Approximate Steady-state Solutions Research Articles

Incorporating different enzyme kinetics in amperometric biosensor for the steady-state conditions: A complete theoretical and numerical approach

The well-defined physical, chemical, and biological interactions that define the behaviour of biosensors are described by nonlinear differential equations. Because these biosensors have so many uses in many domains, it is desirable to represent them mathematically. The enzyme kinetic and the diffusion rate across the enzymatic layer determine the distinct enzymatic processes. The biosensors are modelled by a system of reaction-diffusion equations involving a nonlinear part associated with the reaction rate. The nonlinear equations in this model have been solved analytically (Rajendran-Joy method) and numerically (MATLAB® v2016b software). Approximate steady-state analytical solutions for the substrate and reaction product concentrations and the output current are presented for specific cases of first-order, Michaelis-Menten and ping-pong kinetics. The influence of various parameters on concentrations and currents is discussed. Further, the sensitivity of parameters in current was also analysed. The analytical results are compared with numerically simulated results for various kinetics.

Image analysis for patterns in solutions of reaction–diffusion equations

We consider reaction–diffusion equations in form of a Turing system, i.e., a system of partial differential equations (PDEs) in time and two-dimensional space. The equations model the formation of patterns in animal coats and skins, for example. Stationary solutions represent a final pattern, which depends both on physical parameters and initial values. A spatial finite difference method yields a high-dimensional system of ordinary differential equations (ODEs). Stationary solutions of the ODEs are computed numerically to approximate steady-state solutions of the PDEs. We investigate properties of the patterns using image analysis. The two-dimensional stationary solutions are converted into binary images. This approach allows for the application of methods and algorithms on the binary image. The Euler characteristic yields the number of objects in this case. Properties of each object like area, Feret diameters, circularity, etc. can be computed. Moreover, we examine associated statistical quantities for sample sets of initial values in a Monte-Carlo simulation. Our aim is to analyse the dependence as well as the sensitivity of patterns on the physical parameters. We present results of numerical computations for different selections of those parameters.

AN EFFICIENT ANCIENT CHINESE ALGORITHM TO INVESTIGATE THE DYNAMICS RESPONSE OF A FRACTAL MICROGRAVITY FORCED OSCILLATOR

In this paper, the ancient Chinese algorithm is applied to derive fractal microgravity forced oscillator’s frequency–amplitude response curves using the two-scale fractal transform derivative. Depending on the fractal parameter values, it is shown that the system restoring forces exhibit hardening or softening behavior. Furthermore, the proposed solution approach is simple, efficient, and accurate for obtaining the approximate steady-state solution of a fractal microgravity nonlinear oscillator with a purely nonlinear power-form restoring force.

A modified modeling and dynamical behavior analysis method for fractional-order positive Luo converter

Compared to the integer-order modeling, the fractional-order modeling can achieve higher accuracy for designing and analyzing the DC-DC power converters. However, its applications in pulse width modulation (PWM) converters are limited due to the computational complexities. In this paper, a modified fractional-order modeling methodology for DC-DC converters is proposed, and its effectiveness is verified on the fractional-order positive Luo converters. Instead of using fractional-order calculus, the proposed methodology analyzes the harmonic components of the PWM converters by utilizing the non-linear vector differential equations of the periodically time-variant system. The final solution of the state variables is composed of two parts: the steady-state solution and the transient solution. The approximate steady state solution can be obtained by using the equivalent small parameter (ESP) method and the harmonic balance theory, while the main part of the transient solution can be obtained according to the explicit Grünwald-Letnikov (GL) approximation. In addition, the influence of the fractional orders on the performance of the DC-DC converters, and on the dynamic behaviors of the fractional-orders systems are also discussed in this paper. Compared to the conventional fractional-order numerical models, the proposed model is able to present the time-domain information more precisely, which helps to better reveal and analyze the non-linear behaviors of the DC-DC converters. The effectiveness of the work is demonstrated by the simulation and experimental results of the equivalent circuits built with fractional-order components.

