Search results for: equation error

Authors: Chor Nejmeddine, Ines Ben Omrane, Mohamed Abdelwahed

Abstract:

In this paper, we present a posteriori analysis of the discretization of the heat equation by spectral element method. We apply Euler's implicit scheme in time and spectral method in space. We propose two families of error indicators, both of which are built from the residual of the equation and we prove that they satisfy some optimal estimates. We present some numerical results which are coherent with the theoretical ones.

Keywords: heat equation, spectral elements discretization, error indicators, Euler

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Authors: Abdulrahman Abdulrahman

Abstract:

When the Manning equation is used, a unique value of normal depth in the uniform flow exists for a given channel geometry, discharge, roughness, and slope. Depending on the value of normal depth relative to the critical depth, the flow type (supercritical or subcritical) for a given characteristic of channel conditions is determined whether or not flow is uniform. There is no general solution of Manning's equation for determining the flow depth for a given flow rate, because the area of cross section and the hydraulic radius produce a complicated function of depth. The familiar solution of normal depth for a rectangular channel involves 1) a trial-and-error solution; 2) constructing a non-dimensional graph; 3) preparing tables involving non-dimensional parameters. Author in this paper has derived semi-analytical solution to Manning's equation for determining the flow depth given the flow rate in rectangular open channel. The solution was derived by expressing Manning's equation in non-dimensional form, then expanding this form using Maclaurin's series. In order to simplify the solution, terms containing power up to 4 have been considered. The resulted equation is a quartic equation with a standard form, where its solution was obtained by resolving this into two quadratic factors. The proposed solution for Manning's equation is valid over a large range of parameters, and its maximum error is within -1.586%.

Keywords: channel design, civil engineering, hydraulic engineering, open channel flow, Manning's equation, normal depth, uniform flow

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Authors: Dalvinder Kaur Mangal

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For the past decade, noise corrupted output measurements have been a fundamental research problem to be investigated. On the other hand, the estimation of the parameters for linear dynamic systems when also the input is affected by noise is recognized as more difficult problem which only recently has received increasing attention. Representations where errors or measurement noises/disturbances are present on both the inputs and outputs are usually called errors-in-variables (EIV) models. These disturbances may also have additive effects which are also considered in this paper. Parameter estimation of false EIV problem using equation error, output error and iterative prefiltering identification schemes with and without additive uncertainty, when only the output observation is corrupted by noise has been dealt in this paper. The comparative study of these three schemes has also been carried out.

Keywords: errors-in-variable (EIV), false EIV, equation error, output error, iterative prefiltering, Gaussian noise

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Authors: S. O. Oyamakin, A. U. Chukwu

Abstract:

Richard's growth equation being a generalized logistic growth equation was improved upon by introducing an allometric parameter using the hyperbolic sine function. The integral solution to this was called hyperbolic Richard's growth model having transformed the solution from deterministic to a stochastic growth model. Its ability in model prediction was compared with the classical Richard's growth model an approach which mimicked the natural variability of heights/diameter increment with respect to age and therefore provides a more realistic height/diameter predictions using the coefficient of determination (R2), Mean Absolute Error (MAE) and Mean Square Error (MSE) results. The Kolmogorov-Smirnov test and Shapiro-Wilk test was also used to test the behavior of the error term for possible violations. The mean function of top height/Dbh over age using the two models under study predicted closely the observed values of top height/Dbh in the hyperbolic Richard's nonlinear growth models better than the classical Richard's growth model.

Keywords: height, Dbh, forest, Pinus caribaea, hyperbolic, Richard's, stochastic

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Authors: Itissam Abuiziah

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In several hydraulic systems, it is necessary to calculate the head losses which depend on the resistance flow friction factor in Darcy equation. Computing the resistance friction is based on implicit Colebrook-White equation which is considered as the standard for the friction calculation, but it needs high computational cost, therefore; several explicit approximation methods are used for solving an implicit equation to overcome this issue. It follows that the relative error is used to determine the most accurate method among the approximated used ones. Steel, cast iron and polyethylene pipe materials investigated with practical diameters ranged from 0.1m to 2.5m and velocities between 0.6m/s to 3m/s. In short, the results obtained show that the suitable method for some cases may not be accurate for other cases. For example, when using steel pipe materials, Zigrang and Silvester's method has revealed as the most precise in terms of low velocities 0.6 m/s to 1.3m/s. Comparatively, Halland method showed a less relative error with the gradual increase in velocity. Accordingly, the simulation results of this study might be employed by the hydraulic engineers, so they can take advantage to decide which is the most applicable method according to their practical pipe system expectations.

