formulas for the first passage time density functions of a jump diffusion model. Suppose that the jumps in the model are phase-type distributed. In contrast to existing methods in the literature based on numerical Laplace transform inversion, the proposed formulas are in analytical closed-form and simple to implement. Two different methods are proposed to compute the Laguerre coefficients. Various numerical examples are also given to show the effectiveness of our method.

Infinite series expansion of some finite-time dividend and ruin related functions

In this paper, we consider some finite-time dividend and ruin problems in a class of risk models under the constant dividend barrier strategy. This class of risk models includes Brownian motion risk model and the compound Poisson model with and without diffusion. By using Laguerre series expansion, we derive infinite series expansion formulas for the time T-deferred Laplace transform of the ruin time and time T-deferred expected present valued of dividend payments until ruin.

A new dynamic subgrid-scale model using artificial neural network for compressible flow

The subgrid-scale (SGS) kinetic energy has been used to predict the SGS stress in compressible flow and it was resolved through the SGS kinetic energy transport equation in past studies. In this paper, a new SGS eddy-viscosity model is proposed using artificial neural network to obtain the SGS kinetic energy precisely, instead of using the SGS kinetic energy equation. Using the infinite series expansion and reserving the first term of the expanded term, we obtain an approximated SGS kinetic energy, which has a high correlation with the real SGS kinetic energy. Then, the coefficient of the modelled SGS kinetic energy is resolved by the artificial neural network and the modelled SGS kinetic energy is more accurate through this method compared to the SGS kinetic energy obtained from the SGS kinetic energy equation. The coefficients of the SGS stress and SGS heat flux terms are determined by the dynamic procedure. The new model is tested in the compressible turbulent channel flow. From the a posterior tests, we know that the new model can precisely predict the mean velocity, the Reynolds stress, the mean temperature and turbulence intensities, etc.

On Infiltration and Infiltration Characteristic Times

AbstractIn his seminal paper on the solution of the infiltration equation, Philip (1969), https://doi.org/10.1016/b978-1-4831-9936-8.50010-6 proposed a gravity time, tgrav, to estimate practical convergence time and the time domain validity of his infinite time series expansion, TSE, for describing the transient state. The parameter tgrav refers to a point in time where infiltration is dominated equally by capillarity and gravity as derived from the first two (dominant) terms of the TSE. Evidence suggests that applicability of the truncated two‐term equation of Philip has a time limit requiring higher‐order TSE terms to better describe the infiltration process for times exceeding that limit. Since the conceptual definition of tgrav is valid regardless of the infiltration model used, we opted to reformulate tgrav using the analytic implicit model proposed by Parlange et al. (1982), https://doi.org/10.1097/00010694-198206000-00001 valid for all times and related TSE. Our derived gravity times ensure a given accuracy of the approximations describing transient states, while also providing insight about the times needed to reach steady state. In addition to the roles of soil sorptivity (S) and the saturated (Ks) and initial (Ki) hydraulic conductivities, we explored the effects of a soil specific shape parameter β, involved in Parlange's model and related to the type of soil, on the behavior of tgrav. We show that the reformulated tgrav (notably where F(β) is a β‐dependent function) is about three times larger than the classical tgrav given by . The differences between the classical tgrav,Philip and the reformulated tgrav increase for fine‐textured soils, attributed to the time needed to attain steady‐state infiltration and thus i + infiltration for inferring soil hydraulic properties. Results show that the proposed tgrav is a better indicator of time domain validity than tgrav,Philip. For the attainment of steady‐state infiltration, the reformulated tgrav is suitable for coarse‐textured soils. Still neither the reformulated tgrav nor the classical tgrav,Philip are suitable for fine‐textured soils for which tgrav is too conservative and tgrav,Philip too short. Using tgrav will improve predictions of the soil hydraulic parameters (particularly Ks) from infiltration data compared to tgrav,Philip.

Symbolic iteration method based on computer algebra analysis for Kepler\u2019s equation

The Kepler’s equation of elliptic orbits is one of the most significant fundamental physical equations in Satellite Geodesy. This paper demonstrates symbolic iteration method based on computer algebra analysis (SICAA) to solve the Kepler’s equation. The paper presents general symbolic formulas to compute the eccentric anomaly (E) without complex numerical iterative computation at run-time. This approach couples the Taylor series expansion with higher-order trigonometric function reductions during the symbolic iterative progress. Meanwhile, the relationship between our method and the traditional infinite series expansion solution is analyzed in this paper, obtaining a new truncation method of the series expansion solution for the Kepler’s equation. We performed substantial tests on a modest laptop computer. Solutions for 1,002,001 pairs of (e, M) has been conducted. Compared with numerical iterative methods, 99.93% of all absolute errors δE of eccentric anomaly (E) obtained by our method is lower than machine precision epsilon over the entire interval. The results show that the accuracy is almost one order of magnitude higher than that of those methods (double precision). Besides, the simple codes make our method well-suited for a wide range of algebraic programming languages and computer hardware (GPU and so on).

