scholarly journals Fixed-Point Iterative Method with Eighth-Order Constructed by Undetermined Parameter Technique for Solving Nonlinear Systems

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 863
Author(s):  
Xiaofeng Wang
Keyword(s):  
Nonlinear Systems ◽  
Differential Equations ◽  
Iterative Method ◽  
Computational Cost ◽  
Analytic Solutions ◽  
New Method ◽  
Computational Time ◽  
Parameter Method ◽  
Eighth Order ◽  
Nonlinear Odes

In this manuscript, by using undetermined parameter method, an efficient iterative method with eighth-order is designed to solve nonlinear systems. The new method requires one matrix inversion per iteration, which means that computational cost of our method is low. The theoretical efficiency of the proposed method is analyzed, which is superior to other methods. Numerical results show that the proposed method can reduce the computational time, remarkably. New method is applied to solve the numerical solution of nonlinear ordinary differential equations (ODEs) and partial differential equations (PDEs). The nonlinear ODEs and PDEs are discretized by finite difference method. The validity of the new method is verified by comparison with analytic solutions.

scholarly journals An Iterative Method for Time-Fractional Swift-Hohenberg Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
Wenjin Li ◽  
Yanni Pang
Keyword(s):  
Differential Equations ◽  
Iterative Method ◽  
Fractional Differential Equations ◽  
Initial Value Problems ◽  
Approximate Solutions ◽  
Analytic Solutions ◽  
Cauchy Problems ◽  
Initial Value ◽  
Fractional Differential ◽  
Time Fractional Differential Equations

We study a type of iterative method and apply it to time-fractional Swift-Hohenberg equation with initial value. Using this iterative method, we obtain the approximate analytic solutions with numerical figures to initial value problems, which indicates that such iterative method is effective and simple in constructing approximate solutions to Cauchy problems of time-fractional differential equations.

scholarly journals A new approach for solving systems of fractional differential equations via natural transform

2016 ◽  
Vol 12 (1) ◽  
pp. 5797-5804 ◽  
Author(s):  
A. S Abedl Rady ◽  
S. Z Rida ◽  
A. A. M Arafa ◽  
H. R Abedl Rahim
Keyword(s):  
Nonlinear Systems ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Variational Iteration Method ◽  
New Method ◽  
Transform Method ◽  
Variational Iteration ◽  
Fractional Differential ◽  
Linear And Nonlinear ◽  
Linear And Nonlinear Systems

In this paper, A new method proposed and coined by the authors as the natural variational iteration  transform method(NVITM) is utilized to solve linear and nonlinear systems of fractional differential equations. The new method is a combination of natural transform method and variational iteration method. The solutions of our modeled systems are calculated in the form of convergent power series with easily computable components. The numerical results shows that the approach is easy to implement and accurate when applied to various linear and nonlinear systems of fractional differential equations.

Forced vibration analysis of nonlinear systems using efficient path-following method

2020 ◽  
Vol 26 (7-8) ◽  
pp. 505-519
Author(s):  
Seyed Mojtaba Mousavi ◽  
Mohammad Homayoune Sadr ◽  
Meisam Jelveh
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Nonlinear Dynamical Systems ◽  
Continuation Method ◽  
Computational Cost ◽  
Path Following ◽  
Forced Response ◽  
Degree Of Freedom ◽  
Numerical Continuation ◽  
Computational Time

In this article, nonlinear forced response of dynamical systems is studied using numerical continuation methods. Several methods are available to calculate nonlinear normal modes. Along with the existing analytical methods, recently, numerical methods, especially the pseudo-arclength continuation method, have attracted many researchers. The pseudo-arclength continuation method is a very powerful method which is capable of handling strongly nonlinear systems. However, as mentioned in recently published article reviews, the computational cost of the method has limited its application. In this research, an updating formula is embedded in the pseudo-arclength continuation algorithm to reduce the computational cost. This modified method is called the efficient path-following method. The assumptions and basis of the efficient path-following method algorithm are same as those presented in other references, but none of them have exploited the updating formula of the efficient path-following method to study the forced response of nonlinear dynamical systems. To investigate the capabilities of the method, forced response of a single-degree-of-freedom Duffing system is computed. It is seen that the efficient path-following method has decreased the computational time significantly up to 70%. The results are in very good conformance with those obtained in other references, which shows the accuracy of this method. To study the ability of the efficient path-following method to handle the multi-degree-of-freedom system, a four-degree-of-freedom nonlinear system is considered, and stable and unstable branches of the solution are computed. It is observed that as the nonlinearity of the system gets stronger, the updating formula becomes more effective.

