Fourth Order Research Articles

On a quasilinear parabolic–hyperbolic system arising in MEMS modeling

AbstractA coupled system consisting of a quasilinear parabolic equation and a semilinear hyperbolic equation is considered. The problem arises as a small aspect ratio limit in the modeling of a MEMS device taking into account the gap width of the device and the gas pressure. The system is regarded as a special case of a more general setting for which local well-posedness of strong solutions is shown. The general result applies to different cases including a coupling of the parabolic equation to a semilinear wave equation of either second or fourth order, the latter featuring either clamped or pinned boundary conditions.

ℏ 4 quantum corrections to semiclassical transmission probabilities

The combination of vibrational perturbation theory with the replacement of the harmonic oscillator quantization condition along the reaction coordinate with an imaginary action to be used in the uniform semiclassical approximation for the transmission probability has been shown in recent years to be a practical method for obtaining thermal reaction rates. To date, this theory has been developed systematically only up to second order in perturbation theory. Although it gives the correct leading order term in an ℏ2 expansion, its accuracy at lower temperatures, where tunneling becomes important, is not clear. In this paper, we develop the theory to fourth order in the action. This demands developing the quantum perturbation theory up to sixth order. Remarkably, we find that the fourth order theory gives the correct ℏ4 term in the expansion of the exact thermal rate. The relative magnitude of the fourth order correction as compared to the second order term objectively indicates the accuracy of the second order theory. We also extend the previous modified second order theory to the fourth order case, creating an ℏ2 modified potential for this purpose. The resulting theory is tested on the standard examples—symmetric and asymmetric Eckart potentials and a Gaussian potential. The modified fourth order theory is remarkably accurate for the asymmetric Eckart potential.

Multiple solitons with fission and multi waves interaction solutions of a (3+1)-dimensional combined pKP-BKP integrable equation

Abstract The potential Kadomtsev-Petviashvili (pKP) equation delineates the development of small-amplitude, nonlinear, long waves characterized by a gradual variation in the transverse coordinate. The B-type KP equation outlines the relationships among exponentially localized shapes and was employed as a representation for shallow water waves and plasma physics. In this paper, we consider the combined pKP-BKP integrable equation. We discuss the multiple solitons of a newly proposed (3+1)-dimensional combined pKP-BKP integrable equation. We use the Hirota bilinear (HB) form of the considered equation to deduce fission process in higher order solitons with different orders. Moreover, the breather dynamics and its interaction with other solitons are investigated via HB. The lump solution and its interaction with first order and fourth order kink soliton is studied.

Seasonal shifts in the habitat selection patterns of male American Marten (Martes americana) at a fine spatial scale

Abstract Old-growth forests harbor a large amount of complex structural features that result in a wide array of wildlife habitats. However, intensive forest management is gradually converting old-growth forest into younger, even-aged stands, reducing structural complexity and threatening the persistence of old-growth-dependent species. Maintaining elements of complex stand structure is critical to the conservation of old-growth forest specialists that use different habitat components at different periods of their annual cycle, and it requires a comprehensive understanding of seasonal variation in the habitat needs of these species. However, difficulties in observing free-ranging animals have sometimes limited our ability to assess such variations in habitat requirements, especially for small, elusive species. To address this, we used GPS telemetry collars to describe fine-scale habitat selection patterns of 6 male American Martens (Martes americana) during 2 contrasting periods of the year (snow-free, from mid-April to mid-November; snow-covered, from mid-November to mid-April), an objective formerly hard to achieve using conventional VHF telemetry. We used resource selection functions conducted at the fourth order of selection to compare habitat characteristics found at the sites used by martens (GPS locations, n = 100) to those found on an equal number of available sites (random points, n = 100) within each individual seasonal home range. We conducted vegetation surveys on these 200 sites to describe habitat and built candidate models representing different concurrent hypotheses. Our results showed that proxies of prey availability, predator avoidance, and thermal constraints were the primary factors influencing marten habitat selection during both periods, although their respective importance differed between periods. Martens selected sites with a high density of large-diameter snags (≥30·ha−1), high conifer canopy closure (≥53%), and a dense lateral cover (≥81%) during the snow-free period, but selected sites with a high volume of coarse woody debris (≥64 m3·ha−1) and high conifer canopy closure (≥48%) during the snow-covered period. Our results highlight the importance of contrasting seasonal changes in habitat selection patterns of small carnivores and may help maintain structural attributes in the landscape that are suitable for male American Martens.

