scholarly journals MODELING AFTERSHOCKS BY FRACTIONAL CALCULUS: EXACT DISCRETIZATION VERSUS APPROXIMATE DISCRETIZATION

Fractals ◽  
2021 ◽  
pp. 2140038
Author(s):  
HUA KONG ◽  
GUANG YANG ◽  
CHENG LUO
Keyword(s):  
Fractional Calculus ◽  
Heavy Tails ◽  
Least Square Method ◽  
Least Square ◽  
The Other ◽  
Discrete Fractional Calculus ◽  
Unknown Parameters ◽  
Fractional Difference ◽  
Difference Formula ◽  
The Difference

This paper suggests two fractional differences for aftershock modeling with heavy tails. Discrete fractional calculus is a straightforward tool on isolated time scale. On the other hand, the fractional difference also can be derived by standard finite difference method when the difference formula is convergent. The two methods are both adopted and compared in the results. The unknown parameters are determined by use of the least square method where Ya’an earthquake aftershock data is used. It is reported that the discrete fractional calculus is an exact discretization tool without any loss in the memory effects which leads to better results.

scholarly journals On the nonoscillatory behavior of solutions of three classes of fractional difference equations

2020 ◽  
Vol 40 (5) ◽  
pp. 549-568
Author(s):  
Said Rezk Grace ◽  
Jehad Alzabut ◽  
Sakthivel Punitha ◽  
Velu Muthulakshmi ◽  
Hakan Adıgüzel
Keyword(s):  
Fractional Calculus ◽  
Difference Equations ◽  
Riccati Transformation ◽  
Discrete Fractional Calculus ◽  
Fractional Difference ◽  
Transformation Technique ◽  
Fractional Difference Equations ◽  
Difference Formula

In this paper, we study the nonoscillatory behavior of three classes of fractional difference equations. The investigations are presented in three different folds. Unlike most existing nonoscillation results which have been established by employing Riccati transformation technique, we employ herein an easily verifiable approach based on the fractional Taylor's difference formula, some features of discrete fractional calculus and mathematical inequalities. The theoretical findings are demonstrated by examples. We end the paper by a concluding remark.

A New Insight Into the Grünwald–Letnikov Discrete Fractional Calculus

2019 ◽  
Vol 14 (4) ◽  
Author(s):  
Yiheng Wei ◽  
Weidi Yin ◽  
Yanting Zhao ◽  
Yong Wang
Keyword(s):  
Fractional Calculus ◽  
Discrete Fractional Calculus ◽  
Fractional Difference ◽  
Integer Order ◽  
Convolution Operation ◽  
Order Case ◽  
Insight Into

The primary work of this paper is to investigate some potential properties of Grünwald–Letnikov discrete fractional calculus. By employing a concise and convenient description, this paper not only establishes excellent relationships between fractional difference/sum and the integer order case but also generalizes the Z-transform and convolution operation.

scholarly journals Solution of inverse heat conduction equation with the use of Chebyshev polynomials

2016 ◽  
Vol 37 (4) ◽  
pp. 73-88 ◽  
Author(s):  
Magda Joachimiak ◽  
Andrzej Frąckowiak ◽  
Michał Ciałkowski
Keyword(s):  
Inverse Problem ◽  
Chebyshev Polynomials ◽  
Direct Problem ◽  
Least Square Method ◽  
Least Square ◽  
Random Disturbance ◽  
Inverse Heat Conduction ◽  
Chebyshev Nodes ◽  
The Difference ◽  
The Stability

AbstractA direct problem and an inverse problem for the Laplace’s equation was solved in this paper. Solution to the direct problem in a rectangle was sought in a form of finite linear combinations of Chebyshev polynomials. Calculations were made for a grid consisting of Chebyshev nodes, what allows us to use orthogonal properties of Chebyshev polynomials. Temperature distributions on the boundary for the inverse problem were determined using minimization of the functional being the measure of the difference between the measured and calculated values of temperature (boundary inverse problem). For the quasi-Cauchy problem, the distance between set values of temperature and heat flux on the boundary was minimized using the least square method. Influence of the value of random disturbance to the temperature measurement, of measurement points (distance from the boundary, where the temperature is not known) arrangement as well as of the thermocouple installation error on the stability of the inverse problem was analyzed.

