Exponential Integral Research Articles

Global maximal regularity for equations with degenerate weights

In this paper we are concerned with global maximal regularity estimates for elliptic equations with degenerate weights. We consider both the linear case and the non-linear case. We show that higher integrability of the gradients can be obtained by imposing a local small oscillation condition on the weight and a local small Lipschitz condition on the boundary of the domain. Our results are new in the linear and non-linear case. We show by example that the relation between the exponent of higher integrability and the smallness parameters is sharp even in the linear or the unweighted case.

Energy equality of MHD system under a weaker condition on magnetic field

We prove the energy equality of MHD system in the space founded by Cheskidov et al. (Nonlinearity 21:1233–1252, 2008) and Berselli and Chiodaroli (Nonlinear Anal 192:111704, 2020). It is clarified that the energy equality is established for a larger class of the magnetic field than that of velocity field. Most of the cases, we deal with the energy equality of MHD system in bounded domain. On the other hand, if the spacial integrability exponents of the weak solution are large, it is necessary to use the Besov space, which is suitable for us to handle freely derivatives of the nonlinear convection term. Only in this case we deal with the energy equality of MHD system in the whole space. Our result covers most of previous theorems on validity of the energy equality on the Navier–Stokes equations.

Exponential integrator for stochastic strongly damped wave equation based on the Wong–Zakai approximation

We consider strong convergence of numerical approximations for a stochastic strongly damped wave equation driven by a class of additive space–time noises characterized by a parameter β∈(−1,2]. With the help of the Wong–Zakai (WZ) approximation to the noise, i.e., replacing the driven noise with its finite spectral expansion truncation, we obtain an approximate equation. Based on the consistency and high regularity of the approximate equation, we develop two exponential integrators for time-stepping discretization and use the spectral Galerkin method in space to develop full-discrete schemes. Performing error estimates in the strong sense, we show that the optimal strong order in time of the proposed WZ-approximation-based exponential Euler scheme is min{1,1+β/2−ϵ}, which is |β|/2 order higher than the order of min{1,1+β} in the existing works when β∈(−1,0). Moreover, we prove that the proposed WZ-approximation-based exponential trapezoidal scheme is of min{3/2,1+β/2−ϵ}-order strong convergence in time when β∈[−1/4,2], so it can break the first order barrier and obtain a higher accurate numerical solution. Numerical examples are performed to verify and develop the theoretical findings.

Gauge-Transformed Exponential Integrator for Generalized KdV Equations with Rough Data

Gauge-Transformed Exponential Integrator for Generalized KdV Equations with Rough Data

Semi-discretization and full-discretization with improved accuracy for charged-particle dynamics in a strong nonuniform magnetic field

The aim of this paper is to formulate and analyze numerical discretizations of charged-particle dynamics (CPD) in a strong nonuniform magnetic field. A strategy is firstly performed for the two dimensional CPD to construct the semi-discretization and full-discretization which have improved accuracy. This accuracy is improved in the position and in the velocity when the strength of the magnetic field becomes stronger. This is a better feature than the usual so called ``uniformly accurate methods”. To obtain this refined accuracy, some reformulations of the problem and two-scale exponential integrators are incorporated, and the improved accuracy is derived from this new procedure. Then based on the strategy given for the two dimensional case, a new class of uniformly accurate methods with simple scheme is formulated for the three dimensional CPD in maximal ordering case. All the theoretical results of the accuracy are numerically illustrated by some numerical tests.

Multiple orthogonal polynomials associated with the exponential integral

AbstractWe introduce a new family of multiple orthogonal polynomials satisfying orthogonality conditions with respect to two weights on the positive real line, with the gamma density and a density related to the exponential integral . We give explicit formulas for the type I functions and type II polynomials, their Mellin transform, Rodrigues formulas, hypergeometric series, and recurrence relations. We determine the asymptotic distribution of the (scaled) zeros of the type II multiple orthogonal polynomials and make a connection to random matrix theory. Finally, we also consider two related families of mixed‐type multiple orthogonal polynomials.

Positivity Preserving Exponential Integrators for Differential Riccati Equations

Positivity Preserving Exponential Integrators for Differential Riccati Equations

Strong convergence of a fractional exponential integrator scheme for finite element discretization of time-fractional SPDE driven by fractional and standard Brownian motions

The aim of this work is to provide the first strong convergence result of a numerical approximation of a general time-fractional second order stochastic partial differential equation involving a Caputo derivative in time of order α∈(12,1) and driven simultaneously by a multiplicative standard Brownian motion and additive fBm with Hurst parameter H∈(12,1), more realistic to model the random effects on transport of particles in medium with thermal memory. We prove the existence and uniqueness results, and perform the spatial discretization using the standard finite element and the temporal discretization based on a generalized exponential time differencing method (GETD). We provide the temporal and spatial convergence proofs for our fully discrete scheme and the result shows that the convergence orders depend on the regularity of the initial data, the power of the fractional derivative, and the Hurst parameter H. Numerical results are provided to illustrate our theoretical results.