Obtaining Frequency-Domain Volterra Models From Port-Based Ordinary Differential Equations

2012 ◽  
Vol 134 (4) ◽  
Author(s):  
Eliot Motato ◽  
Clark Radcliffe
Keyword(s):  
Nonlinear Systems ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Frequency Domain ◽  
Nonlinear Models ◽  
General Procedure ◽  
Multiple Input Multiple Output ◽  
Volterra Model ◽  
Nonlinear Odes ◽  
First Case

A frequency-domain Volterra model (FVM) is a nonlinear representation obtained when the multivariable Laplace transform is applied to a sum of multidimensional convolution integrals of increasing order. Two classes of FVMs can be identified. The first class of FVM is the Volterra transfer function (VTF) which has been recognized as a useful tool for nonlinear systems modeling and simulation. The second class of FVM is the Volterra dynamic model (VDM) which has been used in the modular assembly and condensation of port-based nonlinear models. Since physical nonlinear systems are frequently modeled using ordinary differential equations (ODEs), it is of significant value to derive their equivalent FVM representations from a corresponding ODE. Even though methods to obtain VTFs for multiple-input, multiple-output (MIMO) nonlinear ODEs are available, a general procedure to obtain the two classes of FVMs does not exist. In this work, a methodology to obtain the two classes of FVMs from port-based nonlinear ODEs is explained. Two cases are shown. In the first case, the ODEs do not include cross product nonlinearities. In the second case, cross products are included. An example is presented to clarify the idea, and the time response obtained from the nonlinear ODE model is compared to its corresponding third order VTF and its linearized model.

Improving the Computational Cost for Copied Region Detection in Forensic Images

2016 ◽  
Vol 2 (1) ◽  
pp. 55
Author(s):  
Tu Huynh-Kha ◽  
Thuong Le-Tien ◽  
Synh Ha ◽  
Khoa Huynh-Van
Keyword(s):  
Wavelet Transform ◽  
Euclidean Distance ◽  
Research Work ◽  
Computational Cost ◽  
Correlation Coefficients ◽  
Zernike Moments ◽  
Computational Time ◽  
Discrete Wavelet ◽  
Feature Vectors ◽  
Region Detection

This research work develops a new method to detect the forgery in image by combining the Wavelet transform and modified Zernike Moments (MZMs) in which the features are defined from more pixels than in traditional Zernike Moments. The tested image is firstly converted to grayscale and applied one level Discrete Wavelet Transform (DWT) to reduce the size of image by a half in both sides. The approximation sub-band (LL), which is used for processing, is then divided into overlapping blocks and modified Zernike moments are calculated in each block as feature vectors. More pixels are considered, more sufficient features are extracted. Lexicographical sorting and correlation coefficients computation on feature vectors are next steps to find the similar blocks. The purpose of applying DWT to reduce the dimension of the image before using Zernike moments with updated coefficients is to improve the computational time and increase exactness in detection. Copied or duplicated parts will be detected as traces of copy-move forgery manipulation based on a threshold of correlation coefficients and confirmed exactly from the constraint of Euclidean distance. Comparisons results between proposed method and related ones prove the feasibility and efficiency of the proposed algorithm.