Keywords: Colebrook–White, explicit equation, friction factor, hydraulic resistance, implicit equation, Reynolds numbers

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Authors: Johnson Oladele Fatokun, I. P. Akpan

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In this paper, the one-dimensional time dependent Schrödinger equation is discretized by the method of lines using a second order finite difference approximation to replace the second order spatial derivative. The evolving system of stiff ordinary differential equation (ODE) in time is solved numerically by an L-stable trapezoidal-like integrator. Results show accuracy of relative maximum error of order 10-4 in the interval of consideration. The performance of the method as compared to an existing scheme is considered favorable.

Keywords: Schrodinger’s equation, partial differential equations, method of lines (MOL), stiff ODE, trapezoidal-like integrator

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Authors: Kewei Xu, Jens Friedrich, Kevin Dwinger, Wei Fan, Xijin Zhang

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Parallel Compressor Model (PCM) is a simplified approach to predict compressor performance with inlet distortions. In PCM calculation, it is assumed that the sub-compressors’ outlet static pressure is uniform and therefore simplifies PCM calculation procedure. However, if the compressor’s outlet duct is not long and straight, such assumption frequently induces error ranging from 10% to 15%. This paper provides a revised calculation method of PCM that can correct the error. The revised method employs energy equation, momentum equation and continuity equation to acquire needed parameters and replace the equal static pressure assumption. Based on the revised method, PCM is applied on two compression system with different blades types. The predictions of their performance in non-uniform inlet conditions are yielded through the revised calculation method and are employed to evaluate the method’s efficiency. Validating the results by experimental data, it is found that although little deviation occurs, calculated result agrees well with experiment data whose error ranges from 0.1% to 3%. Therefore, this proves the revised calculation method of PCM possesses great advantages in predicting the performance of the distorted compressor with limited exhaust duct.

Keywords: parallel compressor model (pcm), revised calculation method, inlet distortion, outlet unequal pressure distribution

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Authors: Samuel Oluwafemi Oyamakin, Angela Unna Chukwu

Abstract:

Richrads growth equation being a generalized logistic growth equation was improved upon by introducing an allometric parameter using the hyperbolic sine function. The integral solution to this was called hyperbolic Richards growth model having transformed the solution from deterministic to a stochastic growth model. Its ability in model prediction was compared with the classical Richards growth model an approach which mimicked the natural variability of heights/diameter increment with respect to age and therefore provides a more realistic height/diameter predictions using the coefficient of determination (R2), Mean Absolute Error (MAE) and Mean Square Error (MSE) results. The Kolmogorov-Smirnov test and Shapiro-Wilk test was also used to test the behavior of the error term for possible violations. The mean function of top height/Dbh over age using the two models under study predicted closely the observed values of top height/Dbh in the hyperbolic Richards nonlinear growth models better than the classical Richards growth model.

Keywords: height, diameter at breast height, DBH, hyperbolic sine function, Pinus caribaea, Richards' growth model

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Authors: Houcine Naim, Abdelatif Hassini, Noureddine Benabadji, Alex Van Den Bossche

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In this paper, two models to estimate the overall monthly average daily radiation on a horizontal surface were applied to the site of Oran (35.38 ° N, 0.37 °W). We present a comparison between the first one is a regression equation of the Angstrom type and the second model is developed by the present authors some modifications were suggested using as input parameters: the astronomical parameters as (latitude, longitude, and altitude) and meteorological parameters as (relative humidity). The comparisons are made using the mean bias error (MBE), root mean square error (RMSE), mean percentage error (MPE), and mean absolute bias error (MABE). This comparison shows that the second model is closer to the experimental values that the model of Angstrom.