Uniformly accurate nested Picard iterative schemes for nonlinear Schrödinger equation with highly oscillatory potential

The nonlinear Schrödinger equation with a highly oscillatory potential (NLSE-OP) often appears in many multiscale dynamical systems, where the temporal oscillation causes the major numerical difficulties. Recently, the splitting schemes (Su et al., 2020) were analyzed rigorously for solving the NLSE-OP and the error bounds show that they are only uniformly first-order accurate. This obviously can not meet the requirement of high precision in actual calculation. In this paper, in order to obtain a higher uniform convergence order, we study the nested Picard iterative (NPI) schemes for the NLSE-OP with polynomial nonlinearity. Firstly we propose a uniformly accurate first-order exponential wave integrator (EWI) scheme for arbitrary nonlinearity by integrating the potential exactly. Then we construct a uniformly accurate second-order NPI scheme for the NLSE-OP with polynomial nonlinearity. The schemes are fully explicit and very efficient due to the fast Fourier transform (FFT). We give rigorously error analysis and establish error bounds for the numerical solutions without any CFL-type condition constraint. Numerical experiments prove the correctness of our theoretical analysis and the effectiveness of our schemes. Theoretically, our schemes can be extended to uniformly accurate arbitrary high order for the NLSE-OP.

Vanishing exponential integrability for Riesz potentials in Morrey–Orlicz spaces

AbstractOur aim in this note is to establish exponential integrability and vanishing exponential integrability for Riesz potentials of functions in Morrey–Orlicz spaces. Our work generalizes results by Adams and Hurri‐Syrjänen (2003) and the authors (2007) from Orlicz case. We also discuss the size of the exceptional sets by using a capacity and Hausdorff measure.

Vanishing Morrey integrability for Riesz potentials in Morrey-Orlicz spaces

Our aim in this paper is to establish vanishing Morrey integrability for Riesz potentials of functions in Morrey-Orlicz spaces. We discuss the size of the exceptional sets by using a capacity and Hausdorff measure. We also give Trudinger-type exponential Morrey integrability for Riesz potentials of functions in Morrey-Orlicz spaces.

A higher order positivity preserving scheme for the strong approximations of a stochastic epidemic model

By combining a predictor–corrector method with a Lamperti-type transformation, we propose a higher order, explicit, positivity preserving scheme for the stochastic susceptible-infected-susceptible (SIS) epidemic model that takes values in (0,N). The proposed scheme is able to preserve the domain (0,N) of the original SIS model and admits numerical approximations with the exponential integrability. These facts help us to recover the strong convergence rate of order 1.5 for the scheme. Moreover, the dynamic behaviors of numerical approximations are also investigated. It is shown that the numerical scheme is able to reproduce the extinction and persistence properties of the disease under certain assumptions. Numerical experiments are finally presented to verify the theoretical findings.

Sharp bounds for multilinear fractional integral operators on Morrey type spaces: the endpoint cases

In this paper, the Strichartz’s result of the exponential integrability of fractional integral operators is improved. Also, we establish the endpoint boundedness of the multilinear fractional integrals acting on the multi-Morrey spaces. The conclusions relax the restriction that $$p_{i}\ne 1$$ for all $$i=1,\ldots ,m$$ and extend some known results.

A Semi-discrete First-Order Low Regularity Exponential Integrator for the “good” Boussinesq Equation Without Loss of Regularity

A Semi-discrete First-Order Low Regularity Exponential Integrator for the “good” Boussinesq Equation Without Loss of Regularity

A positivity preserving Lamperti transformed Euler–Maruyama method for solving the stochastic Lotka–Volterra competition model

A new positivity preserving numerical scheme is presented for a class of d-dimensional stochastic Lotka–Volterra competitive models, which are characterized by super-linear coefficients and positive solutions. The scheme, dubbed the Lamperti transformed Euler–Maruyama method, approximates the exact solution by integrating a Lamperti-type transformation with an explicit Euler–Maruyama method that has the benefit of being explicit and straightforward to implement. Even though the coefficients of the transformed models grow exponentially and do not satisfy the general monotonicity condition, based on the exponential integrability of the solution, it is proved that the proposed numerical method is of 12-order strong convergence. In particular, when matrix A of the model is a diagonal matrix, the first-order strong convergence is also obtained. Without any step size constraints, the method can preserve long-time dynamical properties such as extinction and pth moment exponential asymptotic stability. Numerical examples are given to support our theoretical conclusions.

Structural Changes in Nonlocal Denoising Models Arising Through Bi-Level Parameter Learning

We introduce a unified framework based on bi-level optimization schemes to deal with parameter learning in the context of image processing. The goal is to identify the optimal regularizer within a family depending on a parameter in a general topological space. Our focus lies on the situation with non-compact parameter domains, which is, for example, relevant when the commonly used box constraints are disposed of. To overcome this lack of compactness, we propose a natural extension of the upper-level functional to the closure of the parameter domain via Gamma-convergence, which captures possible structural changes in the reconstruction model at the edge of the domain. Under two main assumptions, namely, Mosco-convergence of the regularizers and uniqueness of minimizers of the lower-level problem, we prove that the extension coincides with the relaxation, thus admitting minimizers that relate to the parameter optimization problem of interest. We apply our abstract framework to investigate a quartet of practically relevant models in image denoising, all featuring nonlocality. The associated families of regularizers exhibit qualitatively different parameter dependence, describing a weight factor, an amount of nonlocality, an integrability exponent, and a fractional order, respectively. After the asymptotic analysis that determines the relaxation in each of the four settings, we finally establish theoretical conditions on the data that guarantee structural stability of the models and give examples of when stability is lost.