Normal form method in search for periodic solutions of ordinary differential equations. Case of the fourth order

2018 ◽  
pp. 24-34
Author(s):  
V. F. Edneral ◽  
O. D. Timofeevskaya
Keyword(s):  
Normal Form ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Periodic Solutions ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Numerical Calculations ◽  
Computational Time ◽  
Resonant Normal Form ◽  
Normal Form Method

Introduction:The method of resonant normal form is based on reducing a system of nonlinear ordinary differential equations to a simpler form, easier to explore. Moreover, for a number of autonomous nonlinear problems, it is possible to obtain explicit formulas which approximate numerical calculations of families of their periodic solutions. Replacing numerical calculations with their precalculated formulas leads to significant savings in computational time. Similar calculations were made earlier, but their accuracy was insufficient, and their complexity was very high.Purpose:Application of the resonant normal form method and a software package developed for these purposes to fourth-order systems in order to increase the calculation speed.Results:It has been shown that with the help of a single algorithm it is possible to study equations of high orders (4th and higher). Comparing the tabulation of the obtained formulas with the numerical solutions of the corresponding equations shows good quantitative agreement. Moreover, the speed of calculation by prepared approximating formulas is orders of magnitude greater than the numerical calculation speed. The obtained approximations can also be successfully applied to unstable solutions. For example, in the Henon — Heyles system, periodic solutions are surrounded by chaotic solutions and, when numerically integrated, the algorithms are often unstable on them.Practical relevance:The developed approach can be used in the simulation of physical and biological systems.

scholarly journals A divide-and-conquer algorithm for quantum state preparation

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Israel F. Araujo ◽  
Daniel K. Park ◽  
Francesco Petruccione ◽  
Adenilton J. da Silva
Keyword(s):  
Computational Cost ◽  
Quantum Circuit ◽  
Divide And Conquer ◽  
Computational Time ◽  
Quantum Computers ◽  
Dimensional Vector ◽  
Quantum Devices ◽  
Divide And Conquer Algorithm ◽  
Quantum State Preparation ◽  
Quantum Device

AbstractAdvantages in several fields of research and industry are expected with the rise of quantum computers. However, the computational cost to load classical data in quantum computers can impose restrictions on possible quantum speedups. Known algorithms to create arbitrary quantum states require quantum circuits with depth O(N) to load an N-dimensional vector. Here, we show that it is possible to load an N-dimensional vector with exponential time advantage using a quantum circuit with polylogarithmic depth and entangled information in ancillary qubits. Results show that we can efficiently load data in quantum devices using a divide-and-conquer strategy to exchange computational time for space. We demonstrate a proof of concept on a real quantum device and present two applications for quantum machine learning. We expect that this new loading strategy allows the quantum speedup of tasks that require to load a significant volume of information to quantum devices.

scholarly journals Computing Expectiles Using k-Nearest Neighbours Approach

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 645
Author(s):  
Muhammad Farooq ◽  
Sehrish Sarfraz ◽  
Christophe Chesneau ◽  
Mahmood Ul Hassan ◽  
Muhammad Ali Raza ◽  
...  
Keyword(s):  
Computational Cost ◽  
Real Life ◽  
Distance Measures ◽  
Computational Time ◽  
High Dimensional ◽  
Test Error ◽  
Nearest Neighbours ◽  
Comparable Performance ◽  
Asymmetric Least Squares ◽  
Low Computational Cost

Expectiles have gained considerable attention in recent years due to wide applications in many areas. In this study, the k-nearest neighbours approach, together with the asymmetric least squares loss function, called ex-kNN, is proposed for computing expectiles. Firstly, the effect of various distance measures on ex-kNN in terms of test error and computational time is evaluated. It is found that Canberra, Lorentzian, and Soergel distance measures lead to minimum test error, whereas Euclidean, Canberra, and Average of (L1,L∞) lead to a low computational cost. Secondly, the performance of ex-kNN is compared with existing packages er-boost and ex-svm for computing expectiles that are based on nine real life examples. Depending on the nature of data, the ex-kNN showed two to 10 times better performance than er-boost and comparable performance with ex-svm regarding test error. Computationally, the ex-kNN is found two to five times faster than ex-svm and much faster than er-boost, particularly, in the case of high dimensional data.

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