Exponential Integral Research Articles

Exploring exponential time integration for strongly magnetized charged particle motion

A fundamental task in particle-in-cell (PIC) simulations of plasma physics is solving for charged particle motion in electromagnetic fields. This problem is especially challenging when the plasma is strongly magnetized due to numerical stiffness arising from the wide separation in time scales between highly oscillatory gyromotion and overall macroscopic behavior of the system. In contrast to conventional finite difference schemes, we investigated exponential integration techniques to numerically simulate strongly magnetized charged particle motion. Numerical experiments with a uniform magnetic field show that exponential integrators yield superior performance for linear problems (i.e. configurations with an electric field given by a quadratic electric scalar potential) and are competitive with conventional methods for nonlinear problems with cubic and quartic electric scalar potentials.

A bivariate distribution with generalized exponential conditionals

Abstract In this work, we construct a new bivariate statistical distribution by the conditionally specified model approach. The conditional distributions follow the well-known generalized exponential distribution which includes the ordinary exponential distribution and is more flexible than gamma and Weibull in some ways. The newly defined distribution has four parameters that increase the flexibility of the model in data fitting. By equating the dependence parameter to zero, the marginal distributions become independent generalized exponential distributions. The new bivariate distribution depends on the classical exponential integral function which is not difficult to evaluate numerically. The basic properties of the distribution such as distribution functions, moments and stress-strength reliability are derived. The parameters are estimated by the method of maximum likelihood. Two real data fitting applications prove its usefulness in case of negatively correlated bivariate data modelling.

Improved uniform error bounds of a Lawson-type exponential wave integrator method for the Klein-Gordon-Dirac equation

For the Klein-Gordon-Dirac equation (KGDE) with small coupling constant ε∈(0,1], we propose a Lawson-type exponential wave integrator Fourier pseudo-spectral (LEWIFP) method and establish the improved uniform error bounds in the time domain at O(1/ε). We first convert the KGDE to a coupled system and then consider LEWIFP method for the coupled system. The LEWIFP method is proved to be time symmetric which is an important structure in numerical geometric integration. Through careful and rigorous convergence analysis, we establish the error bounds O(hm+ετ2+τ0m) for the full-discretization, where m is determined by the regularity conditions. If the solution is sufficiently smooth with m sufficiently large, we obtain the errors with improved uniform bounds at O(hm+ετ2) in the long-time domain up to O(1/ε). These error bounds are much better than the classical bounds O(hm+τ2) provided by the traditional analysis for the non-Lawson-type exponential wave integrators equipped with Fourier pseudo-spectral method. Combined with the classical analysis tools such as mathematical induction and energy method, we complete our error analysis by adopting the regularity compensation oscillation (RCO) technique which controls the high frequency modes by the regularity of the solution and low frequency modes by phase cancellation. By applying the LEWIFP method to some problems, we show the numerical results to support our error bounds. In addition, the numerical results also show that the discrete mass and energy are stable in the time domain which is long enough. Finally we extend our method to the oscillatory problem.

Dynamic simulation of non-Newtonian boundary layer flow: An enhanced exponential time integrator approach with spatially and temporally variable heat sources

Abstract Scientific inquiry into effective numerical methods for modelling complex physical processes has led to the investigation of fluid dynamics, mainly when non-Newtonian properties and complex heat sources are involved. This paper presents an enhanced exponential time integrator approach to dynamically simulate non-Newtonian boundary layer flow with spatially and temporally varying heat sources. We propose an explicit scheme with second-order accuracy in time, demonstrated to be stable through Fourier series analysis, for solving time-dependent partial differential equations (PDEs). Utilizing this scheme, we construct and solve dimensionless PDEs representing the flow of Williamson fluid under the influence of space- and temperature-dependent heat sources. The scheme discretizes the continuity equation of incompressible fluid and Navier–Stokes, energy, and concentration equations using the central difference in space. Our analysis illuminates how factors affect velocity, temperature, and concentration profiles. Specifically, we observe a rise in temperature profile with enhanced coefficients of space and temperature terms in the heat source. Non-Newtonian behaviours and geographical/temporal variations in heat sources are critical factors influencing overall dynamics. The novelty of our work lies in developing an explicit exponential integrator approach, offering stability and second-order accuracy, for solving time-dependent PDEs in non-Newtonian boundary layer flow with variable heat sources. Our results provide valuable quantitative insights for understanding and controlling complex fluid dynamics phenomena. By addressing these challenges, our study advances numerical techniques for modelling real-world systems with implications for various engineering and scientific applications.

New Sorption Isotherms Derived from a Gamma Distribution of Binding Constants.

New sorption isotherms for heterogeneous sorbents are derived by combining a Gamma distribution of binding constants with a local isotherm defined by a Langmuir or Hill equation. The new "Gamma isotherms" are expressed as Stieltjes transforms of the distribution and involve generalized exponential integrals. The related energy distributions are asymmetric and present a peak corresponding to the mean binding constant. The advantages of the new isotherms are (1) at low pressures or concentrations, with a Langmuir local isotherm, the global "Gamma-Langmuir" isotherm retrieves Henry's law; (2) contrary to the power Freundlich or hypergeometric Freundlich global isotherms, these Gamma isotherms do not need a redefinition of the standard state; (3) with a Hill local isotherm, the global "Gamma-Hill" isotherm allows a separate estimation of the cooperativity and heterogeneity parameters; and (4) the condensation approximation is a good approximation if the local isotherm is Hill and displays a high degree of cooperativity. The Gamma-Langmuir model is applied to three examples from the literature, with rather different Gamma distributions.

Use of rational approximations for phonon density of states in evaluation of neutron thermal scattering law

We propose and explore a new analytical formulation for evaluation of Thermal Scattering Law (TSL) using continuous phonon spectra for neutron inelastic scattering. It involves approximating Phonon Density of States (PDOS) in terms of rational approximations in energy transfer. Debye Waller coefficient calculated with these forms are in reasonable agreement with LEAPR calculated values. TSL can be expanded as a power series, each term of which can be integrated analytically. The method leads to TSL being a series of exponential cosine integrals which again have closed rational forms in energy transfer. Their shapes are analogous to asymmetric resonance line shapes and show maxima for certain values of energy transfer which could correspond to phonon emission or absorption. The series is shown to be convergent and easy to compute as it does not involve any numerical integration. The approach is applied to Beryllium metal and Graphite which have vastly different PDOS shapes, and is shown to be capable of giving most of the trends shown by TSL and cross section results from NJOY nuclear code processing system, even with smaller number of power series terms. Since TSL can be calculated for both negative and positive energy transfer with equal ease, one can easily check for the validity of the approximation obeying Principle of Detailed Balance (PDB). The results overall have shown promise of our formulation in making a case for rapid calculations of TSL and also for using the rational forms in energy transfer for analysis of experimental/calculated PDOS data.

Class of bounds of the generalized Volterra functions

Abstract In the present paper, we prove the monotonicity property of the ratios of the generalized Volterra function. As consequences, new and interesting monotonicity concerning ratios of the exponential integral function, as well as it yields some new functional inequalities including Turán-type inequalities. Moreover, two-side bounding inequalities are then obtained for the generalized Volterra function. The main mathematical tools are some integral inequalities. As applications, a few of upper and lower bound inequalities for the exponential integral function are derived. The various results, which are established in this paper, are presumably new, and their importance is illustrated by several interesting consequences and examples accompanied by graphical representations to substantiate the accuracy of the obtained results. Some potential directions for analogous further research on the subject of the present investigation are indicated in the concluding section.

Exploring localized waves and different dynamics of solitons in (2 + 1)-dimensional Hirota bilinear equation: a multivariate generalized exponential rational integral function approach

Exploring localized waves and different dynamics of solitons in (2 + 1)-dimensional Hirota bilinear equation: a multivariate generalized exponential rational integral function approach

Linearly-fitted energy-mass-preserving schemes for Korteweg–de Vries equations

In this paper, second-order and fourth-order linearly-fitted energy-mass-preserving schemes for Korteweg–de Vries (KdV) equations are proposed. First, the KdV equation is semi-discretized into an oscillatory Hamiltonian system of Ordinary Differential Equations (ODEs) by semi-discretizations of the skew adjoint operator and the Hamiltonian in the Hamiltonian form of the KdV equation. Then the resulting semi-discrete system is integrated by exponential average vector field integrator. The order of convergence and the conservation properties are established for the introduced full-discrete procedures. It shows that the proposed methods are energy-preserving, mass-preserving and qualitatively preserve the dispersive relation well. Numerical experiments are carried out to validate the efficiency and the accuracy of the proposed schemes.

Variable Time-stepping Exponential Integrators for Chemical Reactors with Analytical Jacobians

Chemical combustion problems are known to be stiff and therefore difficult to efficiently integrate in time when numerically simulated. Implicit methods, such as backwards differentiation formula (BDF), are widely considered to be the state-of-the-art methods owing their capability of taking relatively large time-steps while maintaining accurate combustion characteristics. Exponential time integration methods have recently demonstrated the ability to accurately and efficiently solve large scale systems of ordinary differential equations. This study introduces a novel adaptive time stepping exponential integrator named EPI3V. Its performance is measured on spatially homogeneous isobaric reactive mixtures involving three hydrocarbon fuel mechanisms. The full combustion process is simulated using gas compositions with sufficient temperature to obtain auto-ignition. Simulations are run until the steady state is obtained, then a comparison of the computational efficiency and accuracy between a BDF and EPI3V method is made. The novel EPI3V method exhibits comparable computational efficiency to a well-established implementation of the variable time-stepping BDF implicit methods for two of the mechanisms investigated. In certain situations it even demonstrates a slight advantage over the implicit solver. However, in one specific case, the EPI3V shows relative performance degradation compared to the implicit method, but it still converges for this case. These results indicate that exponential time integration methods may be applicable to a larger variety of combustion problems.

