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## Electronic Research Archive

June 2020 , Volume 28 , Issue 2

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*+*[Abstract](918)

*+*[HTML](471)

*+*[PDF](371.26KB)

**Abstract:**

In this paper, we generalize the Riemann-Liouville differential and integral operators on the space of Henstock-Kurzweil integrable distributions,

*+*[Abstract](629)

*+*[HTML](311)

*+*[PDF](329.14KB)

**Abstract:**

We confirm several conjectures of Sun involving quadratic residues modulo odd primes. For any prime

and that

where

where

*+*[Abstract](704)

*+*[HTML](348)

*+*[PDF](417.81KB)

**Abstract:**

In this paper, the initial-boundary value problem for a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity is invsitgated. First, we establish the local well-posedness of solutions by means of the semigroup theory. Then by using ordinary differential inequalities, potential well theory and energy estimate, we study the conditions on global existence and finite time blow-up. Moreover, the lifespan (i.e., the upper bound of the blow-up time) of the finite time blow-up solution is estimated.

*+*[Abstract](544)

*+*[HTML](318)

*+*[PDF](425.86KB)

**Abstract:**

In this paper, our goal is to improve the local well-posedness theory for certain generalized Boussinesq equations by revisiting bilinear estimates related to the Schrödinger equation. Moreover, we propose a novel, automated procedure to handle the summation argument for these bounds.

*+*[Abstract](685)

*+*[HTML](339)

*+*[PDF](429.17KB)

**Abstract:**

The aim of this paper is to investigate the existence of weak solutions for a Kirchhoff-type differential inclusion wave problem involving a discontinuous set-valued term, the fractional

*+*[Abstract](732)

*+*[HTML](329)

*+*[PDF](361.94KB)

**Abstract:**

Nonlinear Klein-Gordon equation with combined power type nonlinearity and critical initial energy is investigated. The qualitative properties of a new ordinary differential equation are studied and the concavity method of Levine is improved. Necessary and sufficient conditions for finite time blow up and global existence of the solutions are proved. New sufficient conditions on the initial data for finite time blow up, based on the necessary and sufficient ones, are obtained. The asymptotic behavior of the global solutions is also investigated.

*+*[Abstract](724)

*+*[HTML](358)

*+*[PDF](457.43KB)

**Abstract:**

In this paper, a rotating blades equation is considered. The arbitrary pre-twisted angle, arbitrary pre-setting angle and arbitrary rotating speed are taken into account when establishing the rotating blades model. The nonlinear PDEs of motion and two types of boundary conditions are derived by the extended Hamilton principle and the first-order piston theory. The well-posedness of weak solution (global in time) for the rotating blades equation with Clamped-Clamped (C-C) boundary conditions can be proved by compactness method and energy method. Strong energy estimates are derived under additional assumptions on the initial data. In addition, the existence and regularity of weak solutions (global in time) for the rotating blades equation with Clamped-Free (C-F) boundary conditions are proved as well.

*+*[Abstract](887)

*+*[HTML](328)

*+*[PDF](375.01KB)

**Abstract:**

Utilizing some properties of multivariate Baskakov–Kantorovich operators and using

*+*[Abstract](445)

*+*[HTML](290)

*+*[PDF](468.15KB)

**Abstract:**

In this paper, we study the Banach

*+*[Abstract](765)

*+*[HTML](554)

*+*[PDF](392.34KB)

**Abstract:**

In this paper, we consider the nonlinear Schrödinger equation on

with

*+*[Abstract](545)

*+*[HTML](654)

*+*[PDF](299.58KB)

**Abstract:**

In the note we study the multipoint Seshadri constants of

*+*[Abstract](685)

*+*[HTML](319)

*+*[PDF](933.36KB)

**Abstract:**

In this paper, a family of potential wells are introduced by means of the modified depths of the potential wells. These potential wells are employed to study the initial-boundary value problem for a wave equation. The expression of the depths of the potential wells is derived. Global existence and finite time blow-up of weak solutions with the subcritical initial energy and the critical initial energy are obtained, respectively. Moreover, some numerical simulations of the depths of the potential wells are carried out.

*+*[Abstract](642)

*+*[HTML](336)

*+*[PDF](377.34KB)

**Abstract:**

In this paper, a hybridized weak Galerkin (HWG) finite element scheme is presented for solving the general second-order elliptic problems. The HWG finite element scheme is based on the use of a Lagrange multiplier defined on the element boundaries. The Lagrange multiplier provides a numerical approximation for certain derivatives of the exact solution. It is worth pointing out that a skew symmetric form has been used for handling the convection term to get the stability in the HWG formulation. Optimal order error estimates are derived for the corresponding HWG finite element approximations. A Schur complement formulation of the HWG method is introduced for implementation purpose.

