Two-dimensional Differential Equations Research Articles

Numerical approximation for two-dimensional neutral parabolic differential equations with delay

In this article, an effective compact difference scheme is proposed to solve two-dimensional neutral delay parabolic differential equations (NDPDEs). Concrete and detailed derivation of the scheme is given. The unique solvability and unconditional stability are obtained. Several numerical experiments indicate that the scheme has second-order accuracy in temporal direction and fourth-order accuracy in spatial direction in the sense of -norms and -norms. All procedures given in the paper can be extended to NDPDEs in three-dimensional case.

Numerical Model Construction with Closed Observables

Performing analysis, optimization and control using simulations of many-particle systems is computationally demanding when no macroscopic model for the dynamics of the variables of interest is available. In case observations on the macroscopic scale can only be produced via legacy simulator code or live experiments, finding a model for these macroscopic variables is challenging. In this paper, we employ time-lagged embedding theory to construct macroscopic numerical models from output data of a black box, such as a simulator or live experiments. Since the state space variables of the constructed, coarse model are dynamically closed and observable by an observation function, we call these variables closed observables. The approach is an online-offline procedure, as model construction from observation data is performed offline and the new model can then be used in an online phase, independent of the original. We illustrate the theoretical findings with numerical models constructed from time series of a two-dimensional ordinary differential equation system, and from the density evolution of a transport-diffusion system. Applicability is demonstrated in a real-world example, where passengers leave a train and the macroscopic model for the density flow onto the platform is constructed with our approach. If only the macroscopic variables are of interest, simulation runtimes with the numerical model are three orders of magnitude lower compared to simulations with the original fine scale model. We conclude with a brief discussion of possibilities of numerical model construction in systematic upscaling, network optimization and uncertainty quantification.

Energy trapping of thickness-extensional modes in thin film bulk acoustic wave filters

This paper presents the thickness-extensional vibration of a rectangular piezoelectric thin film bulk acoustic wave filter with two pairs of electrodes symmetrically deposited on the center of the zinc oxide film. The two-dimensional scalar differential equations which were first derived to describe in-plane vibration distribution by Tiersten and Stevens are employed. The Ritz method with trigonometric functions as basis functions is used based on a variational formulation developed in our previous paper. Free vibration resonant frequencies and corresponding modes are obtained. The modes may separate into symmetric and antisymmetric ones for such a structurally symmetric filter. Trapped modes with vibrations mainly under the driving electrodes are exhibited. The six corner-type regions of the filter neglected by Tiersten and Stevens for an approximation are taken into account in our analysis. Results show that their approximation can lead to an inaccuracy on the order of dozens of ppm for the fundamental mode, which is quite significant in filter operation and application.

Dissection Method Applications for Complex Shaped Membranes and Plates

Dissection (disintegration) for Poisson's two-dimensional problems differential equation takes us to 2nd-order ordinary differential equations, the thing that simplifies a solution algorithm and makes it programmable. This algorithm illustrated by examples of membranes and calculations of bend plates. New obtained results are compared with those known and stated in the references listed at the last part of the article. The new algorithm obtained by dissecting ordinary differential equations of the 1st- order, based on applying the method of successive approximations and the generalized equations of finite difference method.Solution convergence are illustrated in both methods, also; the out coming algorithm shows the great combination between both techniques. The super advantage of the new developed algorithm is the capability of calculating plates sophistically shaped, which stands as a great benefit for engineers and designers.

Optical properties of magnetoexcitons in double quantum dots

Here we investigate the influence of a constant magnetic field on the energy levels and optical properties of excitons in a double quantum disk (quantum molecule). Taking into account the cylindrical symmetry of the double disk we calculate the wave functions and excitonic energies when the external constant magnetic field is applied along the symmetry axis. Having the eigenfunctions and eigenvalues and using the long-wave approximation we can compute all the optical functions. The double quantum dot is considered as one system rather than two interacting dots separated by narrow barrier. The screened Coulomb interaction between an electron and a hole is assumed. Since in the given structure the separation of the relative- and center-of-mass motion of the electron and the hole is not possible, we use an approach where the six-dimensional eigenvalue problem is transformed into the equivalent eigenvalue problem given by the system of the coupled two-dimensional second order differential equations. The so obtained differential equations are solved numerically. As an example, we give detailed results for a InP/InGaP double quantum dot. Satisfactory agreement with the available experimental data is obtained.