Identification of Nonlinear Systems by Haar Wavelet

2004 ◽  
Author(s):  
Shy-Leh Chen ◽  
Keng-Chu Ho
Keyword(s):  
Nonlinear Systems ◽  
Least Square Method ◽  
Haar Wavelet ◽  
State Equation ◽  
Least Square ◽  
Algebraic Equations ◽  
Function Form ◽  
Unknown Parameters ◽  
Complete Orthogonal Basis ◽  
Unknown System

This study addresses the identification of autonomous nonlinear systems. It is assumed that the function form in the nonlinear system is known, leaving some unknown parameters to be estimated. It is also assumed that the free responses of the system can be measured. Since Haar wavelets can form a complete orthogonal basis for the appropriate function space, they are used to expand all signals. In doing so, the state equation can be transformed into a set of algebraic equations in unknown parameters. The technique of Kronecker product is utilized to simplify the expressions of the associated algebraic equations. Together with the least square method, the unknown system parameters are estimated. Several simulation examples verify the analysis.

Three weighted residual methods based on Jeffery-Hamel flow

2014 ◽  
Vol 24 (3) ◽  
pp. 654-668 ◽  
Author(s):  
D.D. Ganji ◽  
Mohammad Hatami
Keyword(s):  
Analytical Methods ◽  
Design Methodology ◽  
Least Square Method ◽  
Numerical Approach ◽  
Least Square ◽  
The Other ◽  
Governing Equations ◽  
Science And Engineering ◽  
Content Type ◽  
Linear Differential

Purpose – The purpose of this paper is to demonstrate the eligibility of the weighted residual methods (WRMs) applied to Jeffery-Hamel Flow. Selecting the most appropriate method among the WRMs and discussing about Jeffery-Hamel flow's treatment in divergent and convergent channels are the other important purposes of the present research. Design/methodology/approach – Three analytical methods (collocation, Galerkin and least square method) have been applied to solve the governing equations. The reliability of the methods is also approved by a comparison made between the forth order Runge-Kutta numerical method. Findings – The obtained solutions revealed that WRMs can be simple, powerful and efficient techniques for finding analytical solutions in science and engineering non-linear differential equations. Originality/value – It could be considered as a first endeavor to use the solution of the Jeffery-Hamel flow using these kind of analytical methods along with the numerical approach.

Application of Photoelectric Sensor in Vehicle Power Control System

2020 ◽  
Vol 15 (6) ◽  
pp. 700-706
Author(s):  
Yifan Zhao ◽  
Mengyu Wang ◽  
Kai Wang
Keyword(s):  
Power Control ◽  
Electric Energy ◽  
Least Square Method ◽  
Least Square ◽  
Motor Speed ◽  
Time Calculation ◽  
The Mean ◽  
The Difference ◽  
Artificial Neural Network Ann ◽  
Mean Square Errors

Due to its characteristics of using clean electric energy and bringing no damage to the environment, electric vehicles (EVs) have become a new developmental direction for the automotive industry. Its reliability issues have also attracted the attention of experts and professionals. In the field of automotive power control, from the perspective of motor control, this study uses the photoelectric sensors (PSs) as the research objects and elaborates on the measurement principles of motor speed with PSs. Meanwhile, a diagnosis scheme is proposed for various faults in the measurement. Among them, the measurement speed is converted by the photoelectric signal, and the measured waveform is amplified. In the fault detection process, the Radial Basis Function (RBF) artificial neural network (ANN) is analyzed. By using this method, the difference in the motor speed detected by the sensor is calculated to determine the cause of the failure. The test uses the least-square method to compare the tested motor speed with the actual motor speed. The results show that PSs can measure the motor speed of EVs. As for the motor failures, the mean square errors (MSEs) of motor speeds generated by different faults are compared to determine the fault points according to the speed changes. In addition, the cause of motor failure can be determined by the real-time calculation of the speed differences. The above tests fully prove the effectiveness of measuring the speed of electric motors by PSs; therefore, PSs have broad application prospects in vehicle power control systems.

scholarly journals SOME FURTHER EXTENSIONS CONSIDERING DISCRETE PROPORTIONAL FRACTIONAL OPERATORS

Fractals ◽  
2021 ◽  
pp. 2240026
Author(s):  
SAIMA RASHID ◽  
SOBIA SULTANA ◽  
YELIZ KARACA ◽  
AASMA KHALID ◽  
YU-MING CHU
Keyword(s):  
Fractional Calculus ◽  
Existence And Uniqueness ◽  
Complex Dynamics ◽  
Fractional Operator ◽  
Fractional Operators ◽  
Discrete Fractional Calculus ◽  
Dynamic Features ◽  
Fractional Difference ◽  
Fractional Systems ◽  
Special Cases