Keywords: meteorology, global radiation, Angstrom model, Oran

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Authors: Chen-Yin Kuo, Yung-Hsin Lee

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Previous studies applying residual income valuation (RIV) model generally use panel data and single-equation model to forecast stock prices. Unlike these, this paper uses Taiwan longitudinal data to estimate multi-equation time-series models such as Vector Autoregressive (VAR), Vector Error Correction Model (VECM), and conduct out-of-sample forecasting. Further, this work assesses their forecasting performance by two instruments. In favor of extant research, the major finding shows that VECM outperforms other three models in forecasting for three stock sectors over entire horizons. It implies that an error correction term containing long-run information contributes to improve forecasting accuracy. Moreover, the pattern of composite shows that at longer horizon, VECM produces the greater reduction in errors, and performs substantially better than VAR.

Keywords: residual income valuation model, vector error correction model, out of sample forecasting, forecasting accuracy

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Authors: Shuting Ji, Yueming Zhang, Jing Zhao

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The output error of the globoidal cam mechanism can be considered as a relevant indicator of mechanism performance, because it determines kinematic and dynamical behavior of mechanical transmission. Based on the differential geometry and the rigid body transformations, the mathematical model of surface geometry of the globoidal cam is established. Then we present the analytical expression of the output error (including the transmission error and the displacement error along the output axis) by considering different manufacture and assembly errors. The effects of the center distance error, the perpendicular error between input and output axes and the rotational angle error of the globoidal cam on the output error are systematically analyzed. A globoidal cam mechanism which is widely used in automatic tool changer of CNC machines is applied for illustration. Our results show that the perpendicular error and the rotational angle error have little effects on the transmission error but have great effects on the displacement error along the output axis. This study plays an important role in the design, manufacture and assembly of the globoidal cam mechanism.

Keywords: globoidal cam mechanism, manufacture error, transmission error, automatic tool changer

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Authors: Ranajay Bhowmick

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Analysis of stresses for an infinitesimal tetrahedron leads to a situation where we obtain a cubic equation consisting of three stress invariants. This cubic equation, when solved for a definite condition, gives the principal stresses directly without requiring any cumbersome and time-consuming trial and error methods or iterative numerical procedures. Since the failure criterion of different materials are generally expressed as functions of principal stresses, an attempt has been made in this study to incorporate the solutions of the cubic equation in the form of principal stresses, obtained for a definite condition, into some of the established failure theories to determine their modified descriptions. It has been observed that the failure theories can be represented using the quadratic stress invariant and the orientation of the principal plane.

Keywords: cubic equation, stress invariant, trigonometric, explicit solution, principal stress, failure criterion

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Authors: Shazalina Mat Zin, Ahmad Abd. Majid, Ahmad Izani Md. Ismail, Muhammad Abbas

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The generalized wave equation models various problems in sciences and engineering. In this paper, a new three-time level implicit approach based on cubic trigonometric B-spline for the approximate solution of wave equation is developed. The usual finite difference approach is used to discretize the time derivative while cubic trigonometric B-spline is applied as an interpolating function in the space dimension. Von Neumann stability analysis is used to analyze the proposed method. Two problems are discussed to exhibit the feasibility and capability of the method. The absolute errors and maximum error are computed to assess the performance of the proposed method. The results were found to be in good agreement with known solutions and with existing schemes in literature.

Keywords: collocation method, cubic trigonometric B-spline, finite difference, wave equation

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Authors: Satyanarayana Engu, Ahmed Mohd, V. Murugan

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We study the large time asymptotics of solutions to the Cauchy problem for a forced Burgers equation (FBE) with the initial data, which is continuous and summable on R. For which, we first derive explicit solutions of FBE assuming a different class of initial data in terms of Hermite polynomials. Later, by violating this assumption we prove the existence of a solution to the considered Cauchy problem. Finally, we give an asymptotic approximate solution and establish that the error will be of order O(t^(-1/2)) with respect to L^p -norm, where 1≤p≤∞, for large time.