High order finite difference method for time-space fractional differential equations with Caputo and Riemann-Liouville derivatives

We consider high order finite difference methods for two-dimensional fractional differential equations with temporal Caputo and spatial Riemann-Liouville derivatives in this paper. We propose a scheme and show that it converges with second order in time and fourth order in space. The accuracy of our proposed method can be improved by Richardson extrapolation. Approximate solution is obtained by the generalized minimal residual (GMRES) method. A preconditioner is proposed to improve the efficiency for the implementation of the GMRES method.

Energy trapping of thickness-extensional modes in thin film bulk acoustic wave resonators

We perform a theoretical analysis on a rectangular trapped-energy piezoelectric resonator operating with thickness-extensional modes of a thin zinc oxide film on a silicon layer. The driving electrode covers the central part of the resonator only. The two-dimensional scalar differential equation for such a structure derived by Tiersten and Stevens is employed which can describe in-plane mode variations. A variational formulation is developed which provides the theoretical foundation for the Ritz method used in our analysis. Free vibration frequencies and modes are obtained and examined. Modes with vibrations trapped under the driving electrode are found. An approximation used to obtain the classical result of the resonator by Tiersten and Stevens, i.e., the neglect of the four corner regions of the resonator, is removed in our variational analysis. It is shown that their approximation causes a frequency difference on the order of dozens of parts per million for the fundamental thickness-extensional operating mode, quite significant in resonator design and operation.

Some high-order difference schemes for the distributed-order differential equations

Two difference schemes are derived for both one-dimensional and two-dimensional distributed-order differential equations. It is proved that the schemes are unconditionally stable and convergent in an L1(L∞) norm with the convergence orders O(τ2+h2+Δα2) and O(τ2+h4+Δα4), respectively, where τ, h and Δα are the step sizes in time, space and distributed-order variables. Several numerical examples are given to confirm the theoretical results.

Two alternating direction implicit difference schemes with the extrapolation method for the two-dimensional distributed-order differential equations

The Grünwald formula is used to solve the two-dimensional distributed-order differential equations. Two alternating direction implicit (ADI) difference schemes are derived. It is proved that the schemes are unconditionally stable and convergent with the convergence orders O(τ+h12+h22+Δα2) and O(τ+h14+h24+Δα4) in H1 norm, respectively, where τ,h1,h2 and Δα are the step sizes in time, space in x- and y-direction, and distributed order. The extrapolation method is applied to improve the approximate accuracy to the orders O(τ2∣lnτ∣2+h12+h22+Δα2) and O(τ2∣lnτ∣2+h14+h24+Δα4), respectively. Several numerical examples are given to confirm the theoretical results.

INTEGRATION OF GIS TECHNOLOGY FOR FUZZY-DETERMINISTIC MODELING CONDITIONS OF FORMATION AND OPERATION KEGEYLI GROUNDWATER IS ABSTRACTED

Offers scientific and methodological basics for application of fuzzy-deterministic simulation geofiltration processes in single layers on the basis of the application of GIS technology in terms of dominance fuzzy initial information. GIS model of Kegeyli withdrawal of groundwater (KWGV) is carried out on the basis of software including ArcGIS (ArcCatalog, ArcMap). The problems of a situational analysis of the results clearly-deterministic simulation. The proposed information model and information model KWGV filtration process to ensure that the relationship between the object model and the geofiltration, represented as a system of fuzzy-deterministic two-dimensional (quasi three-dimensional) nonlinear differential equations filtering and organizing computational experiments (CE). In this context, discusses aspects of geoinformation approach in the process of fuzzy-deterministic simulation of operating conditions Kegeyli abstraction of groundwater. It is proposed software system that focuses on the study of hydro geological objects, both local and regional scale using data from heterogeneous. Formalization of fuzzy algorithms parameters of underground hydrosphere, the relationship abstraction of groundwater and its model. The algorithms select the solution in the formation and exploitation of groundwater resources, the principles of the information model of the region of the filtering process. The results of solving the problem of formalization of parameters and forecast the state of groundwater Kegeyli withdrawals of groundwater.

Finite difference schemes for two-dimensional time-space fractional differential equations

In this paper, finite difference schemes for differential equations with both temporal and spatial fractional derivatives are studied. When the order of the time fractional derivative is in , an alternating direction implicit (ADI) scheme with second-order accuracy in both space and time is constructed. For equations with time fractional derivatives of order lying in , a scheme is derived and solved by the generalized minimal residual method. We also propose a preconditioner to improve the efficiency for the implementation of the scheme in this situation.

Dynamics of Neural Systems with Discrete and Distributed Time Delays

In real-world systems, interactions between elements do not happen instantaneously, due to the time required for a signal to propagate, reaction times of individual elements, and so forth. Moreover, time delays are normally nonconstant and may vary with time. This means that it is vital to introduce time delays in any realistic model of neural networks. In order to analyze the fundamental properties of neural networks with time-delayed connections, we consider a system of two coupled two-dimensional nonlinear delay differential equations. This model represents a neural network, where one subsystem receives a delayed input from another subsystem. An exciting feature of the model under consideration is the combination of both discrete and distributed delays, where distributed time delays represent the neural feedback between the two subsystems, and the discrete delays describe the neural interaction within each of the two subsystems. Stability properties are investigated for different commonly used distribution kernels, and the results are compared to the corresponding results on stability for networks with no distributed delays. It is shown how approximations of the boundary of the stability region of a trivial equilibrium can be obtained analytically for the cases of delta, uniform, and weak gamma delay distributions. Numerical techniques are used to investigate stability properties of the fully nonlinear system, and they fully confirm all analytical findings.

New Numerical Solution of von Karman Equation of Lengthwise Rolling

The calculation of average material contact pressure to rolls base on mathematical theory of rolling process given by Karman equation was solved by many authors. The solutions reported by authors are used simplifications for solution of Karman equation. The simplifications are based on two cases for approximation of the circular arch: (a) by polygonal curve and (b) by parabola. The contribution of the present paper for solution of two-dimensional differential equation of rolling is based on description of the circular arch by equation of a circle. The new term relative stress as nondimensional variable was defined. The result from derived mathematical models can be calculated following variables: normal contact stress distribution, front and back tensions, angle of neutral point, coefficient of the arm of rolling force, rolling force, and rolling torque during rolling process. Laboratory cold rolled experiment of CuZn30 brass material was performed. Work hardening during brass processing was calculated. Comparison of theoretical values of normal contact stress with values of normal contact stress obtained from cold rolling experiment was performed. The calculations were not concluded with roll flattening.

A note on flux integrals over smooth regular domains

We provide new bounds on a flux integral over the portion of the boundary of one regular domain contained inside a second regular domain, based on properties of the second domain rather than the first one. This bound is amenable to numerical computation of a flux through the boundary of a domain, for example, when there is a large variation in the normal vector near a point. We present applications of this result to occupational measures and two-dimensional differential equations, including a new proof that all minimal invariant sets in the plane are trivial.

A higher-order compact LOD method and its extrapolations for nonhomogeneous parabolic differential equations

A higher-order compact locally one-dimensional (LOD) finite difference method for two-dimensional nonhomogeneous parabolic differential equations is proposed. The resulting scheme consists of two one-dimensional tridiagonal systems, and all computations are implemented completely in one spatial direction as for one-dimensional problems. The solvability and the stability of the scheme are proved almost unconditionally. The error estimates are obtained in the discrete H1,L2 and L∞ norms, and show that the proposed compact LOD method has the accuracy of the second-order in time and the fourth-order in space. Two Richardson extrapolation algorithms are presented to increase the accuracy to the fourth-order and the sixth-order in both time and space when the time step is proportional to the spatial mesh size. Numerical results demonstrate the accuracy of the compact LOD method and the high efficiency of its extrapolation algorithms.

