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Entropy
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Wavelets, Fractals and Information Theory I
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Wavelets, Fractals and Information Theory I

A special issue of Entropy (ISSN 1099-4300).

Deadline for manuscript submissions: closed (20 December 2015) | Viewed by 123706

Special Issue Editor

Prof. Dr. Carlo Cattani

E-Mail Website
Guest Editor
Engineering School (DEIM), University of Tuscia, Largo dell'Università, 01100 Viterbo, Italy
Interests: wavelets; fractals; fractional and stochastic equations; numerical and computational methods; mathematical physics; nonlinear systems; artificial intelligence
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Wavelet Analysis and Fractals are playing fundamental roles in various applications in Science, Engineering, and Information Theory.

In information theory, the entropy encoding might be considered a sort of compression in a quantization process, and this can be further investigated by using the wavelet compression. There are many types of entropy definitions that are very useful in the Engineering and Applied Sciences, such as the Shannon-Fano entropy, the Kolmogorov entropy, etc. However, only entropy encoding is optimal for the complexity of large data analysis, such as in data storage. In fact, the principal advantage of modeling a complex problem via wavelet analysis is the minimization of the memory space for storage or transmission. Moreover, this kind of approach reveals some new aspects and promising perspectives in many other kinds of applied and theoretical problems. For instance, in engineering, the best way to model the traffic in wireless communication is based on fractal geometry, whereas the data are efficiently studied through wavelet basis.

This Special Issue will also be an opportunity for extending the research fields of image processing, differential/integral equations, number theory and special functions, image segmentation, the sparse component analysis approach, generalized multiresolution analysis, and entropy as a measure in all aspects of the theoretical and practical studies of Mathematics, Physics, and Engineering.

The main topics of this Special Issue include (but are not limited to):

  • Entropy encoding, wavelet compression, and information theory.
  • Fractals, Non-differentiable functions. Theoretical and applied analytical problems of fractal type, fractional equations.
  • Wavelet Analysis, integral transforms and applications.
  • Wavelet-fractal entropy encoding and computational mathematics, including in image processing.
  • Wavelet-fractal approach.

Prof. Dr. Carlo Cattani
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

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Published Papers (18 papers)