Keywords: Burgers equation, Cole-Hopf transformation, Hermite polynomials, large time asymptotics

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Authors: Somveer Singh, Vineet Kumar Singh

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This work contributes a numerical method based on Legendre wavelet approximation for the treatment of partial integro-differential equation (PIDE). Operational matrices of Legendre wavelets reduce the solution of PIDE into the system of algebraic equations. Some useful results concerning the computational order of convergence and error estimates associated to the suggested scheme are presented. Illustrative examples are provided to show the effectiveness and accuracy of proposed numerical method.

Keywords: legendre wavelets, operational matrices, partial integro-differential equation, viscoelasticity

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Authors: Carla E. O. de Moraes, Gladson O. Antunes, Mauro A. Rincon

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The aim of this paper is to study the internal stabilization of the Bernoulli-Euler equation numerically. For this, we consider a square plate subjected to a feedback/damping force distributed only in a subdomain. An algorithm for obtaining an approximate solution to this problem was proposed and implemented. The numerical method used was the Finite Difference Method. Numerical simulations were performed and showed the behavior of the solution, confirming the theoretical results that have already been proved in the literature. In addition, we studied the validation of the numerical scheme proposed, followed by an analysis of the numerical error; and we conducted a study on the decay of the energy associated.

Keywords: Bernoulli-Euler plate equation, numerical simulations, stability, energy decay, finite difference method

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Authors: Guesh Simretab Gebremedhin, Saumya Rajan Jena

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In this work, a septic B-spline scheme has been used to simplify the process of solving an approximate solution of the generalized Rosenau-regularized long-wave equation (GR-RLWE) with initial boundary conditions. The resulting system of first-order ODEs has dealt with Butcher’s fifth order Runge-Kutta (BFRK) approach without using finite difference techniques for discretizing the time-dependent variables at each time level. Here, no transformation or any kind of linearization technique is employed to tackle the nonlinearity of the equation. Two test problems have been selected for numerical justifications and comparisons with other researchers on the basis of efficiency, accuracy, and results of the two invariants Mᵢ (mass) and Eᵢ (energy) of some motion that has been used to test the conservative properties of the proposed scheme.

Keywords: septic B-spline scheme, Butcher's fifth order Runge-Kutta approach, error norms, generalized Rosenau-RLW equation

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Authors: Lawrence A. Farinola

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Four point implicit schemes for the approximation of the first and pure second order derivatives for the solution of the Dirichlet problem for one dimensional Schrodinger equation with respect to the time variable t were constructed. Also, special four-point implicit difference boundary value problems are proposed for the first and pure second derivatives of the solution with respect to the spatial variable x. The Grid method is also applied to the mixed second derivative of the solution of the Linear Schrodinger time-dependent equation. It is assumed that the initial function belongs to the Holder space C⁸⁺ᵃ, 0 < α < 1, the Schrodinger wave function given in the Schrodinger equation is from the Holder space Cₓ,ₜ⁶⁺ᵃ, ³⁺ᵃ/², the boundary functions are from C⁴⁺ᵃ, and between the initial and the boundary functions the conjugation conditions of orders q = 0,1,2,3,4 are satisfied. It is proven that the solution of the proposed difference schemes converges uniformly on the grids of the order O(h²+ k) where h is the step size in x and k is the step size in time. Numerical experiments are illustrated to support the analysis made.

Keywords: approximation of derivatives, finite difference method, Schrödinger equation, uniform error

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Authors: Riki Dutta, Sagardeep Talukdar, Gautam Kumar Saharia, Sudipta Nandy

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Fokas-Lenells equation (FLE) is one of the integrable nonlinear equations use to describe the propagation of ultrashort optical pulses in an optical medium. A 2x2 Lax pair has been introduced for the FLE and from that solving the Riccati equation yields infinitely many conserved quantities. Thereafter for a new field function (S) of the Landau-Lifshitz (LL) system, a gauge equivalence of the FLE with the generalised LL equation has been derived. We hope our findings are useful for the application purpose of FLE in optics and other branches of physics.

