Low-order Equations Research Articles

Rational Solutions to the Fourth Equation of the Nonlinear Schrödinger Hierarchy

This study concerns the research of rational solutions to the hierarchy of the nonlinear Schrödinger equation. In particular, we are interested in the equation of order 4. Rational solutions to the fourth equation of the NLS hierarchy are constructed and explicit expressions of these solutions are given for the first order. These solutions depend on multiple real parameters. We study the associated patterns of these solutions in the (x,t) plane according to the different values of their parameters. This work allows us to highlight the phenomenon of rogue waves, such as those seen in the case of lower-order equations such as the nonlinear Schrödinger equation, the mKdV equation, or the Hirota equation.

An extended AKNS eigenvalue problem and its affiliated integrable Hamiltonian hierarchies

This paper aims to study an extended 4 × 4 AKNS eigenvalue problem and construct its affiliated integrable hierarchies of bi-Hamiltonian models over the real field. The Lax pair framework serves as the fundamental tool, ensuring integrability through bi-Hamiltonian structures with hereditary recursion operators. We compute illustrative examples of lower-order equations to demonstrate the affiliated integrable hierarchies.

Investigation of the Oscillatory Properties of Fourth-Order Delay Differential Equations Using a Comparison Approach with First- and Second-Order Equations

This paper investigates the oscillatory behavior of solutions to fourth-order functional differential equations (FDEs) with multiple delays and a middle term. By employing a different comparison method approach with lower-order equations, the study introduces enhanced oscillation criteria. A key strength of the proposed method is its ability to reduce the complexity of the fourth-order equation by converting it into first- and second-order forms, allowing for the application of well-established oscillation theories. This approach not only extends existing criteria to higher-order FDEs but also offers more efficient and broadly applicable results. Detailed comparisons with previous research confirm the method’s effectiveness and broader relevance while demonstrating the feasibility and significance of our results as an expansion and improvement of previous results.

Research on Model Reduction of AUV Underwater Support Platform Based on Digital Twin

Digital twin technology, as a data-driven and model-driven innovation means, plays a crucial role in the process of digital transformation and intelligent upgrading of the marine industry, helping the industry to move towards a new stage of more intelligent and efficient development. In order to solve the defects of the Autonomous Underwater Vehicle (AUV) underwater support platform structure deformation field, digital twin technology and model reduction technology are applied to an AUV underwater support platform, and a five-dimensional digital twin model of the AUV underwater support platform is studied, including five dimensions: physical world, digital world, twin data center, service application, and data connection. The digital twin of the subsea support platform is established by using the digital twin modeling technology. The POD method is used to calculate the deformation field matrix of the support structure of the subsea support platform under the 0–5 sea state, and the corresponding eigenvalues and eigenvectors are obtained. By intercepting the eigenvectors corresponding to the eigenvalues of the high energy proportion, the low-order equation is constructed, and the reduced-order model under each sea state can be quickly solved. The experimental results show that the model reduction technology can greatly shorten the model solving time, and the calculated results are highly consistent with the simulation results of the finite element full-order model, which can realize the rapid analysis of the deformation response of the subsea support platform structure, and provide a theoretical basis and technical support for the subsequent simulation, state evaluation, visual monitoring, and predictive maintenance.

A “Consistent” Quasidiffusion Method for Solving Particle Transport Problems

The quasidiffusion (QD) method is an established and efficient iterative technique for solving particle transport problems. Each QD iteration consists of a high-order SN sweep, followed by a low-order QD calculation. QD has two defining characteristics: (1) its iterations converge rapidly for any spatial grid and (2) the converged scalar fluxes from the high-order SN sweep and the low-order QD calculation differ, by spatial truncation errors, from each other and from the scalar flux solution of the SN equations. In this paper, we show that by including a transport consistency factor in the low-order equation, the converged high-order and low-order scalar fluxes become equal to each other and to the converged SN scalar flux. However, the inclusion of the transport consistency factor has a negative impact on the convergence rate. We present numerical results that demonstrate the effect of the transport consistency factor on stability.