Estimation in the partially observed stochastic Morris–Lecar neuronal model with particle filter and stochastic approximation methods

Parameter estimation in multidimensional diffusion models with only one coordinate observed is highly relevant in many biological applications, but a statistically difficult problem. In neuroscience, the membrane potential evolution in single neurons can be measured at high frequency, but biophysical realistic models have to include the unobserved dynamics of ion channels. One such model is the stochastic Morris-Lecar model, defined by a nonlinear two-dimensional stochastic differential equation. The coordinates are coupled, that is, the unobserved coordinate is nonautonomous, the model exhibits oscillations to mimic the spiking behavior, which means it is not of gradient-type, and the measurement noise from intracellular recordings is typically negligible. Therefore, the hidden Markov model framework is degenerate, and available methods break down. The main contributions of this paper are an approach to estimate in this ill-posed situation and nonasymptotic convergence results for the method. Specifically, we propose a sequential Monte Carlo particle filter algorithm to impute the unobserved coordinate, and then estimate parameters maximizing a pseudo-likelihood through a stochastic version of the Expectation-Maximization algorithm. It turns out that even the rate scaling parameter governing the opening and closing of ion channels of the unobserved coordinate can be reasonably estimated. An experimental data set of intracellular recordings of the membrane potential of a spinal motoneuron of a red-eared turtle is analyzed, and the performance is further evaluated in a simulation study.

Stability of difference schemes for Bitsadze-Samarskii type nonlocal boundary value problem involving integral condition

In this study, the stable difference schemes for the numerical solution of Bitsadze-Samarskii type nonlocal boundary-value problem involving integral condition for the elliptic equations are studied. The second and fourth orders of the accuracy difference schemes are presented. A procedure of modified Gauss elimination method is used for solving these difference schemes for the two-dimensional elliptic differential equation. The method is illustrated by numerical examples.

Bifurcations of Tumor-Immune Competition Systems with Delay

A tumor-immune competition model with delay is considered, which consists of two-dimensional nonlinear differential equation. The conditions for the linear stability of the equilibria are obtained by analyzing the distribution of eigenvalues. General formulas for the direction, period, and stability of the bifurcated periodic solutions are given for codimension one and codimension two bifurcations, including Hopf bifurcation, steady-state bifurcation, and B-T bifurcation. Numerical examples and simulations are given to illustrate the bifurcations analysis and obtained results.

A parallel algorithm for the two-dimensional time fractional diffusion equation with implicit difference method.

It is very time consuming to solve fractional differential equations. The computational complexity of two-dimensional fractional differential equation (2D-TFDE) with iterative implicit finite difference method is O(M x M y N 2). In this paper, we present a parallel algorithm for 2D-TFDE and give an in-depth discussion about this algorithm. A task distribution model and data layout with virtual boundary are designed for this parallel algorithm. The experimental results show that the parallel algorithm compares well with the exact solution. The parallel algorithm on single Intel Xeon X5540 CPU runs 3.16–4.17 times faster than the serial algorithm on single CPU core. The parallel efficiency of 81 processes is up to 88.24% compared with 9 processes on a distributed memory cluster system. We do think that the parallel computing technology will become a very basic method for the computational intensive fractional applications in the near future.

Variational formulation of the Stevens-Tiersten equation and application in the analysis of rectangular trapped-energy quartz resonators

The two-dimensional scalar differential equation for transversely varying thickness modes in quartz crystal resonators operating with thickness-shear modes is formulated into variational form for trapped-energy resonators with both electroded and unelectroded regions. A theoretical analysis of rectangular trapped-energy resonators of singly rotated quartz is performed using the Ritz method based on the variational formulation. Free vibration resonant frequencies and modes are obtained. The results show the existence of trapped modes under the electrodes. The effects of various geometric and physical parameters on the trapped modes are examined. It is also found that the classical frequency prediction given by Tiersten and Smythe from an approximate analysis using the scalar differential equation has an inaccuracy on the order of 100 ppm for the fundamental mode, significant in resonator design.