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Research

906 KiB  
Article
Entropy and Fractal Antennas
by Emanuel Guariglia
Entropy 2016, 18(3), 84; https://doi.org/10.3390/e18030084 - 04 Mar 2016
Cited by 167 | Viewed by 13572
Abstract
The entropies of Shannon, Rényi and Kolmogorov are analyzed and compared together with their main properties. The entropy of some particular antennas with a pre-fractal shape, also called fractal antennas, is studied. In particular, their entropy is linked with the fractal geometrical shape [...] Read more.
The entropies of Shannon, Rényi and Kolmogorov are analyzed and compared together with their main properties. The entropy of some particular antennas with a pre-fractal shape, also called fractal antennas, is studied. In particular, their entropy is linked with the fractal geometrical shape and the physical performance. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)
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4556 KiB  
Article
Wavelet Entropy-Based Traction Inverter Open Switch Fault Diagnosis in High-Speed Railways
by Keting Hu, Zhigang Liu and Shuangshuang Lin
Entropy 2016, 18(3), 78; https://doi.org/10.3390/e18030078 - 01 Mar 2016
Cited by 24 | Viewed by 6176
Abstract
In this paper, a diagnosis plan is proposed to settle the detection and isolation problem of open switch faults in high-speed railway traction system traction inverters. Five entropy forms are discussed and compared with the traditional fault detection methods, namely, discrete wavelet transform [...] Read more.
In this paper, a diagnosis plan is proposed to settle the detection and isolation problem of open switch faults in high-speed railway traction system traction inverters. Five entropy forms are discussed and compared with the traditional fault detection methods, namely, discrete wavelet transform and discrete wavelet packet transform. The traditional fault detection methods cannot efficiently detect the open switch faults in traction inverters because of the low resolution or the sudden change of the current. The performances of Wavelet Packet Energy Shannon Entropy (WPESE), Wavelet Packet Energy Tsallis Entropy (WPETE) with different non-extensive parameters, Wavelet Packet Energy Shannon Entropy with a specific sub-band (WPESE3,6), Empirical Mode Decomposition Shannon Entropy (EMDESE), and Empirical Mode Decomposition Tsallis Entropy (EMDETE) with non-extensive parameters in detecting the open switch fault are evaluated by the evaluation parameter. Comparison experiments are carried out to select the best entropy form for the traction inverter open switch fault detection. In addition, the DC component is adopted to isolate the failure Isolated Gate Bipolar Transistor (IGBT). The simulation experiments show that the proposed plan can diagnose single and simultaneous open switch faults correctly and timely. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)
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1838 KiB  
Article
Improved LMD, Permutation Entropy and Optimized K-Means to Fault Diagnosis for Roller Bearings
by Zongli Shi, Wanqing Song and Saied Taheri
Entropy 2016, 18(3), 70; https://doi.org/10.3390/e18030070 - 25 Feb 2016
Cited by 54 | Viewed by 5892
Abstract
A novel bearing vibration signal fault feature extraction and recognition method based on the improved local mean decomposition (LMD), permutation entropy (PE) and the optimized K-means clustering algorithm is put forward in this paper. The improved LMD is proposed based on the self-similarity [...] Read more.
A novel bearing vibration signal fault feature extraction and recognition method based on the improved local mean decomposition (LMD), permutation entropy (PE) and the optimized K-means clustering algorithm is put forward in this paper. The improved LMD is proposed based on the self-similarity of roller bearing vibration signal extending the right and left side of the original signal to suppress its edge effect. After decomposing the extended signal into a set of product functions (PFs), the PE is utilized to display the complexity of the PF component and extract the fault feature meanwhile. Then, the optimized K-means algorithm is used to cluster analysis as a new pattern recognition approach, which uses the probability density distribution (PDD) to identify the initial centroid selection and has the priority of recognition accuracy compared with the classic one. Finally, the experiment results show the proposed method is effectively to fault extraction and recognition for roller bearing. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)
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1572 KiB  
Article
Modelling the Spread of River Blindness Disease via the Caputo Fractional Derivative and the Beta-derivative
by Abdon Atangana and Rubayyi T. Alqahtani
Entropy 2016, 18(2), 40; https://doi.org/10.3390/e18020040 - 26 Jan 2016
Cited by 108 | Viewed by 6454
