Fundamenta Informaticae - Volume 151, issue 1-4

Authors: Mohyud-Din, Syed Tauseef | Ali, Ayyaz

Article Type: Research Article

Abstract: We study a nonlinear generalized Sawada-Kotera equation of fractional order via the exp(–ϕ (η ))–expansion method and Shifted modified Chebyshev Wavelet technique. We obtain abundant exact solutions and approximate solution of the equation. The results of the study shows that the exp(–ϕ (η ))–expansion method is very effective and proficient for solving nonlinear fractional partial differential equations. The solitary wave solutions are obtained through the hyperbolic, trigonometric, exponential and rational functions. Graphical representations along with the numerical data reinforce the efficacy of the used procedure. The specified idea is very expedient for fractional PDEs, and could be extended to other …physical problems. Results of the proposed methods show an excellent conformity with the exact solution of the considered problem. Show more

Keywords: Exp(–ϕ(η))–expansion method, Legendre Wavelets Method, Generalized Sawada-Kotera equation, Soliton solutions, Fractional calculus, Caputo’s Fractional Derivative

DOI: 10.3233/FI-2017-1486

Citation: Fundamenta Informaticae, vol. 151, no. 1-4, pp. 173-190, 2017

Authors: Wang, Shuihua | Rao, Ravipudi Venkata | Chen, Peng | Zhang, Yudong | Liu, Aijun | Wei, Ling

Article Type: Research Article

Abstract: (Aim ) Abnormal breast can be diagnosed using the digital mammography. Traditional manual interpretation method cannot yield high accuracy. (Method ) In this study, we proposed a novel computer-aided diagnosis system for detecting abnormal breasts in mammogram images. First, we segmented the region-of-interest. Next, the weighted-type fractional Fourier transform (WFRFT) was employed to obtain the unified time-frequency spectrum. Third, principal component analysis (PCA) was introduced and used to reduce the spectrum to only 18 principal components. Fourth, feed-forward neural network (FNN) was utilized to generate the classifier. Finally, a novel algorithm-specific parameter free approach, Jaya, was employed to train the …classifier. (Results ) Our proposed WFRFT + PCA + Jaya-FNN achieved sensitivity of 92.26% ± 3.44%, specificity of 92.28% ± 3.58%, and accuracy of 92.27% ± 3.49%. (Conclusions ) The proposed CAD system is effective in detecting abnormal breