Two-dimensional Differential Equations Research Articles

Solve the linear quaternion-valued differential equations having multiple eigenvalues

The theory of two-dimensional linear quaternion-valued differential equations (QDEs) was recently established {see the work of Kou and Xia [Stud. Appl. Math. 141(1), 3–45 (2018)]}. They observed some profound differences between QDEs and ordinary differential equations. Also, an algorithm to evaluate the fundamental matrix by employing the eigenvalues and eigenvectors was presented in the work of Kou and Xia [Stud. Appl. Math. 141(1), 3–45 (2018)]. However, the fundamental matrix can be constructed providing that the eigenvalues are simple. If the linear system has multiple eigenvalues, how to construct the fundamental matrix? In particular, if the number of independent eigenvectors might be less than the dimension of the system, that is, the numbers of the eigenvectors are not enough to construct a fundamental matrix, how to find the “missing solutions”? The main purpose of this paper is to answer this question.

Modeling and simulation of temperature distribution for planar cross-flow solid oxide fuel cell

Solid oxide fuel cell is a device that can convert chemical energy directly into electricity. Its advantages, such as high efficiency, low emission, quiet operation, fuel flexibility, bring about the broad application prospect. However, it is difficult to obtain the temperature distribution in the existing planar cross flow solid oxide fuel cell through the experiment. In this paper, a control orient two-dimensional differential equation model is established for a planar cross flow solid oxide fuel cell based on the finite node method, and an iterative algorithm for calculating the real time voltage of the fuel cell is proposed for the model. Based on the model, the temperature distribution of the fuel cell in the test and system configuration is simulated. The simulation results show that the model can reflect the thermoelectric characteristics of the planar cross flow solid oxide fuel cell, especially the temperature distribution of the fuel cell. The SOFC temperature distribution modeling in this paper is helpful for the development of temperature distribution observers and design related control methods in later studies.

Thermal and aerodynamic characteristics of tube furnaces with wall burners of two types

Tube furnaces of the petrochemical and oil refining industries are energy-intensive devices. The differential method for calculating the complex heat transfer in the furnaces is used to study the flow fields and the temperature of the combustion products in the radiation chambers of technological tube furnaces with two types of wall burners, which are placed on several levels on the sidewall. In a furnace designed for the conversion of methane mixed with steam in the reaction tubes, acoustic burners for burning natural gas near the wall are installed in three tiers on the side lined sidewalls of the radiation chamber. In the primary reforming furnace of natural gas for the production of ammonium nitrate, wall-type burners are installed on six levels. The method used is based on the joint numerical integration of the two-dimensional differential equations of energy transfer by radiation using the S2-approximation of the discrete ordinate method, the energy conservation equations, the equations of turbulent motion of a gas mixture, the k-e turbulence model and the two-stage gas fuel combustion model. The calculated fields of temperature and flue gas flow obtained with the help of a computer program implementing the proposed method are discussed.

Stability and Hopf Bifurcation Analysis in a Delayed Myc/E2F/miR-17-92 Network Involving Interlinked Positive and Negative Feedback Loops

MiR-17-92 plays an important role in regulating the levels of the Myc/E2F protein. In this paper, we consider a coupling network between Myc/E2F/miR-17-92 delayed negative feedback loop and Myc/E2F positive feedback loop described by a two-dimensional delay differential equation. Based on linear stability analysis and bifurcation theory, sufficient conditions for stability of equilibria and oscillatory behaviors via Hopf bifurcation are derived when choosing time delay as well as negative feedback strength associated with oscillations as bifurcation parameters, respectively. Furthermore, direction and stability of Hopf bifurcation of time delay are studied by using the normal form method and center manifold theorem. Finally, several numerical simulations are performed to verify the results we obtained.

Two alternating direction implicit spectral methods for two-dimensional distributed-order differential equation

In this paper, two alternating direction implicit Galerkin-Legendre spectral methods for distributed-order differential equation in two-dimensional space are developed. It is proved that the schemes are unconditionally stable and convergent with the convergence orders O(Δt + σ2 + N−m) and O(Δt2 + σ2 + N−m), respectively, where Δt, σ, N, and m are the time step size, step size in distributed-order variable, polynomial degree, and regularity in the space variable of the exact solution, respectively. Moreover, the applicability and accuracy of the two schemes are demonstrated by numerical experiments to support our theoretical analysis.

Preliminary Results: Simple Model of Evolutionary Dynamic on Insecticide Resistance in Mosquitoes

The information on insecticide resistance in mosquitoes as diseases vectors is a critical issue for the success of mosquito population control. This is because the resistance can reduce the effectiveness of insecticides. Based on the resistance level to insecticides, the mosquitoes are divided into susceptible and resistant groups. Genetically, the existence of susceptible and resistant mosquitoes are determined by the frequency of alleles. In this paper, a dynamical model is constructed based on the genetic information to observe the effect of selection by exposure to insecticides. The model is a two-dimensional nonlinear differential equation system. In the analysis of this mathematical model, we consider two important factors in the evolution of resistance, i.e., the fitness value and the effectiveness of the insecticide. The condition of coexistence and stability are obtained here. The simulations are performed to illustrate the effect of insecticide usage continuously on mosquito resistance. Based on analysis and simulation, the fitness value of intermediate mosquitoes plays an essential role in maintaining the coexistence of the mosquito population. Furthermore, the resistant mosquitoes increases significantly following the pattern of increasing the dose of exposure to insecticides.