Critical Evolution of Finite Perturbations of a Water Evaporation Surface in Porous Media

It is shown that the approximate steady-state solutions, which satisfy the model dissipative equation that describes the process of water evaporation in the neighborhood of the instability threshold of a phase transition interface, determine localized damped finite-amplitude perturbations when a certain condition is fulfilled. These steady-state solutions can be used for forecasting the scenario of the development of a perturbation with sufficient accuracy if this perturbation has no common points with any steady-state solution. If the initial position of the phase transition front is located between the spectrally stable solution and any of the steady-state solutions, this front damps. If the initial position of the front is located above at least one of the spectrally unstable steady-state solutions, then the solution is catastrophically restructured.

Dynamic response of a piecewise linear single-degree-of-freedom oscillator with fractional-order derivative

In this paper, the dynamic response of a piecewise linear single-degree-of-freedom oscillator with fractional-order derivative is studied. First, a mathematical model of the single-degree-of-freedom system is established, and the approximate steady-state solution associated with the amplitude–frequency equation is obtained based on the averaging method. Then, the amplitude–frequency response equations are used for stability analysis, and the stability condition is founded. To validate the correctness and precision, the approximate solutions determined by the analytical method are compared with the solutions based on the numerical integration method. It is found that the approximate solutions and the numerical solutions are in excellent agreement. Finally, the effects of system parameters, such as fractional-order coefficient, order, clearance, and piecewise stiffness, on the complex dynamical behaviors of the piecewise linear single-degree-of-freedom oscillator are studied. The results show that the system parameters not only influence resonance amplitude and resonance frequency but also affect the size of the unstable region.

Resonance Analysis of Fractional-Order Mathieu Oscillator

The resonant behavior of fractional-order Mathieu oscillator subjected to external harmonic excitation is investigated. Based on the harmonic balance (HB) method, the first-order approximate analytical solutions for primary resonance and parametric-forced joint resonance are obtained, and the higher-order approximate steady-state solution for parametric-forced joint resonance is also obtained, where the unified forms of the fractional-order term with fractional order between 0 and 2 are achieved. The correctness of the approximate analytical results is verified by numerical results. The effects of the fractional order and parametric excitation frequency on the resonance response of the system are analyzed in detail. The results show that the HB method is effective to analyze dynamic response in a fractional-order Mathieu system.

Exact steady states of the periodically forced and damped Duffing oscillator

We present an analytical methodology to compute the exact steady states of the strongly nonlinear Duffing oscillator with arbitrary viscous damping and under the action of a time-periodic excitation. In contrast to current analytical approaches that consider harmonic excitations to produce approximate steady-state solutions, the excitations considered in this work are multi-harmonic and are computed in closed form. We consider the exact fundamental resonance plot of this strongly nonlinear system and present an exact depiction of the dynamic balance between the internal and external forces leading to it in terms of rotating vectors in the complex plane with modulated amplitudes and modulated frequencies of rotation. Moreover, an infinity of exact steady-state solutions seems to be possible, each corresponding to a different time-periodic excitation. Generalization of the presented methodology to systems with other type of nonlinearities or with many degrees of freedom is presumably straightforward.

Higher-order approximate steady-state solutions for strongly nonlinear systems by the improved incremental harmonic balance method

Combining the harmonic balance method with the incremental harmonic balance approach, an improved incremental harmonic balance method is presented to obtain the higher-order approximate steady-state solutions for strongly nonlinear systems, which can simplify the calculation process for high-order nonlinear terms. Taking a strongly nonlinear Duffing oscillator with cubic nonlinearity and a strongly nonlinear Duffing oscillator with quintic nonlinearity as examples, the forced vibrations under harmonic excitation are investigated. Based on the first-order approximate analytical solutions obtained by the harmonic balance method, the higher-order approximate solutions are obtained by the improved incremental harmonic balance method. The correctness of the approximate analytical results is verified by the numerical results. The comparison results show that the approximations obtained by the improved incremental harmonic balance method agree with the numerical solutions well, and the improved method is effective to analyze the dynamical response for strongly nonlinear systems.