Three weak solutions for a class of fourth order p(x)-Kirchhoff type problem with Leray-Lions operators

In this work, we study the multiplicity of a weak solution for a fourth order p(x)-Kirchhoff type problem involving the Leray-Lions type operators with no flux boundary condition. By using variational approach and critical point theory, we determine an open interval of parameters for which our problem admits at least three distinct weak solutions.

The $L^{p}$-boundedness of wave operators for fourth order Schrödinger operators on $\R^{4}$

The $L^{p}$-boundedness of wave operators for fourth order Schrödinger operators on $\R^{4}$

Solution of infinite system of fourth order differential equations in $\ell_{p}$ space.

The main objective of this paper is to investigate the solution of infinite system of fourth order differential equations in $\ell_p$ space by using measures of noncompactness. The result is demonstrated with an example.

Convergence analysis of a second order numerical scheme for the Flory–Huggins-Cahn-Hilliard-Navier–Stokes system

We present an optimal rate convergence analysis for a second order accurate in time, fully discrete finite difference scheme for the Cahn-Hilliard-Navier–Stokes (CHNS) system, combined with logarithmic Flory–Huggins energy potential. The numerical scheme has been recently proposed, and the positivity-preserving property of the logarithmic arguments, as well as the total energy stability, have been theoretically justified. In this paper, we rigorously prove second order convergence of the proposed numerical scheme, in both time and space. Since the CHNS is a coupled system, the standard ℓ∞(0,T;ℓ2)∩ℓ2(0,T;Hh2) error estimate could not be easily derived, due to the lack of regularity to control the numerical error associated with the coupled terms. Instead, the ℓ∞(0,T;Hh1)∩ℓ2(0,T;Hh3) error analysis for the phase variable and the ℓ∞(0,T;ℓ2) analysis for the velocity vector, which shares the same regularity as the energy estimate, is more suitable to pass through the nonlinear analysis for the error terms associated with the coupled physical process. Furthermore, the highly nonlinear and singular nature of the logarithmic error terms makes the convergence analysis even more challenging, since a uniform distance between the numerical solution and the singular limit values of is needed for the associated error estimate. Many highly non-standard estimates, such as a higher order asymptotic expansion of the numerical solution (up to the third order accuracy in time and fourth order in space), combined with a rough error estimate (to establish the maximum norm bound for the phase variable), as well as a refined error estimate, have to be carried out to conclude the desired convergence result. To our knowledge, it will be the first work to establish an optimal rate convergence estimate for the Cahn-Hilliard-Navier–Stokes system with a singular energy potential.

Numerical Analysis of the Blade Coating Process Using Non‐Newtonian Nanofluid with Magnetohydrodynamic (MHD) and Slip Effects

AbstractThe coating process is widely used in various industries to enhance the production quality and efficiency. This study gives a comprehensive analysis of non‐isothermal blade coating of non‐Newtonian nanofluid incorporating magnetic, thermophoresis, and Brownian effects. The mathematical equations derived from mass, momentum, and energy conservation laws are initially streamlined by means of lubrication approximation theory (LAT). Subsequently, these dimensionless equations are solved in dimensionless form numerically using fourth order Runge–Kutta and Newton–Raphson methods. This study includes the effects of the slip parameter, magnetohydrodynamic (MHD) and other material parameters on the coating thickness (), blade load, velocity, temperature, concentration, and pressure profiles through graphs and tables. The velocity of molten polymer increases near the substrate while it decreases near the blade surface as the slip parameter increases. The temperature distribution increases as the Brinkman number rises, with the maximum temperature occurring in the nip region of the flow. The coating thickness and load‐carrying force for both plane and exponential coater increase with higher values of the magnetohydrodynamic (MHD) parameter.

