Higher-order Equations Research Articles

The linearized classical Boussinesq system on the half‐line

AbstractThe linearization of the classical Boussinesq system is solved explicitly in the case of nonzero boundary conditions on the half‐line. The analysis relies on the unified transform method of Fokas and is performed in two different frameworks: (i) by exploiting the recently introduced extension of Fokas's method to systems of equations and (ii) by expressing the linearized classical Boussinesq system as a single, higher order equation, which is then solved via the usual version of the unified transform. The resulting formula provides a novel representation for the solution of the linearized classical Boussinesq system on the half‐line. Moreover, thanks to the uniform convergence at the boundary, the novel formula is shown to satisfy the linearized classical Boussinesq system as well as the prescribed initial and boundary data via a direct calculation.

Blow-up rates for a higher-order semilinear parabolic equation with nonlinear memory term

ABSTRACT In this paper, we establish blow-up rates for a higher-order semilinear parabolic equation with nonlocal in time nonlinearity with no positive assumption on the solution. We also give Liouville-type theorem for higher-order semilinear parabolic equation with infinite memory nonlinear term which plays the main tools to prove our blow-up rate result. Finally, we study the well-posedness of mild solutions.

Initial boundary value problem for a class of higher-order n-dimensional nonlinear pseudo-parabolic equations

In this paper, we study the initial boundary value problem for a class of higher-order n-dimensional nonlinear pseudo-parabolic equations which do not have positive energy and come from the soil mechanics, the heat conduction, and the nonlinear optics. By the mountain pass theorem we first prove the existence of nonzero weak solution to the static problem, which is the important basis of evolution problem, then based on the method of potential well we prove the existence of global weak solution to the evolution problem.

A Combination of the Orthogonal Polynomials with Least –Squares Method for Solving High-Orders Fredholm-Volterra Integro-Differential Equations

This study introduced new technique which is based on a combination of the least-squares technique (LST) with Chebyshev and Legendre polynomials for finding the approximate solutions of higher-order linear Fredholm-Volterra integro-differential equations (FVIDEs) subject to the mixed conditions. Two examples of second and third-order linear FVIDEs are considered to illustrate the proposed method, the numerical results are comprised to demonstrate the validity and applicability of this technique, and comparisons with the exact solution are made. These results have shown that the competence and accuracy of the present technique.

Well posedness results for higher-order neutral stochastic differential equations driven by Poisson jumps and Rosenblatt process

In this article, we investigate the existence, uniqueness and stability of mild solutions for a class of higher-order nonautonomous neutral stochastic differential equations (NSDEs) with infinite delay driven by Poisson jumps and Rosenblatt process in Hilbert space. More precisely, using semigroup theory and successive approximation method, we establish a set of sufficient conditions for obtained the required result. Further, the result is deduced to study the higher-order autonomous system. Finally, examples are provided to demonstrate the obtain results.

Дослiдження розв’язкiв дисперсiйнихрiвнянь старшого порядку зφ-субгауссовими початковими умовами

In this paper, there are studied sample paths properties of stochastic processes representing solutions of higher-order dispersive equations with random initial conditions given by φ-sub-Gaussian harmonizable processes. The main results are the bounds for the rate of growth of such stochastic processes considered over unbounded domains. The class of φ-sub-Gaussian processes with φ(x) = |x|^α/α, 1 < α <= 2, is a natural generalization of Gaussian processes. For such initial conditions the bounds for the distribution of supremum of solutions can be calculated in rather simple form. The bounds for the rate of growth of solution to higher-order partial differential equations with random initial conditions in the case of general φ were obtained in [9], the derivation was based on the sults stated in [1]. Here we use another approach, which allows us, for the particular case φ(x) = |x|^α/α, α є (1, 2], to present the expressions for the bounds in the closed form.