Trefftz method in solving Fourier-Kirchhoff equation for two-phase flow boiling in a vertical rectangular minichannel

This paper presents the results of investigations into flow boiling heat transfer in an asymmetrically heated vertical minichannel of 1.7 mm depth. The heated element for FC-72 flowing in the minichannel was an alloy plate 0.45 mm thick, microstructured on one side, in direct contact with the flowing fluid. The computational part of the study contains approximate steady state solutions of the heat transfer problems described by Poisson.s equation and the energy equation for the heated plate and the fluid, respectively. For both equations, the boundary conditions were specified on the basis of experimental data. Temperature of the outer plate surface, measured by infrared thermography, and heat losses to ambient air were included in the calculations. For the energy equation we assumed parabolic profile of fluid velocity and the equality of temperatures and heat fluxes at the interface between the heated surface and the fluid. The void fraction was taken from a single-phase flow model. Two-dimensional temperature distributions were obtained by the Trefftz method and, due to the Robin condition at the interface between them, it was possible to calculate the heat transfer coefficient. Its values were compared to those obtained by other correlations known from literature.

Performance Analysis of Device-to-Device Communications with Dynamic Interference Using Stochastic Petri Nets

In this paper, we study the performance of Device-to-Device (D2D) communications with dynamic interference. In specific, we analyze the performance of frequency reuse among D2D links with dynamic data arrival setting. We first consider the arrival and departure processes of packets in a non-saturated buffer, which result in varying interference on a link based on the change of its backlogged state. The packet-level system behavior is then represented by a coupled processor queuing model, where the service rate varies with time due to both the fast fading and the dynamic interference effects. In order to analyze the queuing model, we formulate it as a Discrete Time Markov Chain (DTMC) and compute its steady-state distribution. Since the state space of the DTMC grows exponentially with the number of D2D links, we use the model decomposition and some iteration techniques in Stochastic Petri Nets (SPNs) to derive its approximate steady state solution, which is used to obtain the approximate performance metrics of the D2D communications in terms of average queue length, mean throughput, average packet delay and packet dropping probability of each link. Simulations are performed to verify the analytical results under different traffic loads and interference conditions.

Application of a line implicit method to fully coupled system of equations for turbulent flow problems

For unstructured finite volume methods, we present a line implicit Runge–Kutta method applied as smoother in an agglomerated multigrid algorithm to significantly improve the reliability and convergence rate to approximate steady-state solutions of the Reynolds-averaged Navier–Stokes equations. To describe turbulence, we consider a one-equation Spalart–Allmaras turbulence model. The line implicit Runge–Kutta method extends a basic explicit Runge–Kutta method by a preconditioner given by an approximate derivative of the residual function. The approximate derivative is only constructed along predetermined lines which resolve anisotropies in the given grid. Therefore, the method is a canonical generalisation of point implicit methods. Numerical examples demonstrate the improvements of the line implicit Runge–Kutta when compared with explicit Runge–Kutta methods accelerated with local time stepping.

Lasing without inversion in a four-level diamond system

We study the lasing without inversion in a four-level diamond configuration in the case of incoherent pumping field within the framework of the bare-state basis. With the strong fields limit, we obtain the approximate steady-state solution, and discuss the dependence of population distribution and system gain on probe detuning and auxiliary field Rabi frequency.

Laser without inversion in a Δ -configuration three-level system with cyclic transition

In the case of incoherent pumping, laser without population inversion in a resonant three-level Δ -configuration system is investigated. By taking the strong coupling field limit, we obtain the approximate steady-state analytical solution of populations and imaginary part of coherences within the dressed-state regime, and discuss the condition of the generating of laser without inversion and the dependences of population distribution and system gain on Rabi frequency of the probe and coherence fields. The results show that the resonant three-level Δ-configuration system is always in the state of no population inversion, and laser without inversion occurs if one of the two fields is strong. When one of the two fields is much stronger than the other one, the gain without inversion is independent of the two Rabi frequencies.