High order compact augmented methods for Stokes equations with different boundary conditions

This paper is devoted to fourth order compact schemes and fast algorithms for solving stationary Stokes equations with different boundary conditions numerically. One of the main ideas is to decouple the Stokes equations into three Poisson equations for the pressure and the velocity via the pressure Poisson equation (PPE). The augmented strategy is utilized to provide numerical boundary conditions for the pressure. Different velocity boundary conditions require different interpolation strategies for the augmented methods. The augmented variable is solved by the GMRES method. A new simple and efficient preconditioning strategy has also been developed to accelerate the convergence of the GMRES iteration. Numerical examples presented in this paper confirmed the designed convergence order and the efficiency of the new methods.

Design and optimization of a two-degrees-of-freedom single-sided vibro-impact nonlinear energy sink for transient vibration suppression of a thin plate

To rapidly suppress the transient vibration response of a thin plate, a two-degrees-of-freedom single-sided vibro-impact nonlinear energy sink (TSSVI NES) is proposed in this paper. Based on the Kirchhoff hypothesis, the coupled equations of motion between the plate and the TSSVI NES are established. The system is modeled by the Galerkin technique and numerically solved using the fourth order Runge-Kutta method. To obtain the optimal parameters of the TSSVI NES, a particle swarm optimization method is adopted. The system response with the TSSVI NES is compared with that of the cubic nonlinear energy sink (CNES) and the single-sided vibro-impact nonlinear energy sink (SSVI NES), showing that the vibration suppression ability of the TSSVI NES is better. The mechanism of rapid response suppression of the TSSVI NES is demonstrated by way of simulations. A parametric study is also conducted to determine the effects of damping, the coefficient of restitution, mass, stiffness, and installation position of TSSVI NES on the efficacy of the vibration control device. The numerical results demonstrate that the performance of the TSSVI NES is robust to the damping, coefficient of restitution, and installation position, but sensitive to mass difference and stiffness difference. Finally, the effect of the TSSVI NES is examined when the plate is excited by seismic loads, demonstrating a good response suppression performance.

Fractal quintic spline solutions for singularly perturbed reaction-diffusion boundary-value problems

In this article, with the help of fractal quintic spline, numerical solutions of singularly perturbed boundary value problem of ordinary differential equations are obtained. To check the accuracy of the proposed method, convergence analysis is derived. The developed method has fourth order convergence. To check the correctness of the proposed method, numerical examples are provided.

Two-dimensional Coupled Asymmetric van der Pol Oscillator

A reliable algorithm based on an adaptation of the standard differential transform method (DTM) is presented, which is the multi-step differential transform method (Ms-DTM) since it may be difficult to directly apply differential transform method (DTM) to obtain the series solutions for the present two- dimensional coupled asymmetric van der Pol oscillator. The solutions of a two dimensional coupled asymmetric van der Pol oscillator were obtained by Ms-DTM. Figurative comparisons between the Ms-DTM and the classical fourth order Runge-Kutta method (RK4) are given. The obtained results reveal that the proposed technique is a promising tool to solve the considered van der Pol oscillator and yield same information on the phase portrait confirming the stability of the system, effectively. It can be said that the considered approach can be easily extended to other nonlinear van der Pol oscillator systems and therefore is widely applicable in engineering and other sciences.

Zirconium phase diagram from ab initio molecular dynamics

The paper presents analysis of precision of a phonon-based free energy calculation technique (Temperature Dependent Effective Potential method) in comparison with the thermodynamic integration method within classical molecular dynamics. It is shown that the use of the second order force constants is insufficient for accurate calculations of free energies at elevated temperatures due to noticeable anharmonic contributions to lattice vibrations. On the other hand, reliable calculations of the crystal phase equilibrium lines requires free energy accuracy better than 1–3 meV per atom. Based upon the obtained results the Temperature Dependent Effective Potential technique is applied to the detailed study of the solid-state Zr phase diagram from ab initio molecular dynamics that may be achieved if force constants up to the fourth order are used. The calculated zirconium phase diagram is in good agreement with experiment. The α−β−ω triple point was determined at P≈6.74 GPa, T≈837 K.