*+*[Abstract](733)

*+*[HTML](290)

*+*[PDF](1291.07KB)

**Abstract:**

In this paper, we introduce a dimension splitting method for simulating the air flow state of the aeroengine turbine fan. Based on the geometric model of the fan blade, the dimension splitting method establishes a semi-geodesic coordinate system. Under such coordinate system, the Navier-Stokes equations are reformulated into the combination of membrane operator equations on two-dimensional manifolds and bending operator equations along the hub circle. Using Euler central difference scheme to approximate the third variable, the new form of Navier-Stokes equations is splitting into a set of two-dimensional sub-problems. Solving these sub-problems by alternate iteration, it follows an approximate solution to Navier-Stokes equations. Furthermore, we conduct a numerical experiment to show that the dimension splitting method has a good performance by comparing with the traditional methods. Finally, we give the simulation results of the pressure and flow state of the fan blade.

*+*[Abstract](470)

*+*[HTML](219)

*+*[PDF](288.54KB)

**Abstract:**

We construct a family of unital non-associative algebras

*+*[Abstract](459)

*+*[HTML](205)

*+*[PDF](358.74KB)

**Abstract:**

This paper is concerned with three-dimensional compressible viscous and heat-conducting micropolar fluid in the domain to the subset of

*+*[Abstract](419)

*+*[HTML](209)

*+*[PDF](347.84KB)

**Abstract:**

In this paper, we study the quasi-neutral limit for the compressible two-fluid Euler–Maxwell equations for well-prepared initial data. Precisely, we proved the solution of the three-dimensional compressible two-fluid Euler–Maxwell equations converges locally in time to that of the compressible Euler equation as

*+*[Abstract](437)

*+*[HTML](225)

*+*[PDF](310.06KB)

**Abstract:**

In this paper, a two-grid finite element scheme for semilinear parabolic integro-differential equations is proposed. In the two-grid scheme, continuous linear element is used for spatial discretization, while Crank-Nicolson scheme and Leap-Frog scheme are ultilized for temporal discretization. Based on the combination of the interpolation and Ritz projection technique, some superclose estimates between the interpolation and the numerical solution in the

*+*[Abstract](585)

*+*[HTML](262)

*+*[PDF](1531.97KB)

**Abstract:**

This paper is concerned with a

*+*[Abstract](594)

*+*[HTML](238)

*+*[PDF](434.86KB)

**Abstract:**

A multiscale finite element method is proposed for addressing a singularly perturbed convection-diffusion model. In reference to the a priori estimate of the boundary layer location, we provide a graded recursion for the mesh adaption. On this mesh, the multiscale basis functions are able to capture the microscopic boundary layers effectively. Then with a global reduction, the multiscale functional space may efficiently reflect the macroscopic essence. The error estimate for the adaptive multiscale strategy is presented, and a high-order convergence is proved. Numerical results validate the robustness of this novel multiscale simulation for singular perturbation with small parameters, as a consequence, high accuracy and uniform superconvergence are obtained through computational reductions.

*+*[Abstract](734)

*+*[HTML](275)

*+*[PDF](310.83KB)

**Abstract:**

Let

*+*[Abstract](856)

*+*[HTML](340)

*+*[PDF](19092.29KB)

**Abstract:**

In this paper, we study an adaptive edge finite element method for time-harmonic Maxwell's equations in metamaterials. *A-posteriori* error estimators based on the recovery type and residual type are proposed, respectively. Based on our *a-posteriori* error estimators, the adaptive edge finite element method is designed and applied to simulate the backward wave propagation, electromagnetic splitter, rotator, concentrator and cloak devices. Numerical examples are presented to illustrate the reliability and efficiency of the proposed *a-posteriori* error estimations for the adaptive method.

*+*[Abstract](574)

*+*[HTML](245)

*+*[PDF](443.55KB)

**Abstract:**

This paper develops efficient numerical algorithms for the optimal control problem constrained by Poisson equations with uncertain diffusion coefficients. We consider both unconstrained condition and box-constrained condition for the control. The algorithms are based upon a multi-mode expansion (MME) for the random state, the finite element approximation for the physical space and the alternating direction method of multipliers (ADMM) or two-block ADMM for the discrete optimization systems. The compelling aspect of our method is that, target random constrained control problem can be approximated to one deterministic constrained control problem under a state variable substitution equality. Thus, the computing resource, especially the memory consumption, can be reduced sharply. The convergence rates of the proposed algorithms are discussed in the paper. We also present some numerical examples to show the performance of our algorithms.

*+*[Abstract](604)

*+*[HTML](238)

*+*[PDF](883.29KB)

**Abstract:**

Efficient numerical methods for solving Poisson equation constraint optimal control problems with random coefficient are discussed in this paper. By applying the finite element method and the Monte Carlo approximation, the original optimal control problem is discretized and transformed into an optimization problem. Taking advantage of the separable structures, Algorithm 1 is proposed for solving the problem, where an alternating direction method of multiplier is used. Both computational and storage costs of this algorithm are very high. In order to reduce the computational cost, Algorithm 2 is proposed, where the multi-modes expansion is introduced and applied. Further, for reducing the storage cost, we propose Algorithm 3 based on Algorithm 2. The main idea is that the random term is shifted to the objective functional, which could be computed in advance. Therefore, we only need to solve a deterministic optimization problem, which could reduce all the costs significantly. Moreover, the convergence analyses of the proposed algorithms are established, and numerical simulations are carried out to test the performances of them.

