Forward Difference Method 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.

An improved discretization‐based kinematic approach for stability analyses of nonuniform c‐φ soil slopes

AbstractThis paper proposes an improved discretization‐based kinematic approach (DKA) with an efficient and robust algorithm to investigate slope stability in nonuniform soils. In an effort to ensure rigorous upper‐bound solutions which may be not satisfied by the initial DKA based on a forward difference method (DKA‐FD), a central and backward difference “point‐to‐point” method (DKA‐CD and DKA‐BD) is proposed to generate discretized points to form a velocity discontinuity surface. Varying (including constant) soil frictional angles along depth are discussed, which can be readily considered in the improved DKA‐CD. Work rate calculations are performed to derive upper‐bound formulations of slope stability number, and critical failure surface is correspondingly obtained at limit state. The comparison with forward and backward difference methods clearly reveals that the improved DKA‐CD could significantly reduce the mesh‐dependency issue and enhance efficacy of slope stability analyses in nonuniform soils.

Partial derivative-based dynamic sensitivity analysis expression for non-linear auto regressive with exogenous (NARX) model[sbnd]case studies on distillation columns and model's interpretation investigation

Constructing the reliable dynamic sensitivity profile for the output variable using the machine learning model is a challenging task; however, the dynamic sensitivity trends are helpful to understand the impact of the input variables on the system's performance. In this paper, we have derived the partial-derivative approach-based sensitivity analysis expression for the non-linear auto regressive with exogenous (NARX) model for the first time. The engineering systems-based case studies, i.e., two distillation columns with five and ten stages, respectively are taken which are commonly found in the chemical processing plants. Two output variables, i.e., liquid composition in tray 2 and tray 4 (Y2 and Y4) of a five-stage distillation column, and liquid composition in tray 7 (Y7) of a ten-stage (higher) distillation column are modelled by NARX with respect to time, feed concentration (Xf) and feed flow rate (Lf). The dynamic sensitivity profiles of the output variables with respect to Xf and Lf for the two distillation columns are plotted by the derived partial derivative-based sensitivity expression on the NARX model. Furthermore, the forward difference method of sensitivity analysis (first principle method) is also applied on the ordinary differential equations of the distillation columns to compute the sensitivity values of the output variables. A good agreement in the dynamic sensitivity values of the output variables with respect to the input variables is found for the two sensitivity analysis techniques thereby demonstrating the effectiveness of the partial-derivative approach for the improved NARX's interpretability performance. This research presents the explicit partial-derivative based sensitivity analysis expression for the NARX model which can be utilised for time-series applications and can provide the insights about the model's interpretation performance.

Variational inequalities of multilayer viscoelastic systems with interlayer Tresca friction: Existence and uniqueness of solution and convergence of numerical solution

The mathematical model of multilayer viscoelastic system based on the actual structure of asphalt pavement and the properties of its numerical solutions are studied. Firstly, based on the partial differential equations, the variational inequality model of the multilayer viscoelastic system is derived, and the existence and uniqueness of its solutions are further proved. Then, based on the finite element theory, the convergence property and error analysis of the semi‐discrete numerical solutions of the variational inequalities are further studied. Finally, the convergence and error analysis of the fully discrete numerical solutions of the variational inequalities are derived by the forward difference method. The conclusion of the above error analysis also confirms the feasibility of studying the mechanical response of asphalt pavement based on variational inequality.

The nonlocal coupled system of Caputo–Fabrizio fractional q‐integro differential equation

This scheme's main goal is to examine the existence, uniqueness, and continuous dependence of solutions for a nonlinear coupled system of fractional q‐integro‐differential equations involving the derivation and integration of fractional Caputo–Fabrizio. The numerical technique methodology of the proposed problem will be introduced. Proving the existence theorem depends on Schauder's fixed‐point theorem. To drive the numerical method, we use the definitions of the fractional derivative and integral of Caputo–Fabrizio and the q‐integral of the Riemann–Liouville type. Then, the integral part will be treated using the trapezoidal method, and the derivative part will be treated using the forward finite difference method. And therefore, the coupled system will be converted into a system of algebraic equation that will be solved together to get the solutions. Finally, we give two examples to illustrate the effectiveness of the suggested approach.