Connecting weakly nonlinear elasticity theories of isotropic hyperelastic materials

Soft materials exhibit significant nonlinear geometric deformations and stress–strain relationships under external forces. This paper explores weakly nonlinear elasticity theories, including Landau’s and Murnaghan’s formulations, advancing understanding beyond linear elasticity. We establish connections between these methods and extend strain-energy functions to the third and fourth orders in power of [Formula: see text], where [Formula: see text] and [Formula: see text], and [Formula: see text] is the perturbation to the deformation gradient tensor [Formula: see text]. Furthermore, we address simplified strain-energy functions applicable to incompressible materials. Through this work, we contribute to a comprehensive understanding of nonlinear elasticity and its relationship to weakly nonlinear elasticity, facilitating the study of moderate deformations in soft material behavior and its practical applications.

New taxa of the superorder Spirillinoida (Foraminifera)

New taxa of the superorder Spirillinoida (Foraminifera)

Stability of Krein-von Neumann self-adjoint operator extension under unbounded perturbations

We have considered a fourth order difference operator defined on the Hilbert space of square summable sequences on N. We investigated the stability of existence of Krein-von Neumann self-adjoint extension of difference operators under bounded and unbounded coefficients. Using asymptotic summation based on discretised Levinson's theorem and appropriate smoothness and decay conditions, we have shown that unlike the case of deficiency indices and discrete spectrum, the existence of positive self-adjoint operator extensions is stable under unbounded perturbations. These results now exhaustively characterise the spectral and structural properties of difference operators with unbounded coefficients.

Liouville-Green formulas for the fourth order matrix differential operators

In this paper, Liouville-Green analytical approximations for the solutions of the fourth order matrix differential equation are constructed. Using the Gronwall’s inequality, we have also determined the error bounds for the Liouville-Green approximations.

Design and Fabrication of 3.5 GHz Band-Pass Film Bulk Acoustic Resonator Filter

With the development of wireless communication, increasing signal processing presents higher requirements for radio frequency (RF) systems. Piezoelectric acoustic filters, as important elements of an RF front-end, have been widely used in 5G-generation systems. In this work, we propose a Sc0.2Al0.8N-based film bulk acoustic wave resonator (FBAR) for use in the design of radio frequency filters for the 5G mid-band spectrum with a passband from 3.4 to 3.6 GHz. With the excellent piezoelectric properties of Sc0.2Al0.8N, FBAR shows a large Keff2 of 13.1%, which can meet the requirement of passband width. Based on the resonant characteristics of Sc0.2Al0.8N FBAR devices, we demonstrate and fabricate different ladder-type FBAR filters with second, third and fourth orders. The test results show that the out-of-band rejection improves and the insertion loss decreases slightly as the filter order increases, although the frequency of the passband is lower than the predicted ones due to fabrication deviation. The passband from 3.27 to 3.47 GHz is achieved with a 200 MHz bandwidth and insertion loss lower than 2 dB. This work provides a potential approach using ScAlN-based FBAR technology to meet the band-pass filter requirements of 5G mid-band frequencies.

Extraction of Solitons in Optical Fibers for the (2+1)-Dimensional Perturbed Nonlinear Schrödinger Equation Via the Improved Modified Extended Tanh Function Technique

The presented research investigates the (2+1)-dimensional perturbed nonlinear Schrödinger model. This model takes into account various effects such as fourth order dispersion, intermodal dispersion, nonlinear dispersion, group velocity dispersion, Kerr nonlinearity and self steepening effects. This model simulates the estimation of optical solitons and a variety of broadcasting networks’ transmission in nonlinear fiber optics. The proposed model is studied using the improved modified extended tanh function technique. For the proposed model, several solitons and other solutions are generated. These solutions including {bright, dark and singular} solitons, singular periodic and Jacobi elliptic solutions. By introducing the two- and three-dimensional graphs, the physical behavior of the retrieved solutions is displayed.