Abstract
Information theory is used in many branches of science and technology. For instance, to inform a set of human beings living in a particular region about the fatality of a disease, one makes use of existing information and then converts it into a [...] Read more.
Information theory is used in many branches of science and technology. For instance, to inform a set of human beings living in a particular region about the fatality of a disease, one makes use of existing information and then converts it into a mathematical equation for prediction. In this work, a model of the well-known river blindness disease is created via the Caputo and beta derivatives. A partial study of stability analysis was presented. The extended system describing the spread of this disease was solved via two analytical techniques: the Laplace perturbation and the homotopy decomposition methods. Summaries of the iteration methods used were provided to derive special solutions to the extended systems. Employing some theoretical parameters, we present some numerical simulations. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)
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2450 KiB  
Article
Wavelet Energy and Wavelet Coherence as EEG Biomarkers for the Diagnosis of Parkinson’s Disease-Related Dementia and Alzheimer’s Disease
by Dong-Hwa Jeong, Young-Do Kim, In-Uk Song, Yong-An Chung and Jaeseung Jeong
Entropy 2016, 18(1), 8; https://doi.org/10.3390/e18010008 - 29 Dec 2015
Cited by 52 | Viewed by 8865
Abstract
Parkinson’s disease (PD) and Alzheimer’s disease (AD) can coexist in severely affected; elderly patients. Since they have different pathological causes and lesions and consequently require different treatments; it is critical to distinguish PD-related dementia (PD-D) from AD. Conventional electroencephalograph (EEG) analysis has produced [...] Read more.
Parkinson’s disease (PD) and Alzheimer’s disease (AD) can coexist in severely affected; elderly patients. Since they have different pathological causes and lesions and consequently require different treatments; it is critical to distinguish PD-related dementia (PD-D) from AD. Conventional electroencephalograph (EEG) analysis has produced poor results. This study investigated the possibility of using relative wavelet energy (RWE) and wavelet coherence (WC) analysis to distinguish between PD-D patients; AD patients and healthy elderly subjects. In EEG signals; we found that low-frequency wavelet energy increased and high-frequency wavelet energy decreased in PD-D patients and AD patients relative to healthy subjects. This result suggests that cognitive decline in both diseases is potentially related to slow EEG activity; which is consistent with previous studies. More importantly; WC values were lower in AD patients and higher in PD-D patients compared with healthy subjects. In particular; AD patients exhibited decreased WC primarily in the γ band and in links related to frontal regions; while PD-D patients exhibited increased WC primarily in the α and β bands and in temporo-parietal links. Linear discriminant analysis (LDA) of RWE produced a maximum accuracy of 79.18% for diagnosing PD-D and 81.25% for diagnosing AD. The discriminant accuracy was 73.40% with 78.78% sensitivity and 69.47% specificity. In distinguishing between the two diseases; the maximum performance of LDA using WC was 80.19%. We suggest that using a wavelet approach to evaluate EEG results may facilitate discrimination between PD-D and AD. In particular; RWE is useful for differentiating individuals with and without dementia and WC is useful for differentiating between PD-D and AD. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)
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3204 KiB  
Article
Mechanical Fault Diagnosis of High Voltage Circuit Breakers Based on Wavelet Time-Frequency Entropy and One-Class Support Vector Machine
by Nantian Huang, Huaijin Chen, Shuxin Zhang, Guowei Cai, Weiguo Li, Dianguo Xu and Lihua Fang
Entropy 2016, 18(1), 7; https://doi.org/10.3390/e18010007 - 26 Dec 2015
Cited by 52 | Viewed by 5424
Abstract
Mechanical faults of high voltage circuit breakers (HVCBs) are one of the most important factors that affect the reliability of power system operation. Because of the limitation of a lack of samples of each fault type; some fault conditions can be recognized as [...] Read more.