Improved uniform error bounds on an exponential wave integrator method for the nonlinear Schrödinger equation with wave operator and weak nonlinearity

In this paper, we establish the improved uniform error bounds for the Lawson-type exponential wave integrator Fourier pseudo-spectral (LEWIFP) method for the nonlinear Schrödinger equation with wave operator (NLSW) and weak nonlinearity characterized by a small constant ε∈(0,1]. We first convert the NLSW to a coupled system and then consider the LEWIFP method for the coupled system. The LEWIFP method is proved to be time symmetric and mass-conservative which is very important from the point of view of numerical geometric integration. Through careful and rigorous convergence analysis, we establish improved uniform error bounds for the full-discretization at O(hm+ε2pτ2) in the long-time domain up to O(1/ε2p) where m is determined by the regularity conditions. These error bounds are much better than the classical error bounds O(hm+τ2) provided by the traditional error analysis. In error analysis, in addition to the classical tools such as mathematical induction and energy method, we adopt the regularity compensation oscillation (RCO) technique to analyze the accumulation of errors carefully. The numerical experiments support our error estimates and demonstrate the discrete mass-conservation. In addition, the numerical results show that the discrete energy is stable in the time domain up to O(1/ε2p).

Soliton unveilings in optical fiber transmission: Examining soliton structures through the Sasa–Satsuma equation

The exploration of higher-order and multicomponent extensions of the nonlinear Schrödinger equation holds significant importance across diverse applications, notably within the field of optics. Among these equations, the integrable Sasa–Satsuma equation stands out for its captivating soliton solutions. The Sasa–Satsuma equation serves as a mathematical representation for describing the propagation of femtosecond light pulses through optical fibers. The breather, multiwave, combined dark–bright, singular, dark, bright and periodic singular optical soliton solutions are derived in this paper by using the 1φ(ϑ),φ′(ϑ)φ(ϑ) and multivariate generalized exponential rational integral function approach. Furthermore, the paper emphasizes the essential fiber parameters essential for the emergence of these structures and offers visual representations of chosen solutions to elucidate their physical characteristics. The novelty of this work lies in the inaugural application of these methods to the Sasa–Satsuma equation and gives a significant advancement in the understanding of optical phenomena. This study not only opens avenues for further research but also introduces a fresh perspective on the Sasa–Satsuma model in the field of nonlinear physics with its applications in optical systems.

Approximation for the Theis Well Function Using Ramanujan’s Series and Bounds for the Exponential Integral

Approximation for the Theis Well Function Using Ramanujan’s Series and Bounds for the Exponential Integral

Spectrally-tuned compact finite-difference schemes with domain decomposition and applications to numerical relativity

Compact finite-difference (FD) schemes specify derivative approximations implicitly, thus to achieve parallelism with domain-decomposition suitable partitioning of linear systems is required. Consistent order of accuracy, dispersion, and dissipation is crucial to maintain in wave propagation problems such that deformation of the associated spectra of the discretized problems is not too severe. In this work we consider numerically tuning spectral error, at fixed formal order of accuracy to automatically devise new compact FD schemes. Grid convergence tests indicate error reduction of at least an order of magnitude over standard FD. A proposed hybrid matching-communication strategy maintains the aforementioned properties under domain-decomposition. Under evolution of linear wave-propagation problems utilizing exponential integration or explicit Runge-Kutta methods improvement is found to remain robust. A first demonstration that compact FD methods may be applied to the Z4c formulation of numerical relativity is provided where we couple our header-only, templated C++ implementation to the highly performant GR-Athena++ code. Evolving Z4c on test-bed problems shows at least an order in magnitude reduction in phase error compared to FD for propagated metric components. Stable binary-black-hole evolution utilizing compact FD together with improved convergence is also demonstrated.

Absent Multiview Semisupervised Classification.

