scholarly journals Some Novel Sixth-Order Iteration Schemes for Computing Zeros of Nonlinear Scalar Equations and Their Applications in Engineering

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
M. A. Rehman ◽  
Amir Naseem ◽  
Thabet Abdeljawad
Keyword(s):  
Open Channel ◽  
Flow Problem ◽  
Flame Temperature ◽  
Real Life ◽  
Sixth Order ◽  
One Dimension ◽  
Interpolation Technique ◽  
Generalized Newton ◽  
Scalar Equations ◽  
Taylor’S Series

In this paper, we propose two novel iteration schemes for computing zeros of nonlinear equations in one dimension. We develop these iteration schemes with the help of Taylor’s series expansion, generalized Newton-Raphson’s method, and interpolation technique. The convergence analysis of the proposed iteration schemes is discussed. It is established that the newly developed iteration schemes have sixth order of convergence. Several numerical examples have been solved to illustrate the applicability and validity of the suggested schemes. These problems also include some real-life applications associated with the chemical and civil engineering such as adiabatic flame temperature equation, conversion of nitrogen-hydrogen feed to ammonia, the van der Wall’s equation, and the open channel flow problem whose numerical results prove the better efficiency of these methods as compared to other well-known existing iterative methods of the same kind.

scholarly journals Numerical Algorithms for Finding Zeros of Nonlinear Equations and Their Dynamical Aspects

2020 ◽  
Vol 2020 ◽  
pp. 1-11 ◽  
Author(s):  
Amir Naseem ◽  
M. A. Rehman ◽  
Thabet Abdeljawad ◽  
Francisco Balibrea
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Numerical Algorithms ◽  
One Dimension ◽  
Second Derivative ◽  
Numerical Examples ◽  
Derivative Free ◽  
Interpolation Technique ◽  
Taylor’S Series

In this paper, we developed two new numerical algorithms for finding zeros of nonlinear equations in one dimension and one of them is second derivative free which has been removed using the interpolation technique. We derive these algorithms with the help of Taylor’s series expansion and Golbabai and Javidi’s method. The convergence analysis of these algorithms is discussed. It is established that the newly developed algorithms have sixth order of convergence. Several numerical examples have been solved which prove the better efficiency of these algorithms as compared to other well-known iterative methods of the same kind. Finally, the comparison of polynomiographs generated by other well-known iterative methods with our developed algorithms has been made which reflects their dynamical aspects.

scholarly journals Some Real-Life Applications of a Newly Constructed Derivative Free Iterative Scheme

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 239 ◽  
Author(s):  
Ramandeep Behl ◽  
M. Salimi ◽  
M. Ferrara ◽  
S. Sharifi ◽  
Samaher Alharbi
Keyword(s):  
Iterative Methods ◽  
Chemical Engineering ◽  
Flame Temperature ◽  
Real Life ◽  
Chemical Reactor ◽  
Basins Of Attraction ◽  
Eighth Order ◽  
Present Scheme ◽  
New Family ◽  
Derivative Free

In this study, we present a new higher-order scheme without memory for simple zeros which has two major advantages. The first one is that each member of our scheme is derivative free and the second one is that the present scheme is capable of producing many new optimal family of eighth-order methods from every 4-order optimal derivative free scheme (available in the literature) whose first substep employs a Steffensen or a Steffensen-like method. In addition, the theoretical and computational properties of the present scheme are fully investigated along with the main theorem, which demonstrates the convergence order and asymptotic error constant. Moreover, the effectiveness of our scheme is tested on several real-life problems like Van der Waal’s, fractional transformation in a chemical reactor, chemical engineering, adiabatic flame temperature, etc. In comparison with the existing robust techniques, the iterative methods in the new family perform better in the considered test examples. The study of dynamics on the proposed iterative methods also confirms this fact via basins of attraction applied to a number of test functions.

An Optimal Reconstruction of Chebyshev–Halley-Type Methods with Local Convergence Analysis

2019 ◽  
Vol 17 (05) ◽  
pp. 1940017
Author(s):  
Ali Saleh Alshomrani ◽  
Ioannis K. Argyros ◽  
Ramandeep Behl
Keyword(s):  
Local Convergence ◽  
Real Life ◽  
Eighth Order ◽  
Dynamical Study ◽  
First Order ◽  
Life Problems ◽  
Local Convergence Analysis ◽  
Lipschitz Conditions ◽  
Scalar Equations ◽  
Basins Of Attractions

Our principle aim in this paper is to present a new reconstruction of classical Chebyshev–Halley schemes having optimal fourth and eighth-order of convergence for all parameters [Formula: see text] unlike in the earlier studies. In addition, we analyze the local convergence of them by using hypotheses requiring the first-order derivative of the involved function [Formula: see text] and the Lipschitz conditions. In addition, we also formulate their theoretical radius of convergence. Several numerical examples originated from real life problems demonstrate that they are applicable to a broad range of scalar equations, where previous studies cannot be used. Finally, a dynamical study of them also demonstrates that bigger and more promising basins of attractions are obtained.

