Triangular splitting implementation of RKN‐type Fourier collocation methods for second‐order differential equations

Wang, Bin

doi:10.1002/mma.4727

Cited by 11 publications

(8 citation statements)

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“…To solve various differential equations, some analytical tools as well as symbolic calculation techniques were established, for instance, fixed point theorems [14][15][16][17], variational methods [18][19][20][21], topological degree method [22][23][24][25], iterative techniques [26][27][28][29], modified Kudryashov method [30,31], exp function method [32,33], sine-Gordon expansion method [34][35][36][37], and complex method [38][39][40][41]. More works about the differential equations can be read in [42][43][44][45][46][47][48][49][50][51][52].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Painlevé Analysis and Abundant Meromorphic Solutions of a Class of Nonlinear Algebraic Differential Equations

Zheng

Meng

2019

Mathematical Problems in Engineering

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In this paper, a class of nonlinear algebraic differential equations (NADEs) is studied. The Painlevé analysis of the NADEs is considered. Abundant meromorphic solutions of the NADEs are obtained by means of the complex method. Then, meromorphic exact solutions of the Schamel-Korteweg-de Vries (S-KdV) equation and (2+1)-dimensional sine-Gordon equation are derived via the applications of the NADEs.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Painlevé Analysis and Abundant Meromorphic Solutions of a Class of Nonlinear Algebraic Differential Equations

Zheng

Meng

2019

Mathematical Problems in Engineering

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“…Functional, delay, difference and neutral integro-differential equations (IDEs) are of great importance especially in the applied sciences, such as mathematics, biology, physics, engineering, electrodynamics, electromagnetics, viscoelasticity, heat and mass transfer (Kürkçü et al 2016(Kürkçü et al , 2017a(Kürkçü et al , b, 2018Yüzbaşı et al 2013;Akyüz-Daşcıoglu 2006;Gülsu et al 2010;Babolian et al 2008;Wang andWang 2013, 2014;Adomian 1994;He 1999He , 2000Wang et al 2016Wang et al , 2017aWang et al , b, c, d, 2008Wang 2018;Liu and Wu 2017;Biazar et al 2011;Reutskiy 2016;Vlasov et al 2015;Dehghan and Shakeri 2008;Heydari et al 2013;Ioannou et al 2012). Most of these equations come to exist when real-life problems are encountered.…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, analytical and numerical methods are presented, but it is important to state that since it is hard to find the analytic solution of IDEs, numerical methods are required. For this aim, the polynomial-based (Kürkçü et al 2016(Kürkçü et al , 2017a(Kürkçü et al , b, 2018Yüzbaşı et al 2013;Akyüz-Daşcıoglu 2006;Gülsu et al 2010;Babolian et al 2008;Wang andWang 2013, 2014), Adomian decomposition (Adomian 1994), He's homotopy perturbation (He 1999) and He's variational iteration (He 2000;Biazar et al 2011), Fourier collocation (Wang et al 2016(Wang et al , 2017aWang 2018), trigonometric collocation and extended Runge-Kutta-Nyström methods have been introduced so far. Especially, in recent years, Yüzbaşı et al (2013) have employed an improved Legendre method to IDEs and a population model.…”

Section: Introductionmentioning

confidence: 99%

“…Akyüz-Daşcıoglu (2006) has constituted the most general form of Fredholm-Volterra IDEs. Wang (2018) has established and analyzed triangular splitting implementation of Runge-Kutta-Nyström-type Fourier collocation methods for second-order differential equations. have applied the efficient implementation of Runge-Kutta-Nyström-type Fourier collocation methods to solve second-order differential equations.…”

Section: Introductionmentioning

confidence: 99%

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An inventive numerical method for solving the most general form of integro-differential equations with functional delays and characteristic behavior of orthoexponential residual function

2019

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In this study, we constitute the most general form of functional integro-differential equations with functional delays. An inventive method based on Dickson polynomials with the parameter-α along with collocation points is employed to solve them. The stability of the solutions is simulated according to an interval of the parameter-α. A useful computer program is developed to obtain the precise values from the method. The residual error analysis is used to improve the obtained solutions. The characteristic behavior of the residual function is established with the aid of the orthoexponential polynomials. We compare the present numerical results of the method with those obtained by the existing methods in tables. Keywords Collocation points • Dickson and orthoexponential polynomials • Error analysis • Matrix method Mathematical Subject Classification 34K06 • 45J05 • 65L60 • 65R20 Communicated by Hui Liang.

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“…There are also a large number of valuable research results, which a large number of researchers have done semantic segmentation and realistic applications based on deep learning. Besides, the deep learning techniques have been applied to various fields of image processing such as other works, making a great breakthrough in traditional image processing.…”

Section: Introductionmentioning

confidence: 99%

Research of animals image semantic segmentation based on deep learning

Liu

et al. 2018

Concurrency and Computation

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It is imperative for us to develop the technology of image semantic segmentation with the increasing demand in the image processing. Nowadays, the development of deep learning is of great significance to the improvement of image segmentation. Furthermore, the paper discussed the relationship between image semantic segmentation and animal image research based on the actual situation, and found that animal image processing technology plays a more important role in the field of protecting precious animals. The end-to-end network training of this paper is consisted of Fully Convolutional Network (FCN) for the front end and Conditional Random Fields as Recurrent Neural Networks (CRF-RNN) for the back end via comparing a variety of research methods. The experiments achieved desired outcome for the semantic segmentation of animal images by utilizing Caffe deep learning framework and explained the implementation details from the aspects of training and testing. KEYWORDS Caffe, CRF-RNN, FCN, image semantic segmentation INTRODUCTIONTraditionally, the segmentation of the image is only directly on the surface layer, such as the appearance features based on texture, color, and shape, and they are separated after aggregating the similar features. Among them, the segmentation method based on regions, because the selection of seed points will lead to low efficiency of the algorithm, the segmented results have a certain flaw and can not reach the expected effect. However, the histogram-based threshold segmentation method ignores certain image space information, and it is very likely to segment a discontinuous and meaningless regions, making the segmentation result less satisfying. Therefore, with the demand for image segmentation technology, the semantic segmentation is indispensable.In semantic segmentation, the method of deep learning can be well escaped from the traditional processing methods, and the limitation of using only the underlying features is eliminated, making process in the level of understanding for image. The deep learning framework of Caffe 1 can handle semantic segmentation well. As the number of treasured animals continues to decline, the application of science and technology to protect them is imminent. In particular, it is necessary to collect a set of quality and distinctive animal image data for a specific animal, and the collection of a large number of data sets depends not only on human resources but also requires the support of the relevant sensors, cameras, and other hardware in the wild places it can automatically detect in the field or other places. One solution is to establish a corresponding index for the existing animal picture database. For the protected animal, it can use image processing technology to quickly identify for these animals, and then semantic segmentation is exploited to make a good recognition of animal images. This will not only increase the efficiency of the staff, but reduce the situation in which Concurrency Computat Pract Exper. 2020;32:e4892. wileyonlinelibrary.com/journal/cp...

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Triangular splitting implementation of RKN‐type Fourier collocation methods for second‐order differential equations

Cited by 11 publications

References 30 publications

Painlevé Analysis and Abundant Meromorphic Solutions of a Class of Nonlinear Algebraic Differential Equations

Painlevé Analysis and Abundant Meromorphic Solutions of a Class of Nonlinear Algebraic Differential Equations

An inventive numerical method for solving the most general form of integro-differential equations with functional delays and characteristic behavior of orthoexponential residual function

Research of animals image semantic segmentation based on deep learning

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