Euler Forward Difference Research Articles

Theoretical analysis for dynamic model of U-shaped electrothermal microactuator based on Chebyshev spectral method

Abstract This paper presents a largely and accurate analysis for dynamic model of U-shaped actuator that considers the material parameters dependent on temperature. The electro-thermal analysis for temperature distribution and thermo-mechanical analysis for deflection is carried out. The electro-thermal dynamic behaviour is described by piecewise one-dimensional (1D) heat transfer partial differential equations (PDE) for each beam of U-shaped actuator. A combination of Chebyshev spectral method for space domain and Euler forward difference method for time domain is used to solve these equations. The convergence test is first done to ensure the result convinced.

Analyzing non-isothermal phase transition problems with natural convection using peridynamic differential operator

In this study, a developed model for non-isothermal phase transition with natural convection is proposed by using peridynamic differential operator (PDDO). The dimensionless governing equations of heat source approach and vorticity-stream function approach are reconstructed into the non-local integral form. The Euler forward difference is used for time integration. The application of the developed PDDO model extends to simulations involving pure phase transition, pure natural convection, and non-isothermal phase transition coupled with natural convection. The validity and accuracy of the developed non-isothermal phase transition model are verified by comparing the PDDO simulation results with results in published literature. This research provides a new alternative approach for simulating phase transition and convection problems.

Analysis of a Discrete Predator-Prey Model for Controlling The Extinction of The Pesut Mahakam (Orcaella brevirostris) with Toxic Effect at The Mahakam River

The Pesut Mahakam (Orcaella brevirostris) is an East Kalimantan animal in critical condition and on the verge of extinction. One of the causes of the extinction of the Mahakam dolphins is the toxic effects of environmental pollution. In this study, an analysis of the discrete predator-prey model using Euler's forward difference between the Pesut Mahakam (Orcaella brevirostris) and its prey was investigated. Dynamic analysis includes determining the equilibrium point, stability analysis of the equilibrium point, and numerical simulation. The value of the Mahakam dolphin growth rate parameter (predator) is obtained by adjusting the curve based on data in the field. Numerical simulations were carried out to describe the results of the analysis. Changes in the value of the toxic effect rate parameter affects the number of Mahakam dolphins and their prey.

Two novel alternative integration schemes for multi-invariants dependent isotropic finite deformation plasticity

Multi-invariants dependent isotropic yield function in terms of first invariant I1 of stress tensor, second J2 and third J3 invariants of deviatoric stress tensor is required for accurate description of plasticity of polycrystalline solids. In the context of multi-invariants dependent multiplicative hyperelasto-plasticity, two integration schemes (IS-1 and IS-2) are formulated and their computational costs are compared for the first time in this paper. IS-1 is fully-implicit in which backward Euler difference approximation of equivalent plastic strain rate is coupled with the exponential approximation of return mapping in principal Kirchhoff stress space and accordingly principal Kirchhoff stresses, equivalent plastic strain and plastic Lagrange multiplier need to be updated iteratively. IS-2 is semi-implicit in which plastic Lagrange multiplier evaluated at current time step and semi-implicit forward Euler difference approximation of equivalent plastic strain rate is coupled with logarithmic approximation of return mapping in Kirchhoff stress space and accordingly only plastic Lagrange multiplier needs to be updated iteratively. For J2 dependent plasticity, computation time involved in numerical simulations for both the integration schemes is nearly the same for same accuracy. For IS-1, computation time involved in simulation for multi-invariants dependent plasticity is significantly greater than that for J2 dependent plasticity due to the computation of the Jacobian matrix in return mapping involving lengthy terms of second order partial derivatives of multi-invariants dependent yield function. For IS-2, computation time involved in simulation for multi-invariants dependent plasticity is nearly the same as that for J2 dependent plasticity. This is due to the fact that the computation of the Jacobian matrix in IS-2 is not required and return mapping involves only first partial derivatives of yield function. Thus, IS-2 is computationally cost effective than IS-1.

Model Predictive Control for First-Order Hyperbolic System Based on Quasi-Shannon Wavelet Basis

In this paper, we consider a class of first-order hyperbolic distributed parameter systems. Our focus is on the design of a new class of model predictive control schemes using a quasi-Shannon wavelet basis. First, the first-order hyperbolic distributed parameter system is transformed into an equivalent system using collocation techniques for the approximation of spatial derivatives and Euler forward difference method for the approximation of the time component. Then, a model reduction method is applied to obtain a reduced-order system on which a nonlinear model predictive controller is designed through solving a nonlinear quadratic programming problem with input constraints. For illustration, the temperature control of a flow-control long-duct heating system is considered to be an example. A comparative simulation study is conducted to demonstrate the effectiveness of the proposed method.