One-sweep moment-based semi-implicit-explicit integration for gray thermal radiation transport

Thermal radiation transport (TRT) is a time dependent, high dimensional partial integro-differential equation. In practical applications such as inertial confinement fusion, TRT is coupled to other physics such as hydrodynamics, plasmas, etc., and the timescales one is interested in capturing are often much slower than the radiation timescale. As a result, TRT is treated implicitly, and due to its stiffness and high dimensionality, is often a dominant computational cost in multiphysics simulations. Here we develop a new approach for implicit-explicit (IMEX) integration of gray TRT in the deterministic SN setting, which requires only one sweep per stage, with the simplest first-order method requiring only one sweep per time step. The partitioning of equations is done via a moment-based high-order low-order formulation of TRT, where the streaming operator and first two moments are used to capture the asymptotic stiff regimes of the streaming limit and diffusion limit. Absorption-reemission is treated explicitly, and although stiff, is sufficiently damped by the implicit solve that we achieve stable accurate time integration without incorporating the coupling of the high order and low order equations implicitly. Due to nonlinear coupling of the high-order and low-order equations through temperature-dependent opacities, to facilitate IMEX partitioning and higher-order methods, we use a semi-implicit integration approach amenable to nonlinear partitions. Results are demonstrated on thick Marshak and crooked pipe benchmark problems, demonstrating orders of magnitude improvement in accuracy and wallclock compared with the standard first-order implicit integration typically used.

Leveraging a Small Dataset to Predict Nonlinear Global Loads in Irregular Waves

In this work, a hybrid machine learning method, which uses ML strategies to model high-order force components within a low-order equation of motion, is considered in the context of the global wave-induced loads of a ship in irregular waves. It is shown that the method can make predictions in a range of wave conditions even when the training data set only includes a single seaway. The proposed method offers a data-leveraging technique which may be useful in the design space, where a small data set derived from a high-fidelity source can be leveraged to make similar fidelity predictions in a larger number of wave conditions.

A reduced-order model for nonlinear radiative transfer problems based on moment equations and POD-Petrov-Galerkin projection of the normalized Boltzmann transport equation

A data-driven projection-based reduced-order model (ROM) for nonlinear thermal radiative transfer (TRT) problems is presented. The TRT ROM is formulated by (i) a hierarchy of low-order quasidiffusion (aka variable Eddington factor) equations for moments of the radiation intensity and (ii) the normalized Boltzmann transport equation (BTE). The multilevel system of moment equations is derived by projection of the BTE onto a sequence of subspaces which represent elements of the phase space of the problem. Exact closure for the moment equations is provided by the Eddington tensor. A Petrov-Galerkin (PG) projection of the normalized BTE is formulated using a proper orthogonal decomposition (POD) basis representing the normalized radiation intensity over the whole phase space and time. The Eddington tensor linearly depends on the solution of the normalized BTE. By linear superposition of the POD basis functions, a low-rank expansion of the Eddington tensor is constructed with coefficients defined by the PG projected normalized BTE. The material energy balance (MEB) equation is coupled with the effective gray low-order equations which exist on the same dimensional scale as the MEB equation. The resulting TRT ROM is structure and asymptotic preserving. A detailed analysis of the ROM is performed on the classical Fleck-Cummings (F-C) TRT multigroup test problem in 2D geometry. Numerical results are presented to demonstrate the ROM's effectiveness in the simulation of radiation wave phenomena. The ROM is shown to produce solutions with sufficiently high accuracy while using low-rank approximation of the normalized BTE solution. Essential physical characteristics of supersonic radiation wave are preserved in the ROM solutions.

Positive solutions for the Riemann–Liouville-type fractional differential equation system with infinite-point boundary conditions on infinite intervals

In this paper, we study the existence and uniqueness of positive solutions for a class of a fractional differential equation system of Riemann–Liouville type on infinite intervals with infinite-point boundary conditions. First, the higher-order equation is reduced to the lower-order equation, and then it is transformed into the equivalent integral equation. Secondly, we obtain the existence and uniqueness of positive solutions for each fixed parameter λ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\lambda >0$\\end{document} by using the mixed monotone operators fixed-point theorem. The results obtained in this paper show that the unique positive solution has good properties: continuity, monotonicity, iteration, and approximation. Finally, an example is given to demonstrate the application of our main results.