Keywords: conserved quantities, fokas-lenells equation, landau-lifshitz equation, lax pair

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Authors: Jian-Jun Shu

Abstract:

It is common knowledge that many physical problems (such as non-linear shallow-water waves and wave motion in plasmas) can be described by the Korteweg-de Vries (KdV) equation, which possesses certain special solutions, known as solitary waves or solitons. As a marriage of the KdV equation and the classical Burgers (KdVB) equation, the Korteweg-de Vries-Burgers (KdVB) equation is a mathematical model of waves on shallow water surfaces in the presence of viscous dissipation. Asymptotic analysis is a method of describing limiting behavior and is a key tool for exploring the differential equations which arise in the mathematical modeling of real-world phenomena. By using variable transformations, the asymptotic expansion of the KdVB equation is presented in this paper. The asymptotic expansion may provide a good gauge on the validation of the corresponding numerical scheme.

Keywords: asymptotic expansion, differential equation, Korteweg-de Vries-Burgers (KdVB) equation, soliton

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Authors: E. Marušić-Paloka, I. Pažanin, M. Prša

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Subject of this study is the stationary heat conduction problem through a pipe filled with incompressible viscous fluid. In previous work, we observed the existence and uniqueness theorems for the corresponding boundary-value problem and within we have taken into account the effects of the pipe's dilatation due to the temperature of the fluid inside of the pipe. The main difficulty comes from the fact that flow domain changes depending on the solution of the observed heat equation leading to a non-standard coupled governing problem. The goal of this work is to find solution estimate since the exact solution of the studied problem is not possible to determine. We use an asymptotic expansion in order of a small parameter which is presented as a heat expansion coefficient of the pipe's material. Furthermore, an error estimate is provided for the mentioned asymptotic approximation of the solution for inner area of the pipe. Close to the boundary, problem becomes more complex so different approaches are observed, mainly Theory of Perturbations and Separations of Variables. In view of that, error estimate for the whole approximation will be provided with additional software simulations of gotten situation.

Keywords: asymptotic analysis, dilated pipe, error estimate, heat conduction

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Authors: Martin K. Steiger, Lukas Heisler, Hans-Georg Brachtendorf

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Deep neural networks and their variants form the backbone of many AI applications. Based on the so-called residual networks, a continuous formulation of such models as ordinary differential equations (ODEs) has proven advantageous since different techniques may be applied that significantly increase the learning speed and enable controlled trade-offs with the resulting error at the same time. For the evaluation of such models, high-performance numerical differential equation solvers are used, which also provide the gradients required for training. However, whether classical gradient-based methods are even applicable or which one yields the best results has not been discussed yet. This paper aims to redeem this situation by providing empirical results for different applications.

Keywords: deep neural networks, gradient-based learning, image processing, ordinary differential equation networks

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Authors: Y. Pala, M. O. Ertas

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In this paper, the general Riccati equation is analytically solved by a new transformation. By the method developed, looking at the transformed equation, whether or not an explicit solution can be obtained is readily determined. Since the present method does not require a proper solution for the general solution, it is especially suitable for equations whose proper solutions cannot be seen at first glance. Since the transformed second order linear equation obtained by the present transformation has the simplest form that it can have, it is immediately seen whether or not the original equation can be solved analytically. The present method is exemplified by several examples.

Keywords: Riccati equation, analytical solution, proper solution, nonlinear

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Authors: A. B. Bolkhir, A. Elshafie, T. K. Yousif

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This paper highlights the methods of error estimation in finite element analysis (FEA) results. It indicates that the modeling error could be eliminated by performing finite element analysis with successively finer meshes or by extrapolating response predictions from an orderly sequence of relatively low degree of freedom analysis results. In addition, the paper eliminates the round-off error by running the code at a higher precision. The paper provides application in finite element analysis results. It draws a conclusion based on results of application of methods of error estimation.

Keywords: finite element analysis (FEA), discretization error, round-off error, mesh refinement, richardson extrapolation, monotonic convergence

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Authors: Sharon-Yasotha Veerayah-Mcgregor, Valipuram Manoranjan

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A backward or inverse problem is known to be an ill-posed problem due to its instability that easily emerges with any slight change within the conditions of the problem. Therefore, only a limited number of numerical approaches are available to solve a backward problem. This paper considers the Nagumo equation, an equation that describes impulse propagation in nerve axons, which also models population growth with the Allee effect. A creative operator splitting numerical scheme is constructed to solve the inverse Nagumo equation. Computational simulations are used to verify that this scheme is stable, accurate, and efficient.