A novel noise-tolerant Zhang neural network for time-varying Lyapunov equation

The Zhang neural network (ZNN) has become a benchmark solver for various time-varying problems solving. In this paper, leveraging a novel design formula, a noise-tolerant continuous-time ZNN (NTCTZNN) model is deliberately developed and analyzed for a time-varying Lyapunov equation, which inherits the exponential convergence rate of the classical CTZNN in a noiseless environment. Theoretical results show that for a time-varying Lyapunov equation with constant noise or time-varying linear noise, the proposed NTCTZNN model is convergent, no matter how large the noise is. For a time-varying Lyapunov equation with quadratic noise, the proposed NTCTZNN model converges to a constant which is reciprocal to the design parameter. These results indicate that the proposed NTCTZNN model has a stronger anti-noise capability than the traditional CTZNN. Beyond that, for potential digital hardware realization, the discrete-time version of the NTCTZNN model (NTDTZNN) is proposed on the basis of the Euler forward difference. Lastly, the efficacy and accuracy of the proposed NTCTZNN and NTDTZNN models are illustrated by some numerical examples.

A class of parabolic evolutionary quasivariational inequalities in contact mechanics

In this paper, we obtain an existence and uniqueness of the solution for a class of parabolic evolutionary quasivariational inequalities in contact mechanics under some mild conditions. We also study an error estimate for the parabolic evolutionary quasivariational inequality by employing the forward Euler difference scheme and the element-free Galerkin spatial approximation.

A power-type varying gain discrete-time recurrent neural network for solving time-varying linear system

Many practical engineering problems can be described as an online time-varying linear system (TVLS), and thus solving TVLS is very important in control theory and control engineering. In this paper, a novel power-type varying gain discrete-time recurrent neural network (PVG-DTRNN) is proposed to solve the TVLS problem. Compared with the state-of-art method, i.e., the fixed-parameter discrete-time zeroing neural network (FP-DTZNN), the proposed PVG-DTRNN has better convergent rate and higher accuracy. To do so, a vector error function is firstly defined. Secondly, a power-type gain implicit dynamic model is derived and needs to be further discretized. Thirdly, by using Euler forward-difference rule, a discretized dynamic model is designed. In order to get the explicit dynamic model, the Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton method is utilized to estimate the inverse of the Hessian matrix. Comparisons of computer simulations verify the effectiveness and superiority of the proposed PVG-DTRNN models.

Numerical Simulation of PDEs by Local Meshless Differential Quadrature Collocation Method

In this paper, a local meshless differential quadrature collocation method based on radial basis functions is proposed for the numerical simulation of one-dimensional Klein–Gordon, two-dimensional coupled Burgers’, and regularized long wave equations. Both local and global meshless collocation procedures are used for spatial discretization, which convert the mentioned partial differential equations into a system of ordinary differential equations. The obtained system has been solved by the forward Euler difference formula. An upwind technique is utilized in the case of the convection-dominated coupled Burgers’ model equation. Having no need for the mesh in the problem domain and being less sensitive to the variation of the shape parameter as compared to global meshless methods are the salient features of the local meshless method. Both rectangular and non-rectangular domains with uniform and scattered nodal points are considered. Accuracy, efficacy, and the ease of implementation of the proposed method are shown via test problems.

A numerical Haar wavelet-finite difference hybrid method for linear and non-linear Schrödinger equation

In this research work, we proposed a Haar wavelet collocation method (HWCM) for numerical solution of linear and nonlinear Schrödinger equations. The nonlinear term present in the model equation is linearized by a linearization technique. The Time derivative in the Schrödinger equation is approximated by forward Euler difference formula while the space derivatives are approximated by Haar function, which convert the model equation into system of algebraic equation. The stability analysis of the HWCM is also given. Several test problems are presented to verify the accuracy, stability and capability of the proposed method.

