Exponential Integral Research Articles

A novel high-order explicit exponential integrator scheme for the space–time fractional Bloch–Torrey equation

A novel high-order explicit exponential integrator scheme for the space–time fractional Bloch–Torrey equation

A trust region algorithm for finite-strain plasticity with strongly coupled hardening

As extensions to our Lagrangian finite-strain plasticity framework based on the approximate exponential integrator, we introduce two new algorithms: (i) a fixed-radius trust region root finder for the nonlinear system involving the elastic right Cauchy Green tensor and plastic multiplier (ii) a partitioned approach for the hardening variables, which are determined in a staggered form using the strongly-coupled concept typically adopted for multiphysics problems. This allows the use of intricate hyperelastic laws combined with recent yield functions, which otherwise would involve a laborious treatment, and the use of corresponding work-hardening. Work-hardening would introduce significant nonlinearities in the constitutive system if used in a fully-coupled form. For the partitioned approach, a dynamic relaxation algorithm is adopted. This allows the efficient solution of the two nonlinear equations without significant drifting. Results show that robustness and efficiency are significantly improved. Herein, algorithms are described in detail. Significant testing is performed with imposed strains up to 100 for a carbon steel. This contributes to the robustness of equilibrium iterations. Drifting is also assessed as a function of number of steps in the dynamic relaxation algorithm. Numerical experimentation is performed in the 3D tension test for an anisotropic yield function.

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We present a method for computing actions of the exponential-like φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi $$\\end{document}-functions for a Kronecker sum K of d arbitrary matrices Aμ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_\\mu $$\\end{document}. It is based on the approximation of the integral representation of the φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi $$\\end{document}-functions by Gaussian quadrature formulas combined with a scaling and squaring technique. The resulting algorithm, which we call phiks, evaluates the required actions by means of μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}-mode products involving exponentials of the small sized matrices Aμ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_\\mu $$\\end{document}, without forming the large sized matrix K itself. phiks, which profits from the highly efficient level 3 BLAS, is designed to compute different φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi $$\\end{document}-functions applied on the same vector or a linear combination of actions of φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi $$\\end{document}-functions applied on different vectors. In addition, thanks to the underlying scaling and squaring techniques, the desired quantities are available simultaneously at suitable time scales. All these features allow the effective usage of phiks in the exponential integration context. In fact, our newly designed method has been tested in popular exponential Runge–Kutta integrators of stiff order from one to four, in comparison with state-of-the-art algorithms for computing actions of φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi $$\\end{document}-functions. The numerical experiments with discretized semilinear evolutionary 2D or 3D advection–diffusion–reaction, Allen–Cahn, and Brusselator equations show the superiority of the proposed μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}-mode approach.

On the study of double dispersive equation in the Murnaghan’s rod: Dynamics of diversity wave structures

This article secures the various wave structures of the fractional double dispersive equation, a significant nonlinear equation that describes the propagation of nonlinear waves within the elastic, uniform, and inhomogeneous Murnaghan’s rod. The model under discussion has a wide range of applications in science and engineering. Two recently developed analytical techniques known as the improved generalized Riccati equation mapping method and the multivariate generalized exponential rational integral function method have been applied to the proposed equation for the first time. A variety of solutions have been revealed such that dark, singular, bright-dark, bright, complex, and combined solitons. Furthermore, we include a diverse array of plots that illustrate the physical interpretation of the obtained solutions in relation to a number of significant parameters, thereby highlighting the impact of fractional derivatives. Within the context of the proposed model, these visualizations give a clear understanding of the behavior and characteristics of the solutions. This study’s results have the potential to enhance comprehension of the nonlinear dynamic characteristics exhibited by the specified system and validate the efficacy of the implemented techniques. The achieved results significantly enhance our understanding of nonlinear science and the nonlinear wave fields associated with more complex nonlinear models.

Self-similarity and the direct (enstrophy) cascade in forced two-dimensional fluid turbulence

In the absence of large-scale coherent structures, a widely used statistical theory of two-dimensional turbulence developed by Kraichnan, Leith, and Batchelor (KLB) predicts a power-law scaling for the energy, $E(k)\propto k^\alpha$ with an integral exponent $\alpha ={-3}$ , in the inertial range associated with the direct cascade. A power-law scaling is also observed in the presence of coherent structures, but the scaling exponent becomes fractal and often differs substantially from the value predicted by the KLB theory. Here we present a dynamical theory that sheds new light on the relationship between the spatial and temporal structure of the large-scale flow and the scaling of small-scale structures representing filamentary vorticity. Specifically, we find hyperbolic regions of the large-scale flow to play a key role in the flux of enstrophy between scales. Small-scale vorticity in these regions can be described by dynamically self-similar solutions of the Euler equation, which explains the power-law scaling. Furthermore, we find that correlations between different hyperbolic regions are responsible for the emergence of fractal scaling exponents.