Free vibration analysis of doubly curved shallow shells using the Superposition-Galerkin method

This paper demonstrates the applicability of the Superposition Method for free vibration analysis of doubly curved thin shallow shells of rectangular planform with any possible combination of simply supported and clamped edges. The same building block yields the natural frequencies for 55 combinations of edge conditions. The natural frequency parameters of the shells were obtained using the Superposition-Galerkin Method (SGM) for seven sets of boundary conditions, several different curvature ratios and two aspect ratios. The SGM uses approximate steady state solutions as building blocks but the method proves to be accurate and efficient. It has also been shown that even with approximate building blocks, the monotonic nature of convergence of the natural frequencies with respect to the number of driving coefficients holds, as long as the number of admissible functions in the steady state solution is kept constant. The results for natural frequencies of the seven boundary conditions may be considered as benchmarks.

Application of point implicit Runge–Kutta methods to inviscid and laminar flow problems using AUSM and AUSM+ upwinding

Explicit Runge–Kutta methods preconditioned by a pointwise matrix valued preconditioner can significantly improve the convergence rate to approximate steady state solutions of laminar flows. This has been shown for central discretisation schemes and Roe upwinding. Since the first-order approximation to the inviscid flux assuming constant weighting of the dissipative terms is given by the absolute value of the Roe matrix, the construction of the preconditioner is rather simple compared to other upwind techniques. However, in this article we show that similar improvements in the convergence rates can also be obtained for the AUSM+ scheme. Following the ideas for the central and Roe schemes, the preconditioner is obtained by a first-order approximation to the derivative of the convective flux. Viscous terms are included into the preconditioner considering a thin shear layer approximation. A complete derivation of the derivative terms is shown. In numerical examples, we demonstrate the improved convergence rates when compared with a standard explicit Runge–Kutta method accelerated with local time stepping.

Application of point implicit Runge-Kutta methods to inviscid and laminar flow problems using AUSM and AUSM+ upwinding

Explicit Runge-Kutta methods preconditioned by a pointwise matrix valued preconditioner can significantly improve the convergence rate to approximate steady state solutions of laminar flows. This h...

MAIN RESONANCE ANALYSIS OF A COMPOSITE HELICOPTER DRIVE SHAFT

Main resonance amplitude of a shaft is concerned with the strength calculation. Based on bending motion equation of a tilting fiber composite helicopter drive shaft with two supports obtained under non-inertia movement coordinate system by theorem of the motion of center of mass and the method of Galerkin, an approximate steady-state solution of the main resonance is obtained by method of multiple scales. The maximum amplitude, amplitude jump characteristic and so on are analyzed. Conclusions of that the motion stability of a fiber composite drive shaft is better and the main resonance amplitude is smaller than that of a drive shaft of steel or aluminum are produced. It is presented that the amplitude jump of a shaft is decreased or removed, and the motion stability is obtained by increasing damping. In condition of increasing damping in the place of larger deflection of a shaft, the amplitude of main resonance can be decreased, effect of the damping is fully brought into play and weight of the shaft is optimized.

Application of Jacobian Elliptic Functions to the Analysis of the Steady-State Solution of the Damped Duffing Equation with Driving Force of Elliptic Type

In this work, Jacobian elliptic functions are applied to obtain the approximate steady-state solution of the damped Duffing equation with driving force of elliptic type. We assume that the damping coefficient and the nonlinear term need not be small. Numerical comparison between the approximate solution and the numerical integration indicates that our proposed approximate steady-state solution captures well the amplitude-time vibrational response. We have also found that for a certain range of values of the driving frequency ω f and for a stiff spring, there are one, three, or five possible motions for the elliptic excitation amplitude-frequency response curve.

Beam–beam interaction models under narrow-band random excitation

The aim of this paper is to continue our previous investigations on beam–beam interaction models in particle accelerators by considering these models under narrow-band random excitation. In this investigation the multiple scale and the moment methods are used. The equations of modulation of the amplitude and the phase are derived and perturbation techniques are used to seek approximate steady-state solutions. Two special cases are considered to illustrate this study and excellent agreement is found between analytical and numerical results. Other cases can be similarly studied. The effects of noise and detuning parameters are examined numerically and theoretically and we find that the system demonstrates a diffused limit cycle as the intensity of noise increases. The numerical results show that the multiple scale method is effective for these stochastic systems.