Higher-order Equations Research Articles

Second-order PLS structural equation modeling in scientific research

This study looked at how structural equation modeling using partial least squares of higher order has been used in various fields of science. We used the integrative type of systematic review method for this. We used the Coordination for the Improvement of Higher Education Personnel database. We searched this database for articles with eight different descriptors published between 2010 and 2020. In total, we used 173 articles as the basis for our analysis in our study. We used coding and graph analysis through social network analysis to analyze the data. The main findings show that social science is the most commonly used area for this method, and that most studies do not detail how they conducted the higher-order structural equation through partial least squares modeling. The study also shows which metrics are commonly used and which could be used more effectively for greater reliability.

Dynamical Structures of Multi-Solitons and Interaction of Solitons to the Higher-Order KdV-5 Equation

In this study, we build multi-wave solutions of the KdV-5 model through Hirota’s bilinear method. Taking complex conjugate values of the free parameters, various colliding exact solutions in the form of rogue wave, symmetric bell soliton and rogue waves form; breather waves, the interaction of a bell and rogue wave, and two colliding rogue wave solutions are constructed. To explore the characteristics of the breather waves, localized in any direction, the higher-order KdV-5 model, which describes the promulgation of weakly nonlinear elongated waves in a narrow channel, and ion-acoustic, and acoustic emission in harmonic crystals symmetrically is analyzed. With the appropriate parameters that affect and manage phase shifts, transmission routes, as well as energies of waves, a mixed solution relating to hyperbolic and sinusoidal expression are derived and illustrated by figures. All the single and multi-soliton appeared symmetric about an axis of the wave propagation. The analyzed outcomes are functional in achieving an understanding of the nonlinear situations in the mentioned fields.

General theory of the higher-order linear quaternion q-difference equations through the quaternion determinant algorithm and the characteristic polynomial

The general theory of quaternion q-difference equations is completely different from traditional q-difference equations in the complex space for the special algebraic structure of the quaternion space, such as the non-commutativity of multiplication among elements. In this paper, some properties of basic quaternion q-discrete functions, such as the quaternion q-exponential function, are obtained. The existence, uniqueness, and extension theorems of the solution for higher-order linear quaternion q-difference equations (QQDCEs) are established through constructing quaternion q-stepwise approximation sequences, and the equivalent variable transform between higher-order linear QQDCEs and quaternion linear q-difference equations is given. Moreover, some basic results, such as the Wronskian formula, Liouville formula, and general solution structure theorems of higher-order linear QQDCEs with constant and variable coefficients, are established by applying the quaternion characteristic polynomial and the quaternion determinant algorithm. In addition, some particular and general solution formulas of non-homogeneous QQDCEs are obtained under the non-commutative condition of quaternion elements. Finally, several examples are provided to illustrate the feasibility of our obtained results.

Mixed Problem for Higher-Order Equations with Fractional Derivative and Degeneration in Both Variables

Mixed Problem for Higher-Order Equations with Fractional Derivative and Degeneration in Both Variables

Mesh Optimization for the Acoustic Parabolic Equation

This work is devoted to increasing the computational efficiency of numerical methods for the one-way Helmholtz Equation (higher-order parabolic equation) in a heterogeneous underwater environment. The finite-difference rational Padé approximation of the propagation operator is considered, whose artificial computational parameters are the grid cell sizes and reference sound speed. The relationship between the parameters of the propagation medium and the artificial computational parameters is established. An optimized method for automatic determination of the artificial computational parameters is proposed. The optimization method makes it possible to account for any propagation angle and arbitrary variations in refractive index. The numerical simulation results confirm the adequacy and efficiency of the proposed approach. Automating the selection process of the computational parameters makes it possible to eliminate human errors and avoid excessive consumption of computational resources.

Black and gray box learning of amplitude equations: Application to phase field systems.

We present a data-driven approach to learning surrogate models for amplitude equationsand illustrate its application to interfacial dynamics of phase field systems. In particular, we demonstrate learning effective partial differential equationsdescribing the evolution of phase field interfaces from full phase field data. We illustrate this on a model phase field system, where analytical approximate equationsfor the dynamics of the phase field interface (a higher-order eikonal equationand its approximation, the Kardar-Parisi-Zhang equation) are known. For this system, we discuss data-driven approaches for the identification of equationsthat accurately describe the front interface dynamics. When the analytical approximate models mentioned above become inaccurate, as we move beyond the region of validity of the underlying assumptions, the data-driven equationsoutperform them. In these regimes, going beyond black box identification, we explore different approaches to learning data-driven corrections to the analytically approximate models, leading to effective gray box partial differential equations.