*+*[Abstract](423)

*+*[HTML](209)

*+*[PDF](324.81KB)

**Abstract:**

We confirm a conjectural supercongruence involving Catalan numbers, which is one of the 100 selected open conjectures on congruences of Sun. The proof makes use of hypergeometric series identities and symbolic summation method.

*+*[Abstract](463)

*+*[HTML](208)

*+*[PDF](258.92KB)

**Abstract:**

Using Watson's terminating

*+*[Abstract](527)

*+*[HTML](290)

*+*[PDF](331.11KB)

**Abstract:**

The asymptotic behaviour of an autonomous neural field lattice system with delays is investigated. It is based on the Amari model, but with the Heaviside function in the interaction term replaced by a sigmoidal function. First, the lattice system is reformulated as an infinite dimensional ordinary delay differential equation on weighted sequence state space

*+*[Abstract](681)

*+*[HTML](275)

*+*[PDF](341.92KB)

**Abstract:**

We present here a generalized Girard-Waring identity constructed from recursive sequences. We also present the construction of Binet Girard-Waring identity and classical Girard-Waring identity by using the generalized Girard-Waring identity and divided differences. The application of the generalized Girard-Waring identity to the transformation of recursive sequences of numbers and polynomials is discussed.

*+*[Abstract](635)

*+*[HTML](309)

*+*[PDF](312.55KB)

**Abstract:**

Let

Z.-W. Sun investigated the congruence properties of Apéry numbers and posed some conjectures. For example, Sun conjectured that for any prime

and for any prime

where

*+*[Abstract](493)

*+*[HTML](245)

*+*[PDF](3164.56KB)

**Abstract:**

This article focuses on the development of high-order energy stable schemes for the multi-length-scale incommensurate phase-field crystal model which is able to study the phase behavior of aperiodic structures. These high-order schemes based on the scalar auxiliary variable (SAV) and spectral deferred correction (SDC) approaches are suitable for the *i.e.*, the Allen-Cahn dynamic equation. Concretely, we propose a second-order Crank-Nicolson (CN) scheme of the SAV system, prove the energy dissipation law, and give the error estimate in the almost periodic function sense. Moreover, we use the SDC method to improve the computational accuracy of the SAV/CN scheme. Numerical results demonstrate the advantages of high-order numerical methods in numerical computations and show the influence of length-scales on the formation of ordered structures.

*+*[Abstract](641)

*+*[HTML](379)

*+*[PDF](3941.13KB)

**Abstract:**

In this paper, we introduce the concept of rough semi-uniform spaces as a supercategory of rough pseudometric spaces and approximation spaces. A completion of approximation spaces has been constructed using rough semi-uniform spaces. Applications of rough semi-uniform spaces in the construction of proximities of digital images is also discussed.

*+*[Abstract](485)

*+*[HTML](274)

*+*[PDF](1258.3KB)

**Abstract:**

In this paper, we are concerned with the three-dimensional (3D) geometric body shape generation with several well-selected characteristic values. Since 3D human shapes can be viewed as the support of the electromagnetic sources, we formulate a scheme to regenerate 3D human shapes by inverse scattering theory. With the help of vector spherical harmonics expansion of the magnetic far field pattern, we build on a smart one-to-one correspondence between the geometric body space and the multi-dimensional vector space that consists of all coefficients of the spherical vector wave function expansion of the magnetic far field pattern. Therefore, these coefficients can serve as the shape generator. For a collection of geometric body shapes, we obtain the inputs (characteristic values of the body shapes) and the outputs (the coefficients of the spherical vector wave function expansion of the corresponding magnetic far field patterns). Then, for any unknown body shape with the given characteristic set, we use the multivariate Lagrange interpolation to get the shape generator of this new shape. Finally, we get the reconstruction of this unknown shape by using the multiple-frequency Fourier method. Numerical examples of both whole body shapes and human head shapes verify the effectiveness of the proposed method.

*+*[Abstract](601)

*+*[HTML](231)

*+*[PDF](841.28KB)

**Abstract:**

This paper proposes a near-field shape neural network (NSNN) to determine the shape of a sound-soft cavity based on a single source and several measurements placed on a curve inside the cavity. The NSNN employs the near-field measurements as input, and the output is the shape parameters of the cavity. The self-attention mechanism is employed to obtain the feature information of the near-field data, as well as the correlations among them. The weights and biases of the NSNN are updated through the gradient descent algorithm, which minimizes the error of the reconstructed shape of the cavity. We prove that the loss function sequence related to the weights is a monotonically bounded non-negative sequence, which indicates the convergence of the NSNN. Numerical experiments show that the shape of the cavity can be effectively reconstructed with the NSNN.

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