scholarly journals Ensemble modeling of stochastic unsteady open-channel flow in terms of its time–space evolutionary probability distribution – Part 2: numerical application

2018 ◽  
Vol 22 (3) ◽  
pp. 2007-2021 ◽  
Author(s):  
Alain Dib ◽  
M. Levent Kavvas
Keyword(s):  
Probability Distribution ◽  
Channel Flow ◽  
Open Channel ◽  
Flow Problem ◽  
Planck Equation ◽  
Open Channel Flow ◽  
Computational Time ◽  
Roughness Coefficient ◽  
Time Space ◽  
Flow Variables

Abstract. The characteristic form of the Saint-Venant equations is solved in a stochastic setting by using a newly proposed Fokker–Planck Equation (FPE) methodology. This methodology computes the ensemble behavior and variability of the unsteady flow in open channels by directly solving for the flow variables' time–space evolutionary probability distribution. The new methodology is tested on a stochastic unsteady open-channel flow problem, with an uncertainty arising from the channel's roughness coefficient. The computed statistical descriptions of the flow variables are compared to the results obtained through Monte Carlo (MC) simulations in order to evaluate the performance of the FPE methodology. The comparisons show that the proposed methodology can adequately predict the results of the considered stochastic flow problem, including the ensemble averages, variances, and probability density functions in time and space. Unlike the large number of simulations performed by the MC approach, only one simulation is required by the FPE methodology. Moreover, the total computational time of the FPE methodology is smaller than that of the MC approach, which could prove to be a particularly crucial advantage in systems with a large number of uncertain parameters. As such, the results obtained in this study indicate that the proposed FPE methodology is a powerful and time-efficient approach for predicting the ensemble average and variance behavior, in both space and time, for an open-channel flow process under an uncertain roughness coefficient.

scholarly journals Modeling of broad crested weirs by using dynamic similarity and CFD

2020 ◽  
Vol 2 (2) ◽  
pp. 86-98
Author(s):  
Ali Yildiz ◽  
◽  
Goknur Elif Yarbasi ◽  
Alpaslan Yarar ◽  
Ali Ihsan Marti
Keyword(s):  
Open Channel ◽  
Scale Effects ◽  
Numerical Models ◽  
Real Life ◽  
Mesh Size ◽  
Water Levels ◽  
Hydraulic Structures ◽  
Ansys Fluent ◽  
Geometric Similarity ◽  
Upstream Side

Broad crested weirs and steps are used to regulate the flow in the channel, increase the water level at the upstream side, and measure the discharge. The construction of the broad crested weirs is more practical and also they are more stable compared with the other types of weirs. To serve in accordance with the purpose of their construction, broad crested weirs should be designed and built by considering certain criteria. Before the hydraulic structures are built, model experimental setups are constructed in the laboratory and problems to be encountered are tried to be determined. However, there may be differences between the structure to be built in real life (prototype) and model due to scale effect. These possible differences must be determined and necessary measures must be taken. In this study, the model and prototype of the broad crested weir are constructed in two different open channel systems by using Froude similarity. The geometric similarity between model and prototype is determined as Lr = 4. 44 experimental data were collected from model and prototype. The results obtained from the model and prototype are compared according to hydraulic similarity rules. In addition to the physical experimental setups, numerical models were created using the ANSYS Fluent for the model and prototype separately. By comparing the numerical model and physical experimental setups, optimum mesh size is tried to be determined. According to the results obtained from experimental setups, differences were observed in the position of critical flow depths and downstream water levels due to scale effects.

scholarly journals Application of the Theory of a Single First Order Equation to Traffic Flow

2014 ◽  
Vol 9 (1) ◽  
pp. 175-181
Author(s):  
Harideo Chaudhary
Keyword(s):  
Fluid Mechanics ◽  
Traffic Flow ◽  
Open Channel ◽  
Flow Problem ◽  
Order Equation ◽  
Point Of View ◽  
Highway Traffic ◽  
First Order ◽  
Moving Vehicles ◽  
Mathematical Relationships