The U Family of Distributions: Properties and Applications

Abstract In this article, we develop a new general family of distributions aimed at unifying some well-established lifetime distributions and offering new work perspectives. A special family member based on the so-called modified Weibull distribution is highlighted and studied. It differs from the competition with a very flexible hazard rate function exhibiting increasing, decreasing, constant, upside-down bathtub and bathtub shapes. This panel of shapes remains rare and particularly desirable for modeling purposes. We provide the main mathematical properties of the special distribution, such as a tractable infinite series expansion of the probability density function, moments of several kinds (raw, incomplete, probability weighted…) with discussions on the skewness and kurtosis. The stochastic ordering structure and stress-strength parameter are also considered, as well as the basics of the order statistics. Then, an emphasis is put on the inferential features of the related model. In particular, the estimation of the model parameters is employed by the maximum likelihood method, with a simulation study to confirm the suitability of the approach. Three practical data sets are then analyzed. It is observed that the proposed model gives better fits than other well-known lifetime models derived from the Weibull model.

First order approximation for coupled film and intraparticle pore diffusion to model sorption/desorption batch experiments

A simple first order approximation was derived to model sorption/desorption kinetics of hazardous compounds in batch experiments based on a coupled film and intraparticle diffusion model. The solution is accurate enough to replace infinite series expansions needed in analytical solution for intraparticle diffusion and it accounts for the mass transfer shift from diffusion in the external aqueous boundary layer to the intraparticle pore space. With increasing distribution coefficient (Kd) and intraparticle particle porosity (ε) or decreasing Sherwood number (Sh) this mass transfer shift from film diffusion to intraparticle pore diffusion is delayed. The simple first order approximation equation allows analyses of mass transfer resistances and calculation of characteristic times which is relevant for the planning of batch experiments. The proposed solution is verified by a semi-analytical solution in Laplace space for fractional mass uptakes in the solid phase at equilibrium ranging from 50% to 91%, representing scenarios typically encountered in batch experiments.

Series expansion of Leray–Trudinger inequality

In this paper, we establish an infinite series expansion of Leray–Trudinger inequality, which is closely related with Hardy inequality and Moser Trudinger inequality. Our result extends early results obtained by Mallick and Tintarev [A. Mallick and C. Tintarev. An improved Leray-Trudinger inequality. Commun. Contemp. Math. 20 (2018), 17501034. OP 21] to the case with many logs. It should be pointed out that our result is about series expansion of Hardy inequality under the case $p=n$, which case is not considered by Gkikas and Psaradakis in [K. T. Gkikas and G. Psaradakis. Optimal non-homogeneous improvements for the series expansion of Hardy's inequality. Commun. Contemp. Math. doi:10.1142/S0219199721500310]. However, we can't obtain the optimal form by our method.

An Analytical Method of Electromagnetic Wave Scattering by a Highly Conductive Sphere in a Lossless Medium with Low-Frequency Dipolar Excitation

The current research involves an analytical method of electromagnetic wave scattering by an impenetrable spherical object, which is immerged in an otherwise lossless environment. The highly conducting body is excited by an arbitrarily orientated time-harmonic magnetic dipole that is located at a reasonable remote distance from the sphere and operates at low frequencies for the physical situation under consideration, wherein the wavelength is much bigger than the size of the object. Upon this assumption, the scattering problem is formulated according to expansions of the implicated magnetic and electric fields in terms of positive integer powers of the wave number of the medium, which is linearly associated to the implied frequency. The static Rayleigh zeroth-order case and the initial three dynamic terms provide an excellent approximation for the obtained solution, while terms of higher orders are of minor significance and are neglected, since we work at the low-frequency regime. To this end, Maxwell’s equations reduce to a finite set of interrelated elliptic partial differential equations, each one accompanied by the perfectly electrically conducting boundary conditions on the metal sphere and the necessary limiting behavior as we move towards theoretical infinity, which is in practice very far from the observation domain. The presented analytical technique is based on the introduction of a suitable spherical coordinated system and yields compact fashioned three-dimensional solutions for the scattered components in view of infinite series expansions of spherical harmonic modes. In order to secure the validity and demonstrate the efficiency of this analytical approach, we invoke an example of reducing already known results from the literature to our complete isotropic case.

Differential Transformation Method for Solving Nonlinear Heat Transfer Equations

In this paper, Differential Transformation Method (DTM) is applied for solving the nonlinear differential equations arising in the field of heat transfer. The solution is considered as an infinite series expansion where converges rapidly to the exact solution. The nonlinear convective–radioactive cooling equation and nonlinear equation of conduction heat transfer with the variable physical properties are chosen as illustrative examples and solutions have been found for each case. Results by DTM with other results calculated by Homotopy Perturbation Method are compatible.