Stability of optical soliton in symmetrical three-quantum-dot system under high-order effects

The influence of high-order effects on the stability of the optical soliton in a semiconductor three-quantum-dot molecular system under the excitation of narrow pulse probe light is analyzed analytically by using the multi-scale method. The results show that optical soliton described by the standard nonlinear Schrödinger equation will have a large attenuation in the propagation process, while the optical soliton described by the high-order nonlinear Schrödinger equation has relatively good stability. In addition, numerical simulations of the interaction between optical solitons show that the amplitudes of the two optical solitons described by the standard nonlinear Schrödinger equation attenuate rapidly after the collisions and radiation of more serious dispersion waves, while the shapes of the two optical solitons described by the high-order nonlinear Schrödinger equation hardly changes after the collision. This is mainly because when the incident probe light pulse is narrow enough, the system must be described by a higher-order equation. The physical reason is that the higher-order effects in the equation, including non-instantaneous effects and third-order dispersion effects, cannot be ignored or treated as perturbations. This kind of stable optical soliton has potential application value for future optical information processing and transmission technology.

Design of Multi-frequency Analog Transmitter of Magnetic Source Based on Selective Harmonic Elimination Technology

Electromagnetic waves of different frequencies penetrate formations at different depths, so the frequency component of the transmitted waveform determines the detection range of electromagnetic exploration. The selective harmonic elimination pulse width modulation technology was originally used for harmonic elimination. In this paper, SHEPWM is used for harmonic enhancement and the control of harmonics. In addition, the current SHEPWM technology mostly uses various heuristic algorithms, which are not ideal for solving higher-order nonlinear equations. This paper uses the third-order convergence arithmetic mean Newton iteration method to solve the nonlinear equations. The number of solving nonlinear equations can reach 41, and the controllable energy ratio can reach 40% to 50%. Finally, the amplitude and phase of each frequency component of the current waveform can be controlled, which provides a reference for the SHEPWM technology to enhance harmonics.

An improved flower pollination algorithm for solving nonlinear system of equations

It is difficult to solve a system of nonlinear equations, especially for higher-order nonlinear equations when we do not have an efficient and reliable algorithm, even though much work has been done in this area. Newton's method and its improved form are widely used at present, but their convergence and performance characteristics can be highly sensitive to the initial guess of the solution, and the methods fail if the initial guess of the solution is inopportune. It is difficult to select a good initial guess for most systems of nonlinear equations. For this reason, it is necessary to find an efficient algorithm for systems of nonlinear equations. Metaheuristic optimisation algorithms have been proposed by many researchers to solve systems of nonlinear equations. The flower pollination algorithm (FPA) is a novel metaheuristic optimisation algorithm with quick convergence, but its population diversity and convergence precision can be limited in some applications. To enhance its exploitation and exploration abilities, in this paper, an elite opposition-based flower pollination algorithm (EFPA) has been applied for solving systems of nonlinear equations. The results show that the proposed algorithm is robust, has high convergence rate and precision, and can give satisfactory solutions of nonlinear equations.

Invariant measure of stochastic higher order KdV equation driven by Poisson processes

The current paper is devoted to stochastic damped higher order KdV equation driven by Poisson process. We establish the well-posedness of stochastic damped higher-order KdV equation, and prove that there exists an unique invariant measure for non-random initial conditions. Some discussion on the general pure jump noise case are also provided. Some numerical simulations of the invariant measure are provided to support the theoretical results.

An improved flower pollination algorithm for solving nonlinear system of equations

It is difficult to solve a system of nonlinear equations, especially for higher-order nonlinear equations when we do not have an efficient and reliable algorithm, even though much work has been done in this area. Newton's method and its improved form are widely used at present, but their convergence and performance characteristics can be highly sensitive to the initial guess of the solution, and the methods fail if the initial guess of the solution is inopportune. It is difficult to select a good initial guess for most systems of nonlinear equations. For this reason, it is necessary to find an efficient algorithm for systems of nonlinear equations. Metaheuristic optimisation algorithms have been proposed by many researchers to solve systems of nonlinear equations. The flower pollination algorithm (FPA) is a novel metaheuristic optimisation algorithm with quick convergence, but its population diversity and convergence precision can be limited in some applications. To enhance its exploitation and exploration abilities, in this paper, an elite opposition-based flower pollination algorithm (EFPA) has been applied for solving systems of nonlinear equations. The results show that the proposed algorithm is robust, has high convergence rate and precision, and can give satisfactory solutions of nonlinear equations.

Strong shock as a stringent test for Onsager-Burnett equations.