Solving System of Higher-Order Linear Differential Equations On The Level of Operators

In this paper, I present a method for solving the system of higher-order linear differential equations (HLDEs) with inhomogeneous initial conditions on the level of operators. Using proposed method, we compute the matrix Green’s operator as well as the vector Green’s function of a fully-inhomogeneous initial value problems (IVPs). Examples are discussed to demonstrate the proposed method.

The modulation instability of shallow wake flows based on the higher-order generalized cubic-quintic complex Ginzburg–Landau equation

In the context of the parallel flow hypothesis, we derive a higher-order generalized cubic-quintic complex Ginzburg–Landau (GCQ-CGL) equation to describe the amplitude evolution of shallow wake flow from the dimensionless shallow water equations by using multi-scale analysis, perturbation expansion, and weak nonlinear theory. The evolution model includes not only the slowly changing envelope approximation but also the influence of higher-order dissipation, dispersion, and cubic and quintic nonlinear effects. We give the analytical solution of the higher-order GCQ-CGL equation based on the ansatz and coordinate transformation methods, and we discuss the influence of the higher-order dissipation coefficient on the amplitude and frequency of the wake flow by means of three-dimensional diagrams, contour maps, and plane graphs. The subsequent linear stability analysis gives a theoretical basis for the modulation instability (MI) of plane waves, and the linear theory predicts the instability of any amplitude of the main waves. Finally, we focus on the MI of shallow wake flows. Results show that the MI gain function is internally related to the background wave number, disturbance wave number, background amplitude, disturbance expansion parameter, and dissipation coefficient. The area of the MI decreases as the higher-order dissipation coefficient decreases.

The applications of symbolic computation to exact wave solutions of two HSI-like equations in (2+1)-dimensional

It is renowned that Hirota–Satsuma–Ito (HSI) equation is widely used to study wave dynamics of shallow water. In this work, two new HSI-like equations are investigated which could be extended to diversify problems in natural phenomena and give admirable contributions by applying the generalized exponential rational function method (GERFM). With the aid of symbolic calculations, various constraints on the free parameters are given, while classes of wave solutions are explicitly constructed from the coefficients of the combined non-linear and dissipative terms. After specifying values for free parameters, singular, periodic singular and anti-kink waves are demonstrated in 3D figures to exhibit different kinds of wave propagations. The fact that parameters directly influence the wave amplitude and speed of traveling waves is illustrated. The derived results are innovative and have important applications in the current field of mathematical physics research. Eventually, we show that generalized exponential rational function method is effective and straightforward to solve higher-order and high-dimensional non-linear evolution equations.

Long time dynamics of a single-particle extended quantum walk on a one-dimensional lattice with complex hoppings: a generalized hydrodynamic description

We study a single-particle continuous-time quantum walk on a one-dimensional spatial lattice with complex nearest neighbour (NN) and next-nearest neighbour (NNN) hopping amplitudes. We show that the model allows for controlled information transfer and directional biasing without a biased initial state. Specifically, we show that for an unbiased initial state, complex couplings lead to chiral propagation and a causal cone structure asymmetric about the origin. We provide a hydrodynamic description for the dynamics in the large space–time limit. We obtain a global ‘quasi-stationary state’ which can be described in terms of the local quasi-particle densities satisfying Euler type of hydrodynamic equations and characterized by an infinite set of conservation laws. Further, we show that higher-order hydrodynamic equations can be used to describe the anomalous sub-diffusive scaling near the extremal fronts. The long-time behaviour for any complex NNN hopping with a nonzero real component is similar to that of purely real hopping; at a critical coupling strength, there is a Lifshitz transition where the topology of the causal structure changes from a regime with one causal cone to a regime with two nested (asymmetric) causal cones. For purely imaginary NNN hopping, there is a transition from one causal cone to a regime with two partially overlapping cones which can be attributed to the existence of degenerate maximal fronts. Both the nature of the Lifshitz transition and the scaling behaviour at the critical coupling are different in the two cases. We also discuss possible experimental realizations of such a model.