A 5-instant finite difference formula to find discrete time-varying generalized matrix inverses, matrix inverses, and scalar reciprocals

Finite difference schemes have been widely studied because of their fundamental role in numerical analysis. However, most finite difference formulas in the literature are not suitable for discrete time-varying problems because of intrinsic limitations and their relatively low precision. In this paper, a high-precision 1-step-ahead finite difference formula is developed. This 5-instant finite difference (5-IFD) formula is used to approximate and discretize first-order derivatives, and it helps us to compute discrete time-varying generalized matrix inverses. Furthermore, as special cases of generalized matrix inverses, time-varying matrix inversion, and scalar reciprocals are generally deemed as independent problems and studied separately, which are solved unitedly in this paper. The precision of the 5-IFD formula and the convergence behavior of the corresponding discrete-time models are derived theoretically and shown in numerical experiments. Conventional useful formulas, such as the Euler forward finite difference (EFFD) formula and the 4-instant finite difference (4-IFD) formula are also used for comparisons and to show the superiority of the 5-IFD formula.

Three-step general discrete-time Zhang neural network design and application to time-variant matrix inversion

Abstract In recent decades, neural networks methods have widely applied to many science and engineering fields. Zhang neural network (ZNN) as a special type of recurrent neural networks was proposed by Zhang et al, which has been applied to different time-variant problems solving. ZNNs are usually used to process the continuous-time signal in time-variant systems as continuous-time ZNN (CTZNN) models. Since digit devices and computers are widely applied in science and engineering, it is necessary to develop a general method to discretize CTZNN models to discrete-time ZNN (DTZNN) models. In previous work, Euler forward difference and two types of three-step Zhang et al. discretization (ZeaD) formulas were applied to discretize CTZNN models. In this paper, a three-step general ZeaD formula based on Taylor expansion is designed to approximate the first-order derivative of the target point, and discretize CTZNN models for time-variant matrix inversion. In comparison, the two types of three-step ZeaD formulas are the special cases of the proposed general ZeaD formula. For the situation of the time derivative of objective matrix unknown, two formulas of estimating the derivative are provided, and two other corresponding three-step general DTZNN models are proposed. Theoretical analyses present the stability and convergence of the three general DTZNN models for time-variant matrix inversion. The numerical experiment results substantiate the efficacy and superiority of the three proposed general DTZNN models for time-variant matrix inversion with the theoretical steady-state residual errors, comparing with those of Newton iteration and one-step DTZNN model. In addition, by comparing the numerical results of the two general DTZNN models for the situation of the time derivative of objective matrix unknown with the one of the derivative known, the steady-state residual errors of the formers are slightly bigger than the latter.

A comparative analysis of local meshless formulation for multi-asset option models

A local meshless radial basis function collocation differential quadrature (LMRBFCDQ) is proposed for the numerical solution of a single and multi-asset option pricing PDE models arising in computational finance. Spatial discretization is performed by both local and a standard global meshless collocation procedures coupled with a set of different time integrators based on the forward Euler difference formula (FEDF), the fully Implicit method (FIM), the Crank–Nicolson method (CNM), the explicit Runge–Kutta method of order two (ERK2), the Crank–Nicolson Runge–Kutta method of order two (CNRK2), the fully Implicit Runge–Kutta method of order two (IRK2), the Runge–Kutta method of order four (RK4), the Embedded Runge–Kutta method (RK23). Operator splitting techniques like the ordinary operator splitting (OOS), the Lie–Trotter splitting, the additive splitting and the Strang splitting are also tested for time integration. The proposed hybrid schemes are the amalgamation of the meshless differential quadrature procedure and the finite difference approximations. Different types of radial basis functions (RBFs) i.e. the multiquadric (MQ), the inverse quadric (IQ) and the Gaussian (GA) are utilized for the spatial discretization of the PDE models. Numerical analysis of a range of computational finance related models are shown to demonstrate accuracy, efficiency and ease of implementation of the proposed meshless-finite difference procedure.

Research on the Forward Euler Difference Method for Parabolic Equations

This article is devoted to the forward Euler difference method for the parabolic equation. In this paper, a forward Euler difference scheme is derived. It is shown that the forward Euler difference scheme is convergence and stability. Moreover, a numerical experiment is conducted to illustrate the theoretical results of the presented method.

Dynamics of numerical discretization in a delayed diffusive Nicholson’s blowflies equation

In this article, we study the dynamics of a diffusive Nicholson’s blowflies equation with a finite delay and Dirichlet boundary condition by the forward and backward Euler difference schemes and Crank–Nicolson scheme. The existence of Hopf bifurcation at the equilibrium is obtained. Through analyzing the distribution of the eigenvalues we find that the unstable equilibrium state without dispersion may become stable with dispersion under certain conditions. Our results show that Crank–Nicolson and the backward Euler difference schemes are superior to the forward Euler difference scheme under the means of describing approximately the dynamics of the original system. Finally, numerical examples are provided to illustrate the analytical results.