A new set of thermodynamically consistent Onsager-Burnett equations [Singh, Jadhav, and Agrawal, Phys. Rev. E 96, 013106 (2017)2470-004510.1103/PhysRevE.96.013106] has recently been derived. In this work, we subject these equations to a severe test case of strong shock (Mach number = 134) for a dilute gas system composed of hard-sphere molecules. The numerical results of OBurnett equations for conserved and nonconserved variables are compared against the molecular dynamics and direct simulation Monte Carlo results available in the literature. With no tweaking of the equations in any way, we establish several fundamental aspects of OBurnett equations which other higher-order continuum theories like Burnett and Grad equations lack. In particular, evidence is put forward for smooth shock structures, the existence of heteroclinic trajectory, and positive entropy generation inside the shock at all Mach numbers. With respect to shock profiles of hydrodynamic variables, it is observed that OBurnett equations significantly improve upon the results of Navier-Stokes equations. Further comparison with regularized 13 (R13) equations at lower Mach numbers shows that OBurnett equations capture the more rarefied upstream part better than the R13 equations. These evidences suggest that the OBurnett equations are accurate and form a reliable set of higher-order transport equations. Further, it should now be possible to describe the complex structure of the shock wave correctly, even at a very large Mach number.

ON THE SOLUTION OF SOME HIGHER-ORDER INTEGRO-DIFFERENTIAL EQUATIONS OF SPECIAL FORM

The article is devoted to the solution of boundary value problems for higher-order linear integro-differential equations of Fredholm type with differential and integral operators encompassing powers of an ideal bijective linear differential operator whose inverse is known explicitly. The conditions for existence and uniqueness of solutions are derived and the solutions are delivered in closed form. The approach is based on the view that an integro-differential operator is a perturbed differential operator. The results obtained are of both theoretical and practical importance. The method is elucidated by solving two illustrative problems.

Oscillation of [formula omitted]th order strongly noncanonical delay differential equations

The oscillatory properties of solutions to the nth order delay differential equations Lny(t)+p(t)y(τ(t))=0,where Ln is a disconjugate strongly noncanonical differential operator, p(t)>0, τ(t)≤t, and limt→∞τ(t)=∞, are studied. The main idea is to show that, unlike Trench original, rather theoretical result, simple closed formulas appearing in a uniquely determined canonical representation of Ln are possible, under one (mild) additional condition. As a consequence, new oscillation theorems for strongly noncanonical higher-order equations are formulated. Moreover, our results generalize even several ones obtained for n=3,4.

A novel two-polynomial criteria for higher-order systems stability boundaries detection and control

There are many methods for identifying the stability of complex dynamic systems. Routh and Hurwitz’s criterion is one of the earliest and commonly used analytical tools analysing the stability of dynamic systems. However, it requires tedious and lengthy derivations of all components of the Routh array to solve the stability problem. Therefore, it is not a simple method to define analytically, stability boundaries for the coefficients of the system characteristic equation.The proposed brand-new criterion is an effective alternaztive technique in identifying stabilityhigher-order linear time-invariant dynamic system that binds the coefficients of the system characteristic polynomial at the stability boundaries by means of an additional single constantk. It defines the necessary and sufficient conditions for the absolute stability of higher-order dynamic systems. It also allows the analysing of the system’s precise marginal stabilityor marginal instability condition when the roots are relocated on imaginary jω-axis of s-plane. The criterion proposed bythe authors, in contrast to Routh criteria, simplifies the identification of maximum and minimum stability limits for any coefficient of the higher-order characteristic equation significantly. The derived in the paperstability boundary formulas for the polynomial coefficients are successfully used for the proportional integral derivative (PID) controller with single or multiple gains selections in closed-loop control system.

Computation of Periodic Solutions to Pendulum Type Systems with a Small Parameter

Periodic solutions of pendulum-type ODE systems are considered. Finding such solutions is a classical problem in mechanics. Numerous methods are available for computing periodic solutions, and these methods have existed as long as the problems themselves. However, they were designed for manual calculation, and attempts to program them in computer algebra systems (CAS) are sometimes ineffective. For computing such solutions, we propose a method intended for CAS. The method is based on the use of high-order variational equations and symbolic differentiation. It is shown on a number of examples that all computations are reduced to operations with polynomials.