In this paper, some attempts have been devoted to investigating the dynamic features of discrete fractional calculus (DFC). To date, discrete fractional systems with complex dynamics have attracted the most consideration. By considering discrete [Formula: see text]-proportional fractional operator with nonlocal kernel, this study contributes to the major consequences of the certain novel versions of reverse Minkowski and related Hölder-type inequalities via discrete [Formula: see text]-proportional fractional sums, as presented. The proposed system has an intriguing feature not investigated in the literature so far, it is characterized by the nabla [Formula: see text] fractional sums. Novel special cases are reported with the intention of assessing the dynamics of the system, as well as to highlighting the several existing outcomes. In terms of applications, we can employ the derived consequences to investigate the existence and uniqueness of fractional difference equations underlying worth problems. Finally, the projected method is efficient in analyzing the complexity of the system.

Parametric Identification of Nonlinear Systems by Haar Wavelets: Theory and Experimental Validation

2012 ◽  
Vol 134 (3) ◽  
Author(s):  
Shy-Leh Chen ◽  
Jin-Wei Liang ◽  
Keng-Chu Ho
Keyword(s):  
Nonlinear Systems ◽  
Chaotic System ◽  
Least Square Method ◽  
Parametric Identification ◽  
State Equation ◽  
Least Square ◽  
Haar Wavelets ◽  
Largest Lyapunov Exponent ◽  
Algebraic Equations ◽  
Unknown Parameters

This study addresses the identification of nonlinear systems. It is assumed that the function form in the nonlinear system is known, leaving some unknown parameters to be estimated. Since Haar wavelets can form a complete orthogonal basis for the appropriate function space, they are used to expand all signals. In doing so, the state equation can be transformed into a set of algebraic equations in unknown parameters. The technique of Kronecker product is utilized to simplify the expressions of the associated algebraic equations. Together with the least square method, the unknown system parameters are estimated. The proposed method is applied to the identification of an experimental two-well chaotic system known as the Moon beam. The identified model is validated by comparing the chaotic characteristics, such as the largest Lyapunov exponent and the correlation dimension, of the experimental data with that of the numerical results. The simple least square approach is also performed for comparison. The results indicate that the proposed method can reliably identify the characteristics of the nonlinear chaotic system.

Simulation and Research for Gas Boiler Based on the Vague Set PID

2014 ◽  
Vol 556-562 ◽  
pp. 2101-2104
Author(s):  
Gang Zhao ◽  
Jian Li
Keyword(s):  
Dynamic Performance ◽  
Least Square Method ◽  
Equation Model ◽  
Hot Water ◽  
Least Square ◽  
Vague Set ◽  
Pid Algorithm ◽  
Static Performance ◽  
Self Tuning ◽  
The Difference

Gas hot water boiler is widely used as heating equipment in everyday life. Because gas hot water boiler has the characteristics of nonlinear, large inertia and disturbances, so it is particularly important to build a precise mathematical model. Then the difference equation model of the system is identified by the least square method according to the collected data in this paper. Writing M file in the MATLAB software to get the continuous transfer function, and setting up Vague Set PID simulation, fuzzy self-tuning PID simulation and conventional PID algorithm in SIMULINK. By comparing among the three kinds of adjusting method, We get that Vague Set PID not only in regulation time, overshoot and effect of dynamic performance is superior compared the other two controller models , but also enhance the robustness and adaptability of the system, has a good dynamic, static performance..

Optimal Weighted Least Square Data Fusion Method for Redundant IMU Based on Calculation of Measurement Error

2012 ◽  
Vol 241-244 ◽  
pp. 149-155
Author(s):  
Chuan Xing ◽  
Hai Zhang
Keyword(s):  
Measurement Error ◽  
Data Fusion ◽  
Measurement Errors ◽  
Least Square Method ◽  
Least Square ◽  
Weighted Least Square ◽  
Weighted Matrix ◽  
Measurement Results ◽  
The Difference ◽  
Projection Error

A dodecahedron non-orthogonal redundant IMU configuration was selected as model. To improve fusion accuracy, we proposed an effective calculation method for measurement errors based on the correlation between measurement errors and fusion errors. The method considered the difference between traditional data fusion vector’s projection and measurement results, and then made a conversion from projection error to measurement error. Combined with optimal weighted least square method, measurement error was used to generate an optimal weighted matrix, and this made data fusion errors minimum. Simulations also proved that the fusion result of this method is more accurate than the result of traditional method.

Sign in / Sign up

Export Citation Format

Share Document