Keywords: inverse/backward equation, operator-splitting, Nagumo equation, ill-posed, finite-difference

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Authors: Merouane Salhi, Toufik Zebbiche

Abstract:

When the stagnation pressure of perfect gas increases, the specific heat and their ratio do not remain constant anymore and start to vary with this pressure. The gas doesn’t remain perfect. Its state equation change and it becomes for a real gas. In this case, the effects of molecular size and intermolecular attraction forces intervene to correct the state equation. The aim of this work is to show and discuss the effect of stagnation pressure on supersonic thermodynamical, physical and geometrical flow parameters, to find a general case for real gas. With the assumptions that Berthelot’s state equation accounts for the molecular size and intermolecular force effects, expressions are developed for analyzing supersonic flow for thermally and calorically imperfect gas lower than the dissociation molecules threshold. The designs parameters for supersonic nozzle like thrust coefficient depend directly on stagnation parameters of the combustion chamber. The application is for air. A computation of error is made in this case to give a limit of perfect gas model compared to real gas model.

Keywords: supersonic flow, real gas model, Berthelot’s state equation, Simpson’s method, condensation function, stagnation pressure

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Authors: Saeed Otarod

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In a different simple and straight forward analysis a closed-form integral solution is found for nonhomogeneous second order linear ordinary differential equations, in terms of a particular solution of their corresponding homogeneous part. To find the particular solution of the homogeneous part, the equation is transformed into a simple Riccati equation from which the general solution of non-homogeneouecond order differential equation, in the form of a closed integral equation is inferred. The method works well in manyimportant cases, such as Schrödinger equation for hydrogen-like atoms. A non-homogenous second order linear differential equation has been solved as an extra example

Keywords: explicit, linear, differential, closed form

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Authors: Yuan Yan Tang, Lina Yang, Hong Li

Abstract:

Harmonic model is a very important approximation for the image transform. The harmanic model converts an image into arbitrary shape; however, this mode cannot be described by any fixed functions in mathematics. In fact, it is represented by partial differential equation (PDE) with boundary conditions. Therefore, to develop an efficient method to solve such a PDE is extremely significant in the image transform. In this paper, a novel Integral Equation-Wavelet based method is presented, which consists of three steps: (1) The partial differential equation is converted into boundary integral equation and representation by an indirect method. (2) The boundary integral equation and representation are changed to plane integral equation and representation by boundary measure formula. (3) The plane integral equation and representation are then solved by a method we call wavelet collocation. Our approach has two main advantages, the shape of an image is arbitrary and the program code is independent of the boundary. The performance of our method is evaluated by numerical experiments.

Keywords: harmonic model, partial differential equation (PDE), integral equation, integral representation, boundary measure formula, wavelet collocation

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Authors: Danni Cong, Meiping Wu, Xiaofeng He, Junxiang Lian, Juliang Cao, Shaokuncai, Hao Qin

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Gravity gradient instrument (GGI) is the core of the gravity gradiometer, so the structural error of the sensor has a great impact on the measurement results. In order not to affect the aimed measurement accuracy, limit error is required in the installation of the accelerometer. In this paper, based on the established measuring principle model, the radial installation limit error is calibrated, which is taken as an example to provide a method to calculate the other limit error of the installation under the premise of ensuring the accuracy of the measurement result. This method provides the idea for deriving the limit error of the geometry structure of the sensor, laying the foundation for the mechanical precision design and physical design.

Keywords: gravity gradient sensor, radial installation limit error, accelerometer, uniaxial rotational modulation

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Authors: Tadas Telksnys, Zenonas Navickas, Minvydas Ragulskis

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Necessary and sufficient conditions for the existence of second order solitary solutions to the Hodgkin-Huxley equation are derived in this paper. The generalized multiplicative operator of differentiation helps not only to construct closed-form solitary solutions but also automatically generates conditions of their existence in the space of the equation's parameters and initial conditions. It is demonstrated that bright, kink-type solitons and solitary solutions with singularities can exist in the Hodgkin-Huxley equation.

Keywords: Hodgkin-Huxley equation, solitary solution, existence condition, operator method

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