Random vibration and reliability analysis of fluid-conveying pipe under white noise excitations

Due to the random excitation in the external environment, the fluid-conveying pipe would inevitably undergo random vibration. This paper investigates the random vibration and reliability of simply-supported fluid-conveying pipes under white noise excitations. Dynamic equations of the nonlinear pipe conveying fluid under white noise excitations are modeled by the generalized Hamilton principle. Then probability density functions (PDFs) of the amplitude and energy of the system are determined by using the modified stochastic averaging method. The PDFs of displacement and velocity of the system are also discussed. The accuracy of results is verified by the Monte Carlo method. In addition, the first-passage problem of the pipe system is investigated for the first time. In order to obtain the PDFs of reliability and first failure time of the system, the forward finite difference method is adopted to solve the backward Kolmogorov equation of the system. The effects of different parameters on the amplitude, energy, reliability, and first failure time are investigated. The results show that the amplitude of the system becomes large and the first-passage time is advanced with the increase of excitation intensity or fluid speed. The amplitude of the system is decreased and the first-passage time is delayed with the increase of the damping coefficient.

Adaptive Hybrid Mixed Two-Point Step Size Gradient Algorithm for Solving Non-Linear Systems

In this paper, a two-point step-size gradient technique is proposed by which the approximate solutions of a non-linear system are found. The two-point step-size includes two types of parameters deterministic and random. A new adaptive backtracking line search is presented and combined with the two-point step-size gradient to make it globally convergent. The idea of the suggested method depends on imitating the forward difference method by using one point to estimate the values of the gradient vector per iteration where the number of the function evaluation is at most one for each iteration. The global convergence analysis of the proposed method is established under actual and limited conditions. The performance of the proposed method is examined by solving a set of non-linear systems containing high dimensions. The results of the proposed method is compared to the results of a derivative-free three-term conjugate gradient CG method that solves the same test problems. Fair, popular, and sensible evaluation criteria are used for comparisons. The numerical results show that the proposed method has merit and is competitive in all cases and superior in terms of efficiency, reliability, and effectiveness in finding the approximate solution of the non-linear systems.

Equilibrium space and a pseudo linearization of nonlinear systems

This paper attempts to extend the concept of the equilibrium point to what is called equilibrium space, which can adapt to a system in which there exists an infinite number of equilibrium points. In the context of Lyapunov’s linearization method extended for the equilibrium space, this paper proposes a pseudo linearization, from which we can derive a linear representation for a nonlinear system. The equilibrium state of this pseudo linearization and its stability are shown to be the same as that of the original nonlinear system. As an example of the applicability, the proposed pseudo linearization is applied to derive a discrete-time model for a control moment gyroscope system from a nonlinear continuous-time model. Simulation results show that the discrete-time model derived using the proposed pseudo linearization yields responses that are closer to that of the continuous-time model than the discrete-time model derived by the well-known forward-difference method and the conventional pseudo linear representation method, even with a large sampling interval.

Symmetrization of Peaks in Chiral Chromatography with an Area-Invariant Resolution Enhancement Method.

A majority of enantiomeric separations show some degree of peak asymmetry, which is detrimental to quantitative and semiquantitative chiral analysis. This paper presents a simple and rapid peak symmetrization algorithm for the correction or reduction of peak tailing or fronting in exponentially modified Gaussians. Raw chromatographic data can be symmetrized by adding a correct fraction of the first derivative to the chromatogram. The area remains invariant since the area under the first derivative is zero for a pure Gaussian and numerically close to zero for asymmetric peaks. A method of easily extracting the distortion parameter is provided, as well as insight into how pre-smoothing the data with the "perfect smoother" algorithm can suppress high frequencies effectively. The central difference method is also used to compute the first derivative, reducing root-mean-square noise by up to 28% compared to the standard forward difference method. A survey of 40 chiral separations is presented, demonstrating the range of asymmetry observed in chiral separations. Examples of symmetrization of the peaks from enantiomers in comparable and disproportionate concentrations are also provided. Artifacts of deconvolution are discussed, along with methods to mitigate such artifacts.

Derivative-free Newton's method for solving intuitionistic fuzzy nonlinear equations with an application

In this paper, we present a derivative-free Newton’s method that avoids computing the derivative by generating an approximation of the derivative for the intuitionistic fuzzy nonlinear equation. We first consider transforming the intuitionistic fuzzy quantities into their equivalent membership and non-membership parametric forms and insert the approximation from the forward difference method applied to F'(x_k) = 0 in Newton’s method to avoid computing the Jacobian matrix. Numerical experiments were carried out, which shows that the approach is a good option for computing Jacobian and is an efficient one.