Existence and uh-stability of integral boundary problem for a class of nonlinear higher-order Hadamard fractional Langevin equation via Mittag-Leffler functions

The Langevin equation is a very important mathematical model in describing the random motion of particles. The fractional Langevin equation is a powerful tool in complex viscoelasticity. Therefore, this paper focuses on a class of nonlinear higher-order Hadamard fractional Langevin equation with integral boundary value conditions. Firstly, we employ successive approximation and Mittag-Leffler function to transform the differential equation into an equivalent integral equation. Then the existence and uniqueness of the solution are obtained by using the fixed point theory. Meanwhile, the Ulam-Hyers (UH) stability is proved by inequality technique and direct analysis.

Elliptic and multiple-valued solutions of some higher order ordinary differential equations

<abstract><p>In the present paper, by the complex method, the meromorphic solutions of the higher order ordinary differential equation $ w^{(5)}+aw^{''}+bw^2-cw+d = 0 $ are investigated, where $ a, b, c, d $ are constant complex numbers, and $ b \neq0 $. Furthermore, by Theorem 1.1, we built elliptic and multiple-valued solutions for the higher order ordinary differential equations $ u^{(6)}-u^{(5)}+u'^2-2u'u+u^2+2u'-2u+1 = 0 $ and $ u^{(6)}-u^{(5)}+au^{'''}-au''+bu'^2-2bu'u+bu^2-cu'+cu+d = 0 $. At the end, we give some new meromorphic solutions for two higher-order KdV-like equations.</p></abstract>

Qualitative behavior of a higher-order fuzzy difference equation

<abstract><p>In this paper, we investigate the qualitative behavior of the fuzzy difference equation</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} z_{n+1} = \frac{Az_{n-s}}{B+C\prod\limits_{i = 0}^{s}z_{n-i}} \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>where $ n\in \mathbb{N}_{0} = \; \mathbb{N} \cup \left\{ 0\right\}, \; (z_{n}) $ is a sequence of positive fuzzy numbers, $ A, B, C $ and the initial conditions $ z_{-j}, \; j = 0, 1, ..., s $ are positive fuzzy numbers and $ s $ is a positive integer. Moreover, two examples are given to verify the effectiveness of the results obtained.</p></abstract>

Blow-up of weak solutions for a higher-order heat equation with logarithmic nonlinearity

Blow-up of weak solutions for a higher-order heat equation with logarithmic nonlinearity

Connection of Balancing numbers with solution of a system of two higher-order difference equations

Connection of Balancing numbers with solution of a system of two higher-order difference equations

Translational-Invariant Ultrahigh-Resolution Spaceborne Bistatic SAR Focusing and Image Coordinate Analysis Using Bistatic Polynomial Model

We have proposed a novel process flow for translational-invariant (TI) and ultrahigh-resolution (UHR) bistatic synthetic aperture radar (BSAR) operations, wherein a UHR BSAR image is obtained via coherent integration of sub-aperture images, and its two-dimensional (2D) coordinates are then interpreted. The main contributions of the proposed method are as follows. First, to address additional aliasing of the TI BSAR spectrum after sub-aperture division, linear range walk compensation and its restoration are subsequently performed. This efficiently extends the duration of the Doppler domain after sub-aperture division, accommodating the skewed shape of the TI BSAR spectrum. Second, spatial-variant focusing is achieved via the bistatic polynomial omega-K algorithm (BP-OKA), for which a key consideration is highlighted to perform BP-OKA in the sub-image integration process. In its absence, the same scatterers appeared at different positions in each sub-aperture image, causing failure of coherent accumulation. Finally, to retrieve the range–azimuth coordinates of targets within an illuminated scene, we established a high-order equation using bistatic polynomials, and the 2D image coordinates of targets were determined by solving the equation. This new concept is fully compatible with the proposed BP-OKA and can be used in various spaceborne TI BSAR scenarios. All these contributions were verified using numerical simulations. Accordingly, the proposed processing flow can provide focused SAR images with accurate image coordinate interpretations for TI and UHR BSAR operations.