Hopf bifurcation with additive noise

We consider the dynamics of a two-dimensional ordinary differential equation exhibiting a Hopf bifurcation subject to additive white noise and identify three dynamical phases: (I) a random attractor with uniform synchronisation of trajectories, (II) a random attractor with non-uniform synchronisation of trajectories and (III) a random attractor without synchronisation of trajectories. The random attractors in phases (I) and (II) are random equilibrium points with negative Lyapunov exponents while in phase (III) there is a so-called random strange attractor with positive Lyapunov exponent.We analyse the occurrence of the different dynamical phases as a function of the linear stability of the origin (deterministic Hopf bifurcation parameter) and shear (amplitude-phase coupling parameter). We show that small shear implies synchronisation and obtain that synchronisation cannot be uniform in the absence of linear stability at the origin or in the presence of sufficiently strong shear. We provide numerical results in support of a conjecture that irrespective of the linear stability of the origin, there is a critical strength of the shear at which the system dynamics loses synchronisation and enters phase (III).

"Approximate solution for the two-dimensional Volterra-Integro differential equation of fractional order by using Generalized Deferential Transform method "

"In this work, an efficient generalized differential transform method (GDTM) is proposed for solving the twodimensional Volterra-Integro differential equation (2-DVIDE) of fractional order. The results of the proposed method are compared with exact solution, a numerical example is considered for testing the accuracy and validity of this method."

Approximate solution for the two-dimensional Volterra-Integro differential equation of fractional order by using Generalized Deferential Transform method

Approximate solution for the two-dimensional Volterra-Integro differential equation of fractional order by using Generalized Deferential Transform method

A Runge–Kutta Gegenbauer spectral method for nonlinear fractional differential equations with Riesz fractional derivatives

ABSTRACTA Runge–Kutta Gegenbauer spectral method is proposed to solve an initial-boundary value problem for a nonlinear two-dimensional fractional differential equation with variable coefficients. The solution to the problem at each time step is approximated by a bivariate polynomial based on shifted Gegenbauer polynomials and then the Runge–Kutta method of order 3 is applied to the problem. The convergence rate of the derived method is analysed. Numerical results are presented to verify the effectiveness of the method.

Research on transient heat transfer performance of disc brakes for mining motor vehicle

The type of disc brake has excellent efficiency and strong heat dissipation, which are beneficial to the transport capacity for mining motor vehicles. To ensure the reliability of the brake, the two-dimensional heat transfer differential equation model of disc brake is established, considering the dynamic change of convective heat transfer coefficient. The heat boundary conditions are established through the characteristics of brake disc structure and air condition, which is more accuracy than the traditional simplified method. Based on the PDE module in MATLAB, the mathematical model of heat transfer is solved and the transient temperature field is obtained. The calculation results are verified by the temperature field of the brake disc Link3900 NVH test platform. The results show that the research scheme has high computation accuracy, and can provide important basis, new ideas and advanced methods for the brake of mining motor vehicle in related fields.

Pressure characteristic analysis and estimation of effective length of a hybrid diffuser and modified hybrid diffusers

In this paper, performance simulation of a hybrid diffuser, hybrid diffuser with suction and hybrid diffuser with fence have been carried out. The two-dimensional steady differential equations for conservation of mass and momentum have been solved for the Reynolds number ranging from 100 to 600. During the simulation, a fixed aspect ratio of 2 and a fixed value of half divergence angle of 3.5° have been considered for all the diffuser geometries. At the inlet, fully developed velocity profile has been used. The effect of Reynolds number on average static pressure have been studied in detail, and compared between three diffuser geometries. The effective diffuser lengths for achieving maximum average static pressure at the diffuser exit have been computed for the considered Reynolds numbers.

The Controller Design Using Local Stability Analysis on a Nonlinear Inverted Pendulum

This paper is designed for the purpose of analytic design and numerical simulation of a controller for nonlinear mobile inverted pendulum. It needs to obtain a method to overcome the difficulties in the design problem of controller for nonlinear plant. The method was using local stability analysis of two fixed points in two-dimensional ordinary nonlinear differential equation. The result provides a stable response solution using a special pole placement design, and transient stability is simulated using simulink. The response behavior can be selected according to the desired poles. The results obtained at this work are different from the optimal control problem. Indeed, our results have been summarized in the design method, target response and simulation process.