A novel accelerated convergence method for solving adjoint equations based on modal reduction

The efficiency of adjoint-based aerodynamic shape optimization depends critically on the solution efficiency of adjoint equations. In this letter, we employ the Proper Orthogonal Decomposition (POD) method to analyze the adjoint field samples and project them from the physical space into a low-order modal space. Subsequently, the full-order adjoint equations are reduced to low-order equations using the POD modes. Thus, we can efficiently predict the initial values for pseudo-time marching, thereby accelerating the solution of adjoint equations. Results indicate that the high-order POD modes are crucial for constructing the low-dimensional system. Moreover, this method can be seamlessly integrated with our previously established Dynamic Mode Decomposition (DMD) acceleration method to form a POD+DMD acceleration approach. Application of this approach to the flow past a National Advisory Committee for Aeronautics 0012 airfoil demonstrates a noteworthy 80.9% reduction in iteration numbers when solving the adjoint equations. Even for the airfoil located on the upper boundary of sampling space, the number of iterations is still reduced by 72.6%. Therefore, we believe that the proposed method holds significant promise for improving the efficiency of adjoint-based aerodynamic shape optimization in future research.

ON THE CORRECTNESS OF ONE CLASS OF FOURTH-ORDERSINGULAR DIFFERENTIAL EQUATIONS

The differential equation of fourth-order with variable smooth coefficients, given on the real axis, is investigated. The coefficients of the equation can be unbounded, and the sign of the potential (the lowest coefficient) is not defined. The case is considered that the intermediate term containing the second derivative of the desired function, as an operator, does not obey the operator of the equation. It is known that fourth-order differential equations with variable coefficients are used in various problems of mathematical physics. The simplest biharmonic equation with minor terms is important for its applications in the theory of elasticity, the mechanics of elastic plates and the slow flow of viscous liquids. However, it is a very special case of a general equation with variable coefficients. Some problems of stochastic analysis, oscillation theory, biology and mathematical finance lead to the equation we are studying. In addition, fourth–order singular differential equations are often used as regularizers in the study of lower-order equations, for example, reaction-diffusion equations. In the paper, sufficient solvability conditions of the equation and the maximal regularity inequality of a strong generalized solution are obtained. The restrictions are formulated in terms of the coefficients themselves and rather weak. They do not contain conditions for any derivatives of coefficients and represent relations between the orders of growth of coefficients of different orders at infinity. The weight estimation of the norms of a solution established by us allows us to additionally apply the methods of the theory of functional spaces to the study of further properties of the solution, for example, the approximation of its elements by finite-dimensional spaces.

On the Monotonic and Asymptotic Properties of Positive Solutions to Third-Order Neutral Differential Equations and Their Effect on Oscillation Criteria

The monotonic properties of positive solutions to functional differential equations of the third order are examined in this paper. It is generally known that by optimizing the relationships between a solution and its corresponding function, as well as its derivatives, one can improve the oscillation criterion for neutral differential equations. Based on this, we obtain new relationships and inequalities and test their effect on the oscillation parameters of the studied equation. To obtain the oscillation parameters, we used Riccati techniques and comparison with lower-order equations. Finally, the progress achieved in oscillation theory for third-order equations was measured by comparing our results with previous relevant results.