Solving time-varying nonlinear inequalities using continuous and discrete-time Zhang dynamics

A special kind of neural dynamics, termed Zhang dynamics (ZD), is proposed, generalized and investigated for the online solution of time-varying scalar-valued nonlinear inequalities by following Zhang et al.’s design method. The continuous-time ZD (CTZD) model based on an exponent-type design formula can be guaranteed to exponentially converge to the time-varying solution set of the problem in an error-free manner. For potential hardware implementation on digital circuits, the corresponding discrete-time ZD (DTZD) model is generated through the well-known Euler forward difference rule. Newton-type algorithm is also developed for comparison purposes. In addition, the simplified CTZD (S-CTZD) and DTZD (S-DTZD) models are developed for solving static scalar-valued nonlinear inequalities. Numerical simulative examples further demonstrate and verify the efficacy of the ZD models for solving time-varying and static scalar-valued nonlinear inequalities. Besides, the DTZD model possesses the lower complexity and higher accuracy, as compared with the Newton-type algorithm.

Fully discrete interior penalty discontinuous Galerkin methods for nonlinear parabolic equations

AbstractIn this article, we investigate interior penalty discontinuous Galerkin (IPDG) methods for solving a class of two‐dimensional nonlinear parabolic equations. For semi‐discrete IPDG schemes on a quasi‐uniform family of meshes, we obtain a priori bounds on solutions measured in the L2 norm and in the broken Sobolev norm. The fully discrete IPDG schemes considered are based on the approximation by forward Euler difference in time and broken Sobolev space. Under a restriction related to the mesh size and time step, an hp ‐version of an a priori l∞(L2) and l2(H1) error estimate is derived and numerical experiments are presented.© 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 288–311, 2012

Asymptotic stability of numerical methods for linear delay parabolic differential equations

This paper is concerned with the asymptotic stability property of some numerical processes by discretization of parabolic differential equations with a constant delay. These numerical processes include forward and backward Euler difference schemes and Crank–Nicolson difference scheme which are obtained by applying step-by-step methods to the resulting systems of delay differential equations. Sufficient and necessary conditions for these difference schemes to be delay-independently asymptotically stable are established. It reveals that an additional restriction on time and spatial stepsizes of the forward Euler difference scheme is required to preserve the delay-independent asymptotic stability due to the existence of the delay term. Numerical experiments have been implemented to confirm the asymptotic stability of these numerical methods.

On the method of modified equations. III. Numerical techniques based on the second equivalent equation for the Euler forward difference method

Direct-correction and asymptotic successive-correction methods based on the second equivalent equation are applied to the Euler forward explicit scheme. In direct-correction, the truncation error terms of the second equivalent equation which contain higher-order derivatives together with a starting procedure, are discretized by means of finite differences. Both explicit and implicit direct-correction schemes are presented and their stability regions are studied. The asymptotic successive-correction numerical technique developed in Part II of this series with a consistent starting procedure is applied to the second equivalent equation. Both all-backward and all-centered asymptotic successive-correction methods are presented. The numerical methods introduced in this paper are applied to autonomous and non-autonomous, scalar and systems of ordinary differential equations and compared with the results of second- and fourth-order accurate Runge–Kutta methods. It is shown that the fourth-order Runge–Kutta method is more accurate than the successive-correction techniques for large time steps due to the need for higher-order derivatives of the Euler solution; however, for sufficiently small time steps, but larger enough so that round-off errors are negligible, both methods have nearly the same accuracy.

On the method of modified equations. IV. Numerical techniques based on the modified equation for the Euler forward difference method

The modified equation method is studied as a technique for the development of new numerical techniques for ordinary differential schemes based on the third modified or (simply) modified equation of the explicit Euler forward method. Both direct-correction and successive-correction techniques based on the modified equation are used to obtain higher-order schemes. The resulting numerical techniques are completely explicit, of order of accuracy as high as desired, and self-starting since the truncation error terms in the modified equation have no derivatives. The methods introduced in this paper are applied to autonomous and non-autonomous, scalar and systems of ordinary differential equations and compared with second- and fourth-order accurate Runge–Kutta schemes. It is shown that, for sufficiently small step sizes, the fourth-order direct-correction and successive-correction methods are as accurate as the fourth-order Runge–Kutta scheme.