Air-Gap Flux Oriented Vector Control Based on Reduced-Order Flux Observer for EESM

Electrically excited synchronous motor (EESM) has the characteristics of high order, nonlinear and strong coupling, so it is difficult to be controlled. However, it has the advantages of adjustable power factor, high efficiency, and high precision torque control, so it is widely used in high-power applications. The accuracy of a flux observer influences the speed control system of EESM. Based on state observer in modern control theory and electrical excitation synchronous machine state equation, a reduced-order flux observer is designed. Using the first-order difference method and forward bilinear transformation method, the reduced-order flux observer is discrete, and the stability of the motor system is analyzed. The analysis shows that the stability of the system using the bilinear transformation method is better than that using the first order forward difference method. In motor operation, motor parameters will be affected by the factors of temperature, magnetic saturation, and motor frequency. In this paper, the influence of parameter variation on the motor system is studied by using the variation of the pole distribution. Finally, the speed regulation system using the reduced-order observer is simulated, which verifies the accuracy of the reduced-order flux observer observation.

A Composite Algorithm for Numerical Solutions of Two-Dimensional Coupled Burgers’ Equations

In this study, a new composite algorithm with the help of the finite difference and the modified cubic trigonometric B-spline differential quadrature method is developed. The developed method was applied to two-dimensional coupled Burgers’ equation with initial and Dirichlet boundary conditions for computational modeling. The established algorithm is better than the traditional differential quadrature algorithm proposed in literature due to more smoothness of cubic trigonometric B-spline functions. In the development of the algorithm, the first step is semidiscretization in time with the forward finite difference method. Furthermore, the obtained system is fully discretized by the modified cubic trigonometric B-spline differential quadrature method. Finally, we obtain coupled Lyapunov systems of linear equations, which are analyzed by the MATLAB solver for the system. Moreover, comparative study of these solutions with the numerical and exact solutions which are appeared in the literature is also discussed. Finally, it is found that there is good suitability between exact solutions and numerical solutions obtained by the developed composite algorithm. The technique can be extended for various multidimensional Burgers’ equations after some modifications.

Numerical Solution of First Order Ordinary Differential Equation by Using Runge-Kutta Method

In this paper, the classical fourth-order Runge-Kutta methodis presented for solving the first-order ordinary differential equation. First, the given solution domain is discretizedby using a uniform discretization grid point. Next by applyingthe forward difference method, we discretized the given ordinary differential equation. And formulating a difference equation. Then using this difference equation, the given first-order ordinary differential equation is solved by using the classicalfourth-order Runge-Kutta method at each specified grid point. To validate the applicability of the proposed method, two model examples are considered and solved at each specific grid point on its solution domain. The stability and convergent analysis of the present method is worked by supportedthe theoretical and mathematical statementsand the accuracy of the solution is obtained. The accuracy of the present methodhas been shown in the sense ofmaximumabsolute error and the local behavior of the solution is captured exactly. Numerical and exact solutions have been presented in tables and graphs and the corresponding maximumabsolute errorisalso presented in tables and graphs. The present method approximates the exact solution very well and it is quite efficient and practically well suitedfor solving first-order ordinary differential equations. The numerical result presented in tables and graphsindicates that the approximate solution is in good agreement with the exact solution. Hence the proposed method is accruable to solve ordinary differential equations.

Takagi–Sugeno fuzzy sampled‐data model‐based less conservative stability criterion for a Francis hydropower unit's governing system

In this study, the stability analysis and controller design of a hydropower unit's governing system (HUGS) were studied based on a Takagi–Sugeno (T–S) fuzzy model and constant sampled-data control. First, according to the T–S fuzzy theory, the non-linear Francis hydro-turbine governing system under rigid water hammer is linearised and then the approximate linear system was obtained. Second, a fuzzy sampled-data controller was designed in the case of periodic sampling, and the closed-loop sampling system was discretised. After constructing the Lyapunov function and using the forward difference method, the stabilisation condition was given with less conservatism in a symmetrical linear matrix inequality form. Finally, the numerical simulation results showed that under four different sampling periods, the HUGS can quickly achieve stability and that it has different stability performance. In addition, the superiority of the controller was verified in comparison with traditional proportional–integral–derivative control and fuzzy control techniques.

On the nonstandard numerical discretization of SIR epidemic model with a saturated incidence rate and vaccination

Recently, Hoang and Egbelowo (Boletin de la Sociedad Matemàtica Mexicana, 2020) proposed a nonstandard finite difference scheme (NSFD) to get a discrete SIR epidemic model with saturated incidence rate and constant vaccination. The discrete model was derived by discretizing the right-hand sides of the system locally and the first order derivative is approximated by the generalized forward difference method but with a restrictive denominator function. Their analysis showed that the NSFD scheme is dynamically-consistent only for relatively small time-step sizes. In this paper, we propose and analyze an alternative NSFD scheme by applying nonlocal approximation and choosing the denominator function such that the proposed scheme preserves the boundedness of solutions. It is verified that the proposed discrete model is dynamically-consistent with the corresponding continuous model for all time-step size. The analytical results have been confirmed by some numerical simulations. We also show numerically that the proposed NSFD scheme is superior to the Euler method and the NSFD method proposed by Hoang and Egbelowo (2020).