Few years ago, Lighthill and Whitham (1955) published a lengthy paper dealing with the theory of highway traffic flow .The basic idea in their approach to the problem is that the flow of traffic along a highway is analogous to the flow of a fluid in an open channel or pipe. This point of view replaces a long column of closely spaced discrete moving vehicles with an equivalent continuous moving stream of liquid (e.g. water) or gas (e.g. air). In other words, Lighthill and Whitham analyzed the phenomenon of traffic flow as though it were a problem in fluid mechanics. This approach allows some, though certainly not all, of the physical and mathematical relationships of hydrodynamics and aerodynamics to be utilized in the traffic flow problem. (Bank, p. 272) DOI: http://dx.doi.org/10.3126/jie.v9i1.10681Journal of the Institute of Engineering, Vol. 9, No. 1, pp. 175–181

scholarly journals Global Dynamics of Sixth-Order Fuzzy Difference Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Abdul Khaliq ◽  
Muhammad Adnan ◽  
Abdul Qadeer Khan
Keyword(s):  
Difference Equation ◽  
Equilibrium Point ◽  
Difference Equations ◽  
Numerical Solutions ◽  
Nonnegative Solution ◽  
Real Life ◽  
Characterization Theorem ◽  
Sixth Order ◽  
Global Dynamics ◽  
Life Problems

Across many fields, such as engineering, ecology, and social science, fuzzy differences are becoming more widely used; there is a wide variety of applications for difference equations in real-life problems. Our study shows that the fuzzy difference equation of sixth order has a nonnegative solution, an equilibrium point and asymptotic behavior. y i + 1 = D y i − 1 y i − 2 / E + F y i − 3 + G y i − 4 + H y i − 5 , i = 0,1,2 , … , where y i is the sequence of fuzzy numbers and the parameter D , E , F , G , H and the initial condition y − 5 , y − 4 , y − 3 , y − 2 , y − 1 , y 0 are nonnegative fuzzy number. The characterization theorem is used to convert each single fuzzy difference equation into a set of two crisp difference equations in a fuzzy environment. So, a pair of crisp difference equations is formed by converting the difference equation. The stability of the equilibrium point of a fuzzy system has been evaluated. By using variational iteration techniques and inequality skills as well as a theory of comparison for fuzzy difference equations, we investigated the governing equation dynamics, such as its boundedness, existence, and local and global stability analysis. In addition, we provide some numerical solutions for the equation describing the system to verify our results.

scholarly journals Self-Organizing IoT Device-Based Smart Diagnosing Assistance System for Activities of Daily Living

Sensors ◽  
2021 ◽  
Vol 21 (3) ◽  
pp. 785
Author(s):  
Yu Jin Park ◽  
Seol Young Jung ◽  
Tae Yong Son ◽  
Soon Ju Kang
Keyword(s):  
Daily Living ◽  
Flame Temperature ◽  
Real Life ◽  
Wearable Devices ◽  
Activity Of Daily Living ◽  
Environmental Sensors ◽  
Iot Platform ◽  
Sensor Devices ◽  
Hospital Patients ◽  
Performance Ability

Activity of daily living (ADL) is a criterion for evaluating the performance ability of daily life by recognizing various activity events occurring in real life. However, most of the data necessary for ADL evaluation are collected only through observation and questionnaire by the patient or the patient’s caregiver. Recently, Internet of Things (IoT) device studies using various environmental sensors are being used for ADL collection and analysis. In this paper, we propose an IoT Device Platform for ADL capability measurement. Wearable devices and stationary devices recognize activity events in real environments and perform user identification through various sensors. The user’s ADL data are sent to the network hub for analysis. The proposed IoT platform devices support many sensor devices such as acceleration, flame, temperature, and humidity in order to recognize various activities in real life. In addition, in this paper, using the implemented platform, ADL measurement test was performed on hospital patients. Through this test, the accuracy and reliability of the platform are analyzed.

scholarly journals Ensemble modeling of stochastic unsteady open-channel flow in terms of its time-space evolutionary probability distribution: numerical application

2017 ◽  
Author(s):  
Alain Dib ◽  
M. Levent Kavvas
Keyword(s):  
Probability Distribution ◽  
Channel Flow ◽  
Open Channel ◽  
Flow Problem ◽  
Planck Equation ◽  
Open Channel Flow ◽  
Roughness Coefficient ◽  
Flow Process ◽  
Numerical Application ◽  
Time Space

Abstract. The characteristic form of the Saint–Venant equations was solved in a stochastic setting by using a newly proposed Fokker–Planck Equation (FPE) methodology. This methodology computes the ensemble behavior and variability of a system by directly solving for its time-space evolutionary probability distribution. The new methodology was tested on a stochastic unsteady open-channel flow problem, with an uncertainty arising from the channel’s roughness coefficient. The computed statistical descriptions of the flow variables were compared to the results obtained through Monte Carlo (MC) simulations in order to evaluate the performance of the FPE methodology. The comparisons showed that the proposed methodology can adequately predict the results of the considered stochastic flow problem, including the ensemble averages, variances, and probability density functions in time and space. However, unlike the large number of simulations performed by the MC approach, only one simulation was required by the FPE methodology. Moreover, the total simulation period of the FPE methodology was significantly smaller than that of the MC approach. As such, the results obtained in this study indicate that the proposed FPE methodology is a powerful and time-efficient approach for predicting the ensemble average and variance behavior, in both space and time, for an open-channel flow process under an uncertain roughness coefficient.

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