With the advent of vast data collection ways, data are often with multiple modalities or coming from multiple sources. Traditional multiview learning often assumes that each example of data appears in all views. However, this assumption is too strict in some real applications such as multisensor surveillance system, where every view suffers from some data absent. In this article, we focus on how to classify such incomplete multiview data in semisupervised scenario and a method called absent multiview semisupervised classification (AMSC) has been proposed. Specifically, partial graph matrices are constructed independently by anchor strategy to measure the relationships among between each pair of present samples on each view. And to obtain unambiguous classification results for all unlabeled data points, AMSC learns view-specific label matrices and a common label matrix simultaneously. AMSC measures the similarity between pair of view-specific label vectors on each view by partial graph matrices, and consider the similarity between view-specific label vectors and class indicator vectors based on the common label matrix. To characterize the contributions of different views, the p th root integration strategy is adopted to incorporate the losses of different views. By further analyzing the relation between the p th root integration strategy and exponential decay integration strategy, we develop an efficient algorithm with proved convergence to solve the proposed nonconvex problem. To validate the effectiveness of AMSC, comparisons are made with some benchmark methods on real-world datasets and in the document classification scenario as well. The experimental results demonstrate the advantages of our proposed approach.

High-order linearly implicit exponential integrators conserving quadratic invariants with application to scalar auxiliary variable approach

This paper proposes a framework for constructing high-order linearly implicit exponential integrators that conserve a quadratic invariant. This is then applied to the scalar auxiliary variable (SAV) approach. Quadratic invariants are significant objects that are present in various physical equations and also in computationally efficient conservative schemes for general invariants. For instance, the SAV approach converts the invariant into a quadratic form by introducing scalar auxiliary variables, which have been intensively studied in recent years. In this vein, Sato et al. (Appl. Numer. Math. 187, 71-88 2023) proposed high-order linearly implicit schemes that conserve a quadratic invariant. In this study, it is shown that their method can be effectively merged with the Lawson transformation, a technique commonly utilized in the construction of exponential integrators. It is also demonstrated that combining the constructed exponential integrators and the SAV approach yields schemes that are computationally less expensive. Specifically, the main part of the computational cost is the product of several matrix exponentials and vectors, which are parallelizable. Moreover, we conduct some mathematical analyses on the proposed schemes.

Improved uniform error bounds of Lawson-type exponential integrator method for long-time dynamics of the high-dimensional space fractional sine-Gordon equation

Improved uniform error bounds of Lawson-type exponential integrator method for long-time dynamics of the high-dimensional space fractional sine-Gordon equation

Feynman–Kac formula for parabolic Anderson model in Gaussian potential and fractional white noise

In this paper, we establish a Feynman–Kac formula for the stochastic parabolic Anderson model with Gaussian potential in space and fractional white noise in time with Hurst parameter H > 1/2. We obtain the necesscary and suffcient condition for the integrability of the Gaussian potential and the exponential integrability of the solution which is defined by Feynman–Kac formula. By the smoothing of the fractional white noise and techniques from Malliavin calculus, we prove that the Feynman–Kac representation is a mild solution of the stochastic parabolic Anderson equation.

A second-order exponential integration constraint energy minimizing generalized multiscale method for parabolic problems

This paper investigates an efficient exponential integrator generalized multiscale finite element method for solving a class of time-evolving partial differential equations in bounded domains. The proposed method first performs the spatial discretization of the model problem using constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM). This approach consists of two stages. First, the auxiliary space is constructed by solving local spectral problems, where the basis functions corresponding to small eigenvalues are captured. The multiscale basis functions are obtained in the second stage using the auxiliary space by solving local energy minimization problems over the oversampling domains. The basis functions have exponential decay outside the corresponding local oversampling regions. We shall consider the first and second-order explicit exponential Runge-Kutta approach for temporal discretization and to build a fully discrete numerical solution. The exponential integration strategy for the time variable allows us to take full advantage of the CEM-GMsFEM as it enables larger time steps due to its stability properties. We derive the error estimates in the energy norm under the regularity assumption. Finally, we will provide some numerical experiments to sustain the efficiency of the proposed method.

A modification of explicit time integrator scheme for unsteady power-law nanofluid flow over the moving sheets

This paper introduces an exponential time integrator scheme for solving partial differential equations in time, specifically addressing the scalar time-dependent convection-diffusion equation. The proposed second-order accurate scheme is demonstrated to be stable. It is applied to analyze the heat and mass transfer mixed convective flow of power-law nanofluid over flat and oscillatory sheets. The governing equations are transformed into a dimensionless set of partial differential equations, with the continuity equation discretized using a first-order scheme. The proposed time integrator scheme is employed in the time direction, complemented by second-order central discretization in the space direction for the momentum, energy, and nanoparticle volume fraction equations. Quantitative results indicate intriguing trends, indicating that an increase in the Prandtl number and thermophoresis parameter leads to a decrease in the local Nusselt number. This modified time integrator is a valuable tool for exploring the dynamics of unsteady power-law nanofluid flow over moving sheets across various scenarios. Its versatility extends to the examination of unstable fluid flows. This work improves engineering and technological design and operation in nanofluid dynamics. Improving numerical simulations’ precision and computational efficiency deepens our comprehension of fundamental physics, yielding helpful information for enhancing systems that rely on nanofluids.