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.

Reliability assessment of a planar steel frame subjected to earthquakes in case of an implicit limit-state function

This study determines the probability of failure of a planar steel frame using a proposed framework and Monte Carlo simulation (MCS) considering the uncertainties of the material and cross-section parameters. A new framework (O-FCD method) is proposed by combining the OpenSees software for structural analysis and the first-order reliability method (FORM) implemented using MATLAB for analyzing the structural reliability to determine the failure probability of a case study structure with the implicit limit-state function. The limit-state conditions of this case study structure are defined by the limitations in two scenarios: the total drift defined as the ratio of the maximum top displacement to the building height of 3%, and the maximum interstory drift of 5%. Both static and dynamic loads are applied to compare the results of the O-FCD method and MCS. The investigations show that the O-FCD method can determine well the probability of failure of this case study structure. The absolute errors of the proposed framework and MCS are slightly different under both static and dynamic loads. In addition, the O-FCD method can solve the computationally expensive part of MCS in the case of a lower failure probability. Finally, the O-FCD method uses the finite central difference method to calculate the gradient vector in FORM numerically. The proposed framework can reach the convergence criteria with numerous random variables compared with a framework using the forward finite difference method.

Discrete-Inverse Optimal Control Applied to the Ball and Beam Dynamical System: A Passivity-Based Control Approach

This express brief deals with the problem of the state variables regulation in the ball and beam system by applying the discrete-inverse optimal control approach. The ball and beam system model is defined by a set of four-order nonlinear differential equations that are discretized using the forward difference method. The main advantages of using the discrete-inverse optimal control to regulate state variables in dynamic systems are (i) the control input is an optimal signal as it guarantees the minimum of the Hamiltonian function, (ii) the control signal makes the dynamical system passive, and (iii) the control input ensures asymptotic stability in the sense of Lyapunov. Numerical simulations in the MATLAB environment allow demonstrating the effectiveness and robustness of the studied control design for state variables regulation with a wide gamma of dynamic behaviors as a function of the assigned control gains.

A locally stabilized explicit approach for coupled thermo-mechanical analysis

In this paper, a stabilized, locally defined, explicit approach is considered to analyse coupled thermo-mechanical models. In this sense, modified central and forward difference methods are adaptively applied following the features of the discretized model; thus, a connected formulation between the adopted temporal and spatial solution procedures takes place. The proposed technique enables stabilized decoupled analyses, allowing each “phase” of the coupled model to be handled separately, without considering stability restrictions for their time solutions, providing a very versatile and efficient methodology. In addition, the new formulation is designed as a single-solve framework based on reduced systems of equations, which further greatly improves the efficiency of the method. The new technique is accurate, simple to implement and entirely automated, requiring no decision or expertise from the user. Numerical results are presented at the end of the manuscript, illustrating the performance and effectiveness of the new procedure.

Semi-Closed Form Prices of Barrier Options in the Time-Dependent CEV and CIR Models

The authors continue a series of articles where prices of the barrier options written on the underlying, whose dynamics follow a one-factor stochastic model with time-dependent coefficients and the barrier, are obtained in semi-closed form; see Carr and Itkin (2020) and Itkin and Muravey (2020). This article extends this methodology to the Cox–Ingersoll–Ross model for zero-coupon bonds and to the constant elasticity of variance model for stocks, which are used as the corresponding underlying for the barrier options. The authors describe two approaches. One is a generalization of the method of heat potentials (for the heat equation) to the Bessel process, so it is named “the method of Bessel potentials.” The authors also propose a general scheme for constructing the potential method for any linear differential operator with time-independent coefficients. The second approach is the method of generalized integral transform, which is also extended to the Bessel process. In all cases, a semi-closed solution means that first, we need to numerically solve a linear Volterra equation of the second kind, and then the option price is represented as a one-dimensional integral. The authors demonstrate that their method is computationally more efficient than both the backward and forward finite difference methods, while providing better accuracy and stability. Also, it is shown that these methods do not duplicate, but instead they complement each other, as one provides very accurate results at shorter maturities and the other provides such results at longer maturities. TOPICS:Options, statistical methods Key Findings • This article extends this methodology to the CIR model for zero-coupon bonds and to the CEV model for stocks, which are used as the corresponding underlying for the barrier options. • One approach is a generalization of the method of heat potentials (for the heat equation) to the Bessel process, so we call it “the method of Bessel potentials.” The second approach is the method of generalized integral transform, which is also extended to the Bessel process. • These methods do not duplicate, but instead they complement each other, as one provides very accurate results at shorter maturities and the other provides such results at longer maturities.