Navigating Innovation Success through Projects. Role of CEO Transformational Leadership, Project Management Best Practices, and Project Management Technology Quotient

In an advancing project-based global economy, the ability to leverage innovation, and the adoption of disruptive technologies are critical for product and process innovation success. This study makes the initial attempt to examine the effects of CEO transformational leadership (CTL), project management best practices (PMBP), and project management technology quotient (PMTQ) on multi-dimensional innovation success. Drawing on data of 261 practitioners in the information and communications technology industry in South Korea, the hypothesized relationships were tested using higher-order partial least squares structural equation modeling (PLS-SEM). The study finds supportive evidence on the positive effects of CTL, PMBP, and PMTQ on innovation success. Moreover, PMBP and PMTQ demonstrated a significant moderating influence on the relationship between CTL and innovation success. Besides the development and validation of a new PMTQ scale, the study findings offer novel theoretical predictions, methodological contributions, and implications for practitioners to mitigate the challenges of successful innovations.

Application of Higher-Order Interpolation Perturbation Theory on Postprandial Plasma Glucose Waveform of a Single Lunch Based on GH-Method: Math-Physical Medicine (No. 426)

In this particular research note, the author arbitrarily selects the postprandial plasma glucose (PPG) data and waveforms from his lunch on 3/31/21 as the reference base. He then applies the higher-order, including first-order, second-order, and third-order interpolation perturbation theory to generate three approximated PPG waveforms to compare them against the measured nonlinear PPG waveform. The two major purposes of this study is to check the waveform shape similarity via correlation coefficients and the prediction accuracy via comparison of the averaged PPG values using the “higher order” interpolation perturbation theory. One obvious conclusion can be drawn from this article is that the higher-order perturbation equation provides a better approximate solution. However, in reality, there are very few patients who understand the perturbation theory of quantum mechanics, let alone to be able to conduct the complex mathematical calculations in order to achieve their desired glucose projection.

Application of the First, Second, and Third Order Equations of Interpolation Perturbation Theory from Quantum Mechanics to Predict a Synthesized 3-year Postprandial Plasma Glucose Wave Based on GH-Method: Math-Physical Medicine (No. 460)

In this research note, the author applies the higher-order interpolation perturbation equations from quantum mechanics. They include the first-order, second-order, and third-order, to generate three predicted PPG waveforms. He then utilizes his measured postprandial plasma glucose (PPG) data and a synthesized waveform generated from his 3,422 meals in 3+ years, from 5/5/2018 to 6/1/2021, as the comparison base against the three predicted PPG waveforms. There are two final yardsticks to check this study. The first target is to verify the prediction accuracies of the three perturbed PPG values. The second target is to examine the waveform shape similarity via correlation coefficients between the measured PPG dataset or waveform and the three perturbed PPG dataset or waveform.

Revisiting the Application of Field Dislocation and Disclination Mechanics to Grain Boundaries

We review the mechanical theory of dislocation and disclination density fields and its application to grain boundary modeling. The theory accounts for the incompatibility of the elastic strain and curvature tensors due to the presence of dislocations and disclinations. The free energy density is assumed to be quadratic in elastic strain and curvature and has nonlocal character. The balance of loads in the body is described by higher-order equations using the work-conjugates of the strain and curvature tensors, i.e., the stress and couple-stress tensors. Conservation statements for the translational and rotational discontinuities provide a dynamic framework for dislocation and disclination motion in terms of transport relationships. Plasticity of the body is therefore viewed as being mediated by both dislocation and disclination motion. The driving forces for these motions are identified from the mechanical dissipation, which provides guidelines for the admissible constitutive relations. On this basis, the theory is expressed as a set of partial differential equations where the unknowns are the material displacement and the dislocation and disclination density fields. The theory is applied in cases where rotational defects matter in the structure and deformation of the body, such as grain boundaries in polycrystals and grain boundary-mediated plasticity. Characteristic examples are provided for the grain boundary structure in terms of periodic arrays of disclination dipoles and for grain boundary migration under applied shear.