Применение к радиационной акустике метода Уитхэма для анализа волновых явлений в намагниченной плазме

The exact governing equation of radiative acoustics in radiating grey gas is derived in this work, taking into account the influence of the transverse magnetic field. The radiative MHD is described by three equations of hydrodynamics and two equations of radiation momentum with extensive use of the formalism of radiative thermodynamics. In order to describe the propagation of linear radiative magneto-acoustic perturbation waves with scattering and attenuation, radiation-thermal dissipation conditions, radiation drag force, and magnetic force and Joule heat are introduced into these equations. The Eddington approximation is used, which allows us to study the modes of radiative magneto-hydrodynamic waves in two asymptotic cases - optically thin and thick gas. The exact governing equation derived in this paper allows, using the heuristic Whitham method, to obtain a set of approximate governing equations of lower order, each of which is part of a reliable approximation to the exact equation in a certain region of the independent time variable. The relatively simple form of such equations allows, without formally solving the full problem, to investigate the physical processes occurring in each radiative magneto-hydrodynamic linear travelling wave.

Convergence Properties of a Linear Diffusion-Acceleration Method for k-Eigenvalue Transport Problems

Most methods that use low-order operators to accelerate the iterative solution of transport eigenvalue problems employ nonlinear functionals of the transport solution (such as Eddington tensors) in their low-order equations, which are themselves standard eigenvalue problems. Here, we discuss linear diffusion synthetic acceleration (DSA) for -eigenvalue problems, which belongs to a family of methods that has received less attention than its nonlinear counterparts. We review the history of these linear methods as far as we know it and describe theoretical questions that to our knowledge have remained unanswered. With these methods, a low-order problem is solved after each transport step for an updated eigenvalue and an additive correction to the eigenfunction. These low-order problems are not standard eigenvalue problems, for they contain residuals as fixed sources. The low-order problems admit infinitely many solutions (updated and additive correction to the eigenfunction), and the solution that is obtained depends on the initial guess and iterative method chosen for the low-order problems. Experience has shown that when the low-order problems are solved with a powerlike iteration method and certain initial guesses, they yield solutions that cause rapid convergence to the correct high-order solution. We study the convergence properties of this algorithm applied to two model problems: an infinite homogeneous medium and a one-cell problem. For the infinite homogeneous problem, we present a Fourier analysis of the linear DSA method, which demonstrates that when the low-order problems are solved using a powerlike iteration scheme, the linear DSA scheme provides immediate convergence of the -eigenvalue and rapid convergence of the eigenfunction (much like DSA applied to scattering iterations in fixed-source problems). For the one-cell problems, we find that the linear scheme for -eigenvalue problems performs approximately as well as DSA for fixed-source problems. The latter analysis reveals a quantitative bound on the consistency between low- and high-order operators that is necessary and sufficient for convergence of those problems. With some theoretical foundations for the linear methods now established, we turn to numerical testing. We find, as others have before us using different low-order operators, that the method works well in practice. We provide numerical results from reactor problems in which our linear DSA is approximately as effective as the more widely used nonlinear methods. Our theoretical and numerical results add to the body of evidence that the linear methodology offers a simple path to rapid convergence of -eigenvalue problems, especially for codes that already employ linear low-order operators to converge scattering iterations.

Size-dependent nonlinear bending analysis of nonlocal magneto-electro-elastic laminated nanobeams resting on elastic foundation

Employing Reddy’s third-order shear deformation theory (RTSDT) and nonlocal elasticity theory, a nonlinear bending model of the nonlocal three-layer magneto-electro-elastic (MEE) laminated nanobeam resting on elastic foundation is established by considering von Karman’s geometrically nonlinear equations. Nonlinear higher order partial differential governing equations of MEE laminated nanobeams can be obtained employing Hamilton variational principle. The three-layered MEE laminated nanobeam is considered to have simply supported boundary condition in the present paper. Meanwhile, the electric and magnetic potential distributions in the laminated nanobeam are determined through Maxwell’s magnetic-electro equations and boundary conditions. The governing equations of laminated nanobeams are re-expressed in the dimensionless form by introducing the non-dimensional terms. Employing Galerkin method, the nonlinear higher order partial differential governing equations are simplified into lower order equations. Several cases are explored to indicate the effects of foundation parameters, nonlocal parameter, stacking sequence, external electric voltage and external magnetic potential on bending behaviors of MEE laminated nanobeams.