Modeling of unsteady non-Newtonian blood flow through a stenosed artery: with nanoparticles

Literature survey related to the nanoparticles reveals that nanofluids are getting popularity in hematological treatment. Development in this direction motivated us to write a theoretical study on unsteady blood motion in stenosed vessel with nanoparticles. Geometry of a stenosed arterial section is being written mathematically by an appropriate geometric expression. The constitutive equation of Carreau fluid model is used to characterize the dynamical behavior of the blood. The rheology of the blood is formulated mathematically by coupled partial differential equations. Similarly, the effects of nanoparticles are incorporated mathematically into governing equations by using Buonjiornio’s formulation. Mild stenotic condition is employed to reduce the two-dimensional differential equations to simple form. Numerical technique is being used to obtain the numerical solution to the existing problem. The obtained simulation reveals that the magnitude of velocity shows an accelerating behavior for Brownian motion parameter and shows deceleration trend on increasing the thermophoresis parameter. Similarly, the instantaneous behavior of blood flow pattern is shown through streamlines.

Two-dimensional heat and mass transfer and thermodynamic analyses of porous microreactors with Soret and thermal radiation effects—An analytical approach

Transport of heat and mass and the thermodynamics of porous microreactors with thermal diffusion and radiation effects are investigated analytically. The examined configuration includes an axisymmetric, thick-wall microchannel with an iso-flux thermal boundary condition imposed on the external surfaces. The microchannel is filled with porous materials and accommodates a zeroth order homogenous chemical reaction. Internal radiative heat transfer is modelled in addition to heat convection and conduction, while the local thermal non-equilibrium approach is taken within the porous section of the system. The transport of species is coupled with that of heat via the inclusion of thermodiffusion or Soret effect. Two-dimensional heat and mass transfer differential equations are solved analytically. The results are subsequently used to predict the thermodynamic irreversibilities inside the reactor and a thorough analysis of local and total entropy generation rates is performed. Also, the changes in Nusselt number, calculated on the internal walls of the microreactor, versus various parameters are reported. It is shown that the radiation effects can impact the temperature of the solid phase of the porous medium and lead to alteration of Nusselt number. It is further observed that the transfer of mass is the main source of irreversibility in the system. The findings are of particular use for the design and analysis of the microreactors with homogenous chemical reactions and can be also used for the validation of computational models.

The Implementation of Milstein Scheme in Two-Dimensional SDEs Using the Fourier Method

Multiple stochastic integrals of higher multiplicity cannot always be expressed in terms of simpler stochastic integrals, especially when the Wiener process is multidimensional. In this paper we describe how the Fourier series expansion of Wiener process can be used to simulate a two-dimensional stochastic differential equation (SDE) using Matlab program. Our numerical experiments use Matlab to show how our truncation of Itô’-Taylor expansion at an appropriate point produces Milstein method for the SDE.

A Semi-Analytical Solution for the Thickness-Vibration of Centrally Partially-Electroded Circular AT-Cut Quartz Resonators.

Vibration frequencies and modes for the thickness-shear vibrations of infinite partially-electroded circular AT-cut quartz plates are obtained by solving the two-dimensional (2D) scalar differential equation derived by Tiersten and Smythe. The Mathieu and modified Mathieu equations are derived from the governing equation using the coordinate transform and the collocation method is used to deal with the boundary conditions. Solutions of the resonant frequencies and trapped modes are validated by those results obtained from COMSOL software. The current study provides a theoretical way for figuring out the vibration analysis of circular quartz resonators.

Stabilizing two-dimensional stochastic systems through sliding mode control

This paper presents a stabilizing control for two-dimensional stochastic differential equations. The stability concept in this study is the stability in probability. To ensure such a stability, the control is designed based on the sliding mode technique, and applied to account stochastic systems. This finding has a practical implication—the proposed control can be used to stabilize a real-time automotive electronic throttle valve. The proposed approach is verified by data collected from experiments.

Investigations of the Switching Dynamics in Sliding Mode Control

In this paper, a (switching) sliding mode controller, applied to a mechanical system with additive white process noise, is investigated. The practical relevance of this study is a statistical characterization of system performance in terms of the stationary variance of the control error.The system is modeled with a two-dimensional stochastic differential equation, whose coordinate functions to an extend are analyzed separately. In order to determine the stationary variance of one of the coordinate functions, the auto-correlation function for the other coordinate function is approximated with a Fourier series. Finally, analytical results are compared to results from Euler-Maruyama simulation over a wide range of model parameter settings.

Quantum scattering by nonspherical objects

The problem of scattering by spherically asymmetric potentials is considered where angular momentum not only ceases to be a conserved quantity but is also not a convenient basis for the expansion of wave functions and scattering amplitudes. Numerical solutions of the two-dimensional differential equation for the scattering amplitude of a particle and a coupled pair on a semitransparent disc of finite thickness are obtained. The effect of resonant diffraction is shown. A numerical scheme can be used to describe the scattering of a particle by deformed atomic nuclei.