Review of Vedic Mathematics Yavadunam Tavadunikritya Varganicha Yojayet Sutra for Cubing a Number

This paper discloses secrets and logics of the short formula of ancient Vedic Mathematics Yavadunam Tavadunikritya Varganicha Yojayet Sutra in finding the cube of a positive number. Through the formalization process, we have aimed to bridge the gap between ancient mathematical wisdom and modern theoretical foundations, showcasing the timeless utility of Vedic techniques in mathematical calculations. The sutra has been reviewed using the algebraic explanation in terms of decimal number system of the multiplication process by applying Binomial theorem for the positive integral exponent 3. Some more short-cut techniques have been discussed to explain the sutra and it has been found that they are producing the same result. There are shortcomings that have been found in applying the sutra, which has been solved using some other techniques as a special rule. Later the sutra has been discussed for negative integers also with a note. Before coming to the conclusion, we have discussed some limitations of the sutra, although that has been solved in the paper but with a different process. We have discussed the implications of the sutra for modern computational methods and educational practices, suggesting potential areas for further research.

Explicit exponential Runge–Kutta methods for semilinear time-fractional integro-differential equations

In this work, we consider and analyze explicit exponential Runge–Kutta methods for solving semilinear time-fractional integro-differential equation, which involves two nonlocal terms in time. Firstly, the temporal Runge–Kutta discretizations follow the idea of exponential integrators. Subsequently, we utilize the spectral Galerkin method to introduce a fully discrete scheme. Then, we mainly focus on discussing the one-stage and two-stage methods for solving the proposed semilinear problem. Based on special abstract settings, we perform the convergence analysis for the proposed two different stage methods. In this process, we heavily use estimates about the operator family {S̃(t)}, and in combination with Lipschitz continuous condition. Finally, some numerical experiments confirm theoretical results. Meanwhile, applying this scheme to the related linear problem yields high-order convergence, highlighting the advantages of explicit exponential Runge–Kutta methods.

Sparse-spike seismic inversion with semismooth newton algorithm solver

Seismic prospecting has been widely used in the exploration and development of underground geological resources, such as mineral products (e.x., coal, uranium deposit), oil and gas, groundwater, and so forth. Seismic impedance is a physical characteristic parameter of underground formation, which can be used in lithologic classification, rock characterization, stratigraphic correlation, and further mineral reservoir prediction, reserve estimation, and so forth. To estimate impedance from seismic data, one must perform reflectivity series inversion first. Under a simple exponential integration transformation, the reflectivity series can give the final estimated impedance. Sparse-spike seismic inversion is the most common method to obtain reflectivity series with high resolution. It adopts a sparse regularization to impose sparsity on reflectivity series. From sparse reflectivity series, the final estimated impedance has blocky features to make formation interfaces and geological edges precise, which is very important to accurately delineate the distribution range of mineral resources. The development of sparse-spike seismic inversion is still facing major challenges of fast optimization algorithms in real-life application, especially for massive seismic data in 3D case. Semismooth Newton algorithm (SNA), which is a second order mehtod and has super-linear, even quadratic convergence rate, is used to solve sparse-spike seismic inversion. The proposed algorithm has been compared with common used algorithms through a synthetic seismic trace and a 3D real seismic data volume. The results show that the proposed algorithm has faster convergence rate and fewer computation time. It provides a new effective algorithm to solve sparse-spike seismic inversion.

Padé Approximations and Irrationality Measures on Values of Confluent Hypergeometric Functions

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.

Error estimates of exponential wave integrators for the Dirac equation in the massless and nonrelativistic regime

AbstractWe present exponential wave integrator Fourier pseudospectral (EWI‐FP) methods and establish their error estimates of the fully discrete schemes for the Dirac equation in the massless and nonrelativistic regime. This regime involves a small dimensionless parameter where , and is inversely proportional to the speed of light. The solution exhibits highly oscillatory behavior in time and rapid wave propagation in space in this regime. Specifically, the time oscillations have a wavelength of , while the spatial oscillations have a wavelength of , with a wave speed of . We employ (symmetric) exponential wave integrators for temporal derivatives and Fourier spectral discretization for spatial derivatives. We rigorously derive the error bounds which explicitly depend on the mesh size , the time step and the small dimensionless parameter . The error estimates for the EWI‐FP methods demonstrate that their meshing strategy requirement (‐scalability) necessitates setting and when . Finally, some numerical examples are provided to validate the error bounds.