Numerical Simulation of Flood Routing, Model of Dynamic Equations

Dynamic equations of flow in a rectangular inclined channel are solved numerically for the case where the friction force of the channel wall is neglected by the gravity force. The flow discharge and the cross-section area along the channel are physical quantity that is calculated. In non-dimensional variables, the equations indicate solution in traveling wave occurring for critical flow. Near that type of flow, the perturbation method is applied to get second order equations that are first order partial differential equation with external force from the lower order equations. Numerical solution is obtained by predictor-corrector method, and the effect of these second order equations can be observed to the traveling wave, depending on the type of the flow, su-percritical or subcritical.

Effective action of string theory at order alpha ' in the presence of boundary

Recently, using the assumption that the string theory effective action at the critical dimension is background independent, the classical on-shell effective action of the bosonic string theory at order alpha ' in a spacetime manifold without boundary has been reproduced, up to an overall parameter, by imposing the O(1, 1) symmetry when the background has a circle. In the presence of the boundary, we consider a background which has boundary and a circle such that the unit normal vector of the boundary is independent of the circle. Then the O(1, 1) symmetry can fix the bulk action without using the lowest order equation of motion. Moreover, the above constraints and the constraint from the principle of the least action in the presence of boundary can fix the boundary action, up to five boundary parameters. In the least action principle, we assume that not only the values of the massless fields but also the values of their first derivatives are arbitrary on the boundary. We have also observed that the cosmological reduction of the leading order action in the presence of the Hawking–Gibbons boundary term, produces zero cosmological boundary action. Imposing this as another constraint on the boundary couplings at order alpha ', we find the boundary action up to two parameters. For a specific value for these two parameters, the gravity couplings in the boundary become the Chern–Simons gravity plus another term which has the Laplacian of the extrinsic curvature.

Extremely correlated superconductors

Superconductivity in the t-J model is studied by extending the recently introduced extremely correlated fermi liquid theory. Exact equations for the Greens functions are obtained by generalizing Gor’kov’s equations to include extremely strong local repulsion between electrons of opposite spin. These equations are expanded in a parameter λ representing the fraction of double occupancy, and the lowest order equations are further simplified near Tc, resulting in an approximate integral equation for the superconducting gap. The condition for Tc is studied using a model spectral function embodying a reduced quasiparticle weight Z near half-filling, yielding an approximate analytical formula for Tc. This formula is evaluated using parameters representative of single layer High-Tc systems. In a narrow range of electron densities that is necessarily separated from the Mott–Hubbard insulator at half filling, we find a typical Tc∼102 K.

A high-order/low-order (HOLO) algorithm for preserving conservation in time-dependent low-rank transport calculations

Dynamical low-rank (DLR) approximation methods have previously been developed for time-dependent radiation transport problems. One crucial drawback of DLR is that it does not conserve important quantities of the calculation, which limits the applicability of the method. Here we address this conservation issue by solving a low-order equation with closure terms computed via a high-order solution calculated with DLR. We observe that the high-order solution well approximates the closure term, and the low-order solution can be used to correct the conservation bias in the DLR evolution. We also apply the linear discontinuous Galerkin method for the spatial discretization. We then demonstrate with the numerical results that this so-called high-order/low-order (HOLO) algorithm is conservative without sacrificing computational efficiency and accuracy.

Nonlinear iterative projection methods with multigrid in photon frequency for thermal radiative transfer

This paper presents nonlinear iterative methods for the fundamental thermal radiative transfer (TRT) model defined by the time-dependent multifrequency radiative transfer (RT) equation and the material energy balance (MEB) equation. The iterative methods are based on the nonlinear projection approach and use multiple grids in photon frequency. They are formulated by the high-order RT equation on a given grid in photon frequency and low-order moment equations on a hierarchy of frequency grids. The material temperature is evaluated in the subspace of lowest dimensionality from the MEB equation coupled to the effective grey low-order equations. The algorithms apply various multigrid cycles to visit frequency grids. Numerical results are presented to demonstrate convergence of the multigrid iterative algorithms in TRT problems with large number of photon frequency groups.