Double coset operators and eta-quotients

We study a type of generalized double coset operators which may change the characters of modular forms. For any pair of characters v1 and v2, we describe explicitly those operators mapping modular forms of character v1 to those of v2. We give three applications, concerned with eta-quotients. For the first application, we give many pairs of eta-quotients of small weights and levels, such that there are operators map one eta-quotient to another. We also find out these operators. For the second application, we apply the operators to eta-powers whose exponents are positive integers not greater than 24. This results in recursive formulas of the coefficients of these functions, generalizing Newman's theorem. For the third application, we describe a criterion and an algorithm of whether and how an eta-power of arbitrary integral exponent can be expressed as a linear combination of certain eta-quotients.

A uniformly accurate method for the Klein-Gordon-Dirac system in the nonrelativistic regime

A multiscale time integrator Fourier pseudospectral (MTI-FP) method is presented for discretizing the massive Klein-Gordon-Dirac (KGD) system which involves a small dimensionless parameter 0<ε≤1. In the nonrelativistic limit regime, the KGD system admits rapid oscillations in time as ε→0+. In addition, the nonlinear Yukawa interaction and the indefinite Dirac operator bring other significant difficulties. The main idea of the MTI-FP method is to construct a precise multiscale decomposition by the frequency (MDF) to the solution of the KGD system at each time step and then employ the Fourier pseudospectral discretization for the spatial derivatives followed with the exponential wave integrator (EWI) for the time marching. This approach is explicit, easy to implement and preforms significantly better than the classical methods in the literature. More specifically, we rigorously establish the uniform error bounds at O(τ+hm0−1) for all ε∈(0,1] and optimal quadratic temporal error bounds at O(τ2) in the ε=O(1) regime, where τ is the time step size, h is mesh size and m0 depends on the regularity of the solution. Extensive numerical results demonstrate that our error bounds are optimal and sharp. Finally, we apply the MTI-FP method to numerically study the nonrelativistic limit behaviors of the KGD system when ε→0+.

Analytical Approximations of Well Function by Solving the Governing Differential Equation Representing Unsteady Groundwater Flow in a Confined Aquifer

A solution of the governing equation representing the drawdown in a horizontal confined aquifer, where groundwater flow is unsteady, was first provided by Theis and is famously known as the Theis solution. This solution was given in terms of an exponential integral, also called the well function, for which simple and reliable approximations are preferred due to their practical applications. To that end, several approximations are available in the literature, of which some are based on series approximations for the integral, and others are numerical approximations. This study employs three kinds of homotopy-based methods, namely the homotopy analysis method (HAM), homotopy perturbation method (HPM), and optimal homotopy asymptotic method (OHAM), for analytically solving the governing partial differential equation (PDE). For convenience, the PDE is first converted to a boundary value problem (BVP) using a similarity transformation. Comparing the series approximations obtained using these methods with the Theis solution, it is found that the 10th-order HAM, and just three terms of OHAM-based solutions, provide accurate approximations. On the other hand, the HPM-based solution is found to be accurate only within a small domain. Further, the proposed approximations are compared with several series and numerical approximations available in the literature using the percentage error. The proposed methodology provides accurate approximations of the well function by directly solving the governing differential equation in a general framework and thus can be adapted to other practical situations arising in groundwater flow.

Structure‐preserving exponential wave integrator methods and the long‐time convergence analysis for the Klein‐Gordon‐Dirac equation with the small coupling constant

AbstractRecently, the numerical methods for long‐time dynamics of PDEs with weak nonlinearity (or small potentials) have received more and more attention. For the Klein‐Gordon‐Dirac (KGD) equation with the small coupling constant , we propose two time symmetric and structure‐preserving exponential wave integrator Fourier pseudo‐spectral (TSSPEWIFP) methods under periodic boundary conditions. The new methods are proved to preserve the energy in discrete level and in addition the first one preserves the modified discrete mass. Through rigorous error analysis, we establish uniform error bounds of the numerical solutions at up to the time at for where and are the mesh size and time step, respectively, and depends on the regularity conditions. Compared with the results of existing numerical analysis, our analysis has the advantage of showing the long time numerical errors for the KGD equation with the small coupling constant. The tools for error analysis mainly include cut‐off technique and the standard energy method. We also extend the results on error bounds, structure‐preservation and time symmetry to the oscillatory KGD equation with wavelength at in time which is equivalent to the KGD equation with small coupling constant. Numerical experiments confirm that the theoretical results are correct. Our methods are novel because that to the best of our knowledge there has not been any energy‐preserving exponential wave integrator method and any relevant long time analysis for the KGD equation.