Решение уравнений конвекции-диффузии локальным разрывным методом Галеркина

In the work, the discontinuous Galerkin method (LDG) is applied to solving linear and nonlinear convection-diffusion problems. The idea of this method is to transform high-order equations into a system of first-order equations, and then apply the classical discontinuous Galerkin method (DG) to this system. An orthogonal system of Legendre polynomials is chosen as the system of basis functions. The cases of both continuous and discontinuous solutions of the problem are considered. The dependence of the accuracy and stability of the method on the step of the spatial grid and the number of basis functions is investigated. The application of parallel computing to the algorithm is investigated. In the future, it is proposed to apply the LRG method for solving the problems of modeling gas mixtures flows.

DETERMINATION, ANALYSIS AND SYNTHESIS OF THE ROOTS OF VALID ALGEBRAIC EQUATIONS USING LOGARITHMIC AMPLITUDE-PHASE FREQUENCY CHARACTERISTICS

The article considers the method of approximate determination, analysis and synthesis of the roots of real algebraic equations of high order. The solution of such problems is relevant in the case of designing information-measuring and control systems, studying the dynamics of movement of various mechanisms (industrial robots, quadrocopters, etc.), determining the trajectories of aircraft, etc. The analytical solution of such problems is limited to equations of the third (sometimes fourth) degree, in other cases it is necessary to use either special sequential algorithms or packages of applied computer programs such as "Wolfram.Matematica", which allow only to find the roots of the equations, but not to synthesize them. The proposed method is based on the application for the decomposition of the studied polyomial (corresponding to the equation) into the simplest multipliers corresponding to aperiodic and/or oscillatory links, asymptotic logarithmic amplitude and phase-frequency characteristics. The form and values of the roots of the equation are proposed to be judged by the slopes at the fracture points of the logarithmic amplitude and phase-frequency characteristics of the polyparticle under study. The construction of logarithmic amplitude and phase-frequency characteristics is carried out by discarding the "small" terms of the polygamy at separate frequency intervals. A feature of the method is the possibility of its use both in conjunction with the computer and without it. Manual use of the method assumes that the user has a calculator and a ruler. The method allows to determine not only the roots of real algebraic equations (both real and complex), but also to establish a visual relationship between the coefficients for the terms of the equations with the type and values of the roots and purposefully change the necessary coefficients to change the parameters and type of roots. The possibilities of the method are not limited to solving real algebraic equations with positive coefficients and integer powers, it shows quite satisfactory results for equations with mixed coefficients and fractional powers. The method is quite simple, clear, has a small error in the case of far spaced roots, but in the case of closely spaced roots, its error increases, although it remains quite acceptable. The article presents the substantiation of the method, shows numerous examples of its capabilities, compares the results obtained with the results obtained with the help of the package of applied computer programs "Wolfram.Matematica".

Solitons Solution of Riemann Wave Equation via Modified Exp Function Method

In the areas of tidal and tsunami waves in oceans, rivers, ion and magneto-sound waves in plasmas, electromagnetic waves in transmission lines, homogeneous and stationary media, etc., the Riemann wave equations are attractive nonlinear equations. The modified exp(−Φ(η))-function method is used in this article to show how well it can be applied to extract travelling and solitary wave solutions from higher-order nonlinear evolution equations (NLEEs) using the equations mentioned above. Trigonometric, hyperbolic, and exponential functions solitary wave solutions can be extracted using the above-mentioned technique. By changing specific values of the embedded parameters, we can obtain bell-form soliton, consolidated bell-shape soliton, compacton, singular kink soliton, flat kink shape soliton, smooth singular soliton, and other sorts of soliton solutions. The solutions are graphically illustrated in 3D and 2D for the accuracy of the outcome by using the Wolfram Mathematica 10. The verification of numerical solvers on the stability analysis of the solution is substantially aided by the analytic solutions.

The Whole Is More than Its Parts: A Multidimensional Construct of Values in Consumer Information Search Behavior on SNS

The notion of consumer values has been a key factor in understanding consumer behavior and has attracted the attention of various scholars in e-commerce. The aims of this study are to conceptualize a multidimensional construct of consumer information value (CIV) on social network sites (SNS), to examine the construct empirically, and to investigate the nature of its relationships with consumer information search behavior, i.e., the use of information channels on SNS. Quantitative research was conducted using a representative sample of 612 Facebook users. The use of a higher-order construct and structural equation modeling procedures reveals that the following five value constructs—Economic, Psychological, Social, Functional and Hedonic values—are significant facets of the CIV. It suggests that the CIV model is a powerful tool that directly explains consumer information search behavior and has better explanatory power than the sum of its parts.