Mechanical faults of high voltage circuit breakers (HVCBs) are one of the most important factors that affect the reliability of power system operation. Because of the limitation of a lack of samples of each fault type; some fault conditions can be recognized as a normal condition. The fault diagnosis results of HVCBs seriously affect the operation reliability of the entire power system. In order to improve the fault diagnosis accuracy of HVCBs; a method for mechanical fault diagnosis of HVCBs based on wavelet time-frequency entropy (WTFE) and one-class support vector machine (OCSVM) is proposed. In this method; the S-transform (ST) is proposed to analyze the energy time-frequency distribution of HVCBs’ vibration signals. Then; WTFE is selected as the feature vector that reflects the information characteristics of vibration signals in the time and frequency domains. OCSVM is used for judging whether a mechanical fault of HVCBs has occurred or not. In order to improve the fault detection accuracy; a particle swarm optimization (PSO) algorithm is employed to optimize the parameters of OCSVM; including the window width of the kernel function and error limit. If the mechanical fault is confirmed; a support vector machine (SVM)-based classifier will be used to recognize the fault type. The experiments carried on a real SF6 HVCB demonstrated the improved effectiveness of the new approach. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)
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1777 KiB  
Article
Pathological Brain Detection by a Novel Image Feature—Fractional Fourier Entropy
by Shuihua Wang, Yudong Zhang, Xiaojun Yang, Ping Sun, Zhengchao Dong, Aijun Liu and Ti-Fei Yuan
Entropy 2015, 17(12), 8278-8296; https://doi.org/10.3390/e17127877 - 17 Dec 2015
Cited by 81 | Viewed by 6792
Abstract
Aim: To detect pathological brain conditions early is a core procedure for patients so as to have enough time for treatment. Traditional manual detection is either cumbersome, or expensive, or time-consuming. We aim to offer a system that can automatically identify pathological [...] Read more.
Aim: To detect pathological brain conditions early is a core procedure for patients so as to have enough time for treatment. Traditional manual detection is either cumbersome, or expensive, or time-consuming. We aim to offer a system that can automatically identify pathological brain images in this paper. Method: We propose a novel image feature, viz., Fractional Fourier Entropy (FRFE), which is based on the combination of Fractional Fourier Transform (FRFT) and Shannon entropy. Afterwards, the Welch’s t-test (WTT) and Mahalanobis distance (MD) were harnessed to select distinguishing features. Finally, we introduced an advanced classifier: twin support vector machine (TSVM). Results: A 10 × K-fold stratified cross validation test showed that this proposed “FRFE + WTT + TSVM” yielded an accuracy of 100.00%, 100.00%, and 99.57% on datasets that contained 66, 160, and 255 brain images, respectively. Conclusions: The proposed “FRFE + WTT + TSVM” method is superior to 20 state-of-the-art methods. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)
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7408 KiB  
Article
On the Complex and Hyperbolic Structures for the (2 + 1)-Dimensional Boussinesq Water Equation
by Figen Özpinar, Haci Mehmet Baskonus and Hasan Bulut
Entropy 2015, 17(12), 8267-8277; https://doi.org/10.3390/e17127878 - 17 Dec 2015
Cited by 42 | Viewed by 5197
Abstract
In this study, we have applied the modified exp(−Ω(ξ))-expansion function method to the (2 + 1)-dimensional Boussinesq water equation. We have obtained some new analytical solutions such as exponential function, complex function and hyperbolic function solutions. It has been observed that all analytical [...] Read more.
In this study, we have applied the modified exp(−Ω(ξ))-expansion function method to the (2 + 1)-dimensional Boussinesq water equation. We have obtained some new analytical solutions such as exponential function, complex function and hyperbolic function solutions. It has been observed that all analytical solutions have been verified to the (2 + 1)-dimensional Boussinesq water equation by using Wolfram Mathematica 9. We have constructed the two- and three-dimensional surfaces for all analytical solutions obtained in this paper using the same computer program. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)
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9485 KiB  
Article
An Optimal Segmentation Method Using Jensen–Shannon Divergence via a Multi-Size Sliding Window Technique
by Qutaibeh D. Katatbeh, José Martínez-Aroza, Juan Francisco Gómez-Lopera and David Blanco-Navarro
Entropy 2015, 17(12), 7996-8006; https://doi.org/10.3390/e17127858 - 04 Dec 2015
Cited by 10 | Viewed by 5868
Abstract
In this paper we develop a new procedure for entropic image edge detection. The presented method computes the Jensen–Shannon divergence of the normalized grayscale histogram of a set of multi-sized double sliding windows over the entire image. The procedure presents a good performance [...] Read more.
In this paper we develop a new procedure for entropic image edge detection. The presented method computes the Jensen–Shannon divergence of the normalized grayscale histogram of a set of multi-sized double sliding windows over the entire image. The procedure presents a good performance in images with textures, contrast variations and noise. We illustrate our procedure in the edge detection of medical images. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)