Uniform and optimal error estimates of a nested Picard integrator for the nonlinear Schrödinger equation with wave operator

AbstractWe propose a second‐order nested Picard iterative integrator sine pseudospectral (NPI‐SP) method for the nonlinear Schrödinger equation with wave operator (NLSW) involving a parameter and carry out rigorous error estimates. Actually, the equation propagates wave with wavelength in time, while the amplitude of the leading oscillation is for well‐prepared initial data, and for ill‐prepared initial data, respectively. Based on the exponential integrator and nested Picard iteration, the uniformly accurate (w.r.t. ) NPI‐SP scheme is proposed with the optimal uniform error bounds at in time and spectral accuracy in space for both well‐prepared and ill‐prepared data in ‐norm. This result significantly improves the error bounds of the finite difference methods and exponential wave integrator for the NLSW. Error estimates are rigorously carried out and numerical examples are provided to confirm the theoretical analysis.

Impact of the numerical conversion to optical depth on the transfer of polarized radiation

Making the conversion from the geometrical spatial scale to the optical depth spatial scale is useful in obtaining numerical solutions for the radiative transfer equation. This is because it allows for the use of exponential integrators, while enforcing numerical stability. Such a conversion involves the integration of the total opacity of the medium along the considered ray path. This is usually approximated by applying a piecewise quadrature in each spatial cell of the discretized medium. However, a rigorous analysis of this numerical step is lacking. This work is aimed at clearly assessing the performance of different optical depth conversion schemes with respect to the solution of the radiative transfer problem for polarized radiation, out of the local thermodynamic equilibrium. We analyzed different optical depth conversion schemes and their combinations with common formal solvers, both in terms of the rate of convergence as a function of the number of spatial points and the accuracy of the emergent Stokes profiles. The analysis was performed in a 1D semi-empirical model of the solar atmosphere, both in the absence and in the presence of a magnetic field. We solved the transfer problem of polarized radiation in different settings: the continuum, the photospheric Sr i AA modeled under the assumption of complete frequency redistribution, and the chromospheric Ca i AA taking the partial frequency redistribution effects into account during the modeling. High-order conversion schemes generally outperform low-order methods when a sufficiently high number of spatial grid points is considered. In the synthesis of the emergent Stokes profiles, the convergence rate, as a function of the number of spatial points, is impacted by both the conversion scheme and formal solver. The use of low-order conversion schemes significantly reduces the accuracy of high-order formal solvers. In practical applications, the use of low-order optical depth conversion schemes introduces large numerical errors in the formal solution. To fully exploit high-order formal solvers and obtain accurate synthetic emergent Stokes profiles, it is necessary to use high-order optical depth conversion schemes.

Weak convergence of tamed exponential integrators for stochastic differential equations

We prove weak convergence of order one for a class of exponential based integrators for SDEs with non-globally Lipschitz drift. Our analysis covers tamed versions of Geometric Brownian Motion (GBM) based methods as well as the standard exponential schemes. The numerical performance of both the GBM and exponential tamed methods through four different multi-level Monte Carlo techniques are compared. We observe that for linear noise the standard exponential tamed method requires severe restrictions on the step size unlike the GBM tamed method.

Multicomponent nonlinear fractional Schrödinger equation: On the study of optical wave propagation in the fiber optics

In this article, we investigate the soliton wave dynamics of the fractional three-component coupled nonlinear Schrödinger equation. This equation is considered as a fundamental tool for describing the behavior of optical pulses in optical fibers. There is increasing interest in studying these equations as they can be used to explain a wide range of complex physical phenomena in various scientific and engineering fields, including nonlinear fiber optics, electromagnetic field waves, and signal processing through optical fibers. In order to obtain the required solutions, the first step is to apply the complex wave transformations with β-fractional derivative to obtain the nonlinear ordinary differential equations of the governing model. Furthermore, by applying the novel integration methods known as generalized Arnous method and multivariate generalized exponential rational integral function approach, different type of soliton solutions are extracted. Different sets and values of the physical parameters are used to study the optical soliton solutions of the studied system. Based on a comparative analysis of our results with existing techniques, this study concludes that our proposed solutions are novel. The results described in this work are able to enhance the nonlinear dynamical behavior of a given system and confirm the effectiveness of the methods used. Furthermore, the different graphs are sketched to visualize the solutions behavior with various parametric values. It is anticipated that obtained solutions will be significant in the study of wave propagation and other related fields.

Improved uniform error bounds on a Lawson-type exponential integrator for the long-time dynamics of sine-Gordon equation

Improved uniform error bounds on a Lawson-type exponential integrator for the long-time dynamics of sine-Gordon equation

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.