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1233 KiB  
Article
Wavelet-Tsallis Entropy Detection and Location of Mean Level-Shifts in Long-Memory fGn Signals
by Julio César Ramírez-Pacheco, Luis Rizo-Domínguez and Joaquin Cortez-González
Entropy 2015, 17(12), 7979-7995; https://doi.org/10.3390/e17127856 - 04 Dec 2015
Cited by 2 | Viewed by 5432
Abstract
Long-memory processes, in particular fractional Gaussian noise processes, have been applied as models for many phenomena occurring in nature. Non-stationarities, such as trends, mean level-shifts, etc., impact the accuracy of long-memory parameter estimators, giving rise to biases and misinterpretations of the phenomena. [...] Read more.
Long-memory processes, in particular fractional Gaussian noise processes, have been applied as models for many phenomena occurring in nature. Non-stationarities, such as trends, mean level-shifts, etc., impact the accuracy of long-memory parameter estimators, giving rise to biases and misinterpretations of the phenomena. In this article, a novel methodology for the detection and location of mean level-shifts in stationary long-memory fractional Gaussian noise (fGn) signals is proposed. It is based on a joint application of the wavelet-Tsallis q-entropy as a preprocessing technique and a peak detection methodology. Extensive simulation experiments in synthesized fGn signals with mean level-shifts confirm that the proposed methodology not only detects, but also locates level-shifts with high accuracy. A comparative study against standard techniques of level-shift detection and location shows that the technique based on wavelet-Tsallis q-entropy outperforms the one based on trees and the Bai and Perron procedure, as well. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)
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1404 KiB  
Article
A Novel Method for PD Feature Extraction of Power Cable with Renyi Entropy
by Jikai Chen, Yanhui Dou, Zhenhao Wang and Guoqing Li
Entropy 2015, 17(11), 7698-7712; https://doi.org/10.3390/e17117698 - 13 Nov 2015
Cited by 11 | Viewed by 5352
Abstract
Partial discharge (PD) detection can effectively achieve the status maintenance of XLPE (Cross Linked Polyethylene) cable, so it is the direction of the development of equipment maintenance in power systems. At present, a main method of PD detection is the broadband electromagnetic coupling [...] Read more.
Partial discharge (PD) detection can effectively achieve the status maintenance of XLPE (Cross Linked Polyethylene) cable, so it is the direction of the development of equipment maintenance in power systems. At present, a main method of PD detection is the broadband electromagnetic coupling with a high-frequency current transformer (HFCT). Due to the strong electromagnetic interference (EMI) generated among the mass amount of cables in a tunnel and the impedance mismatching of HFCT and the data acquisition equipment, the features of the pulse current generated by PD are often submerged in the background noise. The conventional method for the stationary signal analysis cannot analyze the PD signal, which is transient and non-stationary. Although the algorithm of Shannon wavelet singular entropy (SWSE) can be used to analyze the PD signal at some level, its precision and anti-interference capability of PD feature extraction are still insufficient. For the above problem, a novel method named Renyi wavelet packet singular entropy (RWPSE) is proposed and applied to the PD feature extraction on power cables. Taking a three-level system as an example, we analyze the statistical properties of Renyi entropy and the intrinsic correlation with Shannon entropy under different values of α . At the same time, discrete wavelet packet transform (DWPT) is taken instead of discrete wavelet transform (DWT), and Renyi entropy is combined to construct the RWPSE algorithm. Taking the grounding current signal from the shielding layer of XLPE cable as the research object, which includes the current pulse feature of PD, the effectiveness of the novel method is tested. The theoretical analysis and experimental results show that compared to SWSE, RWPSE can not only improve the feature extraction accuracy for PD, but also can suppress EMI effectively. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)
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952 KiB  
Article
Multi-Level Wavelet Shannon Entropy-Based Method for Single-Sensor Fault Location
by Qiaoning Yang and Jianlin Wang
Entropy 2015, 17(10), 7101-7117; https://doi.org/10.3390/e17107101 - 20 Oct 2015
Cited by 22 | Viewed by 8275
Abstract
In actual application, sensors are prone to failure because of harsh environments, battery drain, and sensor aging. Sensor fault location is an important step for follow-up sensor fault detection. In this paper, two new multi-level wavelet Shannon entropies (multi-level wavelet time [...] Read more.
In actual application, sensors are prone to failure because of harsh environments, battery drain, and sensor aging. Sensor fault location is an important step for follow-up sensor fault detection. In this paper, two new multi-level wavelet Shannon entropies (multi-level wavelet time Shannon entropy and multi-level wavelet time-energy Shannon entropy) are defined. They take full advantage of sensor fault frequency distribution and energy distribution across multi-subband in wavelet domain. Based on the multi-level wavelet Shannon entropy, a method is proposed for single sensor fault location. The method firstly uses a criterion of maximum energy-to-Shannon entropy ratio to select the appropriate wavelet base for signal analysis. Then multi-level wavelet time Shannon entropy and multi-level wavelet time-energy Shannon entropy are used to locate the fault. The method is validated using practical chemical gas concentration data from a gas sensor array. Compared with wavelet time Shannon entropy and wavelet energy Shannon entropy, the experimental results demonstrate that the proposed method can achieve accurate location of a single sensor fault and has good anti-noise ability. The proposed method is feasible and effective for single-sensor fault location. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)
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744 KiB  
Article
Modified Legendre Wavelets Technique for Fractional Oscillation Equations
by Syed Tauseef Mohyud-Din, Muhammad Asad Iqbal and Saleh M. Hassan
Entropy 2015, 17(10), 6925-6936; https://doi.org/10.3390/e17106925 - 16 Oct 2015
Cited by 23 | Viewed by 5425
Abstract
Physical Phenomena’s located around us are primarily nonlinear in nature and their solutions are of highest significance for scientists and engineers. In order to have a better representation of these physical models, fractional calculus is used. Fractional order oscillation equations are included among [...] Read more.
Physical Phenomena’s located around us are primarily nonlinear in nature and their solutions are of highest significance for scientists and engineers. In order to have a better representation of these physical models, fractional calculus is used. Fractional order oscillation equations are included among these nonlinear phenomena’s. To tackle with the nonlinearity arising, in these phenomena’s we recommend a new method. In the proposed method, Picard’s iteration is used to convert the nonlinear fractional order oscillation equation into a fractional order recurrence relation and then Legendre wavelets method is applied on the converted problem. In order to check the efficiency and accuracy of the suggested modification, we have considered three problems namely: fractional order force-free Duffing–van der Pol oscillator, forced Duffing–van der Pol oscillator and higher order fractional Duffing equations. The obtained results are compared with the results obtained via other techniques. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)
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1699 KiB  
Article
Identification of Green, Oolong and Black Teas in China via Wavelet Packet Entropy and Fuzzy Support Vector Machine
by Shuihua Wang, Xiaojun Yang, Yudong Zhang, Preetha Phillips, Jianfei Yang and Ti-Fei Yuan
Entropy 2015, 17(10), 6663-6682; https://doi.org/10.3390/e17106663 - 25 Sep 2015
Cited by 97 | Viewed by 8059
Abstract
To develop an automatic tea-category identification system with a high recall rate, we proposed a computer-vision and machine-learning based system, which did not require expensive signal acquiring devices and time-consuming procedures. We captured 300 tea images using a 3-CCD digital camera, and then [...] Read more.
To develop an automatic tea-category identification system with a high recall rate, we proposed a computer-vision and machine-learning based system, which did not require expensive signal acquiring devices and time-consuming procedures. We captured 300 tea images using a 3-CCD digital camera, and then extracted 64 color histogram features and 16 wavelet packet entropy (WPE) features to obtain color information and texture information, respectively. Principal component analysis was used to reduce features, which were fed into a fuzzy support vector machine (FSVM). Winner-take-all (WTA) was introduced to help the classifier deal with this 3-class problem. The 10 × 10-fold stratified cross-validation results show that the proposed FSVM + WTA method yields an overall recall rate of 97.77%, higher than 5 existing methods. In addition, the number of reduced features is only five, less than or equal to existing methods. The proposed method is effective for tea identification. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)
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612 KiB  
Article
On the Exact Solution of Wave Equations on Cantor Sets
by Dumitru Baleanu, Hasib Khan, Hossien Jafari and Rahmat Ali Khan
Entropy 2015, 17(9), 6229-6237; https://doi.org/10.3390/e17096229 - 08 Sep 2015
Cited by 25 | Viewed by 5069
Abstract
The transfer of heat due to the emission of electromagnetic waves is called thermal radiations. In local fractional calculus, there are numerous contributions of scientists, like Mandelbrot, who described fractal geometry and its wide range of applications in many scientific fields. Christianto and [...] Read more.
The transfer of heat due to the emission of electromagnetic waves is called thermal radiations. In local fractional calculus, there are numerous contributions of scientists, like Mandelbrot, who described fractal geometry and its wide range of applications in many scientific fields. Christianto and Rahul gave the derivation of Proca equations on Cantor sets. Hao et al. investigated the Helmholtz and diffusion equations in Cantorian and Cantor-Type Cylindrical Coordinates. Carpinteri and Sapora studied diffusion problems in fractal media in Cantor sets. Zhang et al. studied local fractional wave equations under fixed entropy. In this paper, we are concerned with the exact solutions of wave equations by the help of local fractional Laplace variation iteration method (LFLVIM). We develop an iterative scheme for the exact solutions of local fractional wave equations (LFWEs). The efficiency of the scheme is examined by two illustrative examples. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)
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359 KiB  
Article
Active Control of a Chaotic Fractional Order Economic System
by Haci Mehmet Baskonus, Toufik Mekkaoui, Zakia Hammouch and Hasan Bulut
Entropy 2015, 17(8), 5771-5783; https://doi.org/10.3390/e17085771 - 11 Aug 2015
Cited by 123 | Viewed by 6666
Abstract
In this paper, a fractional order economic system is studied. An active control technique is applied to control chaos in this system. The stabilization of equilibria is obtained by both theoretical analysis and the simulation result. The numerical simulations, via the improved Adams–Bashforth [...] Read more.
In this paper, a fractional order economic system is studied. An active control technique is applied to control chaos in this system. The stabilization of equilibria is obtained by both theoretical analysis and the simulation result. The numerical simulations, via the improved Adams–Bashforth algorithm, show the effectiveness of the proposed controller. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)
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852 KiB  
Article
Power-Type Functions of Prediction Error of Sea Level Time Series
by Ming Li, Yuanchun Li and Jianxing Leng
Entropy 2015, 17(7), 4809-4837; https://doi.org/10.3390/e17074809 - 09 Jul 2015
Cited by 11 | Viewed by 5410
Abstract
This paper gives the quantitative relationship between prediction error and given past sample size in our research of sea level time series. The present result exhibits that the prediction error of sea level time series in terms of given past sample size follows [...] Read more.
This paper gives the quantitative relationship between prediction error and given past sample size in our research of sea level time series. The present result exhibits that the prediction error of sea level time series in terms of given past sample size follows decayed power functions, providing a quantitative guideline for the quality control of sea level prediction. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)
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1582 KiB  
Article
Analysis of the Keller–Segel Model with a Fractional Derivative without Singular Kernel
by Abdon Atangana and Badr Saad T. Alkahtani
Entropy 2015, 17(6), 4439-4453; https://doi.org/10.3390/e17064439 - 23 Jun 2015
Cited by 249 | Viewed by 8329
Abstract
Using some investigations based on information theory, the model proposed by Keller and Segel was extended to the concept of fractional derivative using the derivative with fractional order without singular kernel recently proposed by Caputo and Fabrizio. We present in detail the existence [...] Read more.
Using some investigations based on information theory, the model proposed by Keller and Segel was extended to the concept of fractional derivative using the derivative with fractional order without singular kernel recently proposed by Caputo and Fabrizio. We present in detail the existence of the coupled-solutions using the fixed-point theorem. A detailed analysis of the uniqueness of the coupled-solutions is also presented. Using an iterative approach, we derive special coupled-solutions of the modified system and we present some numerical simulations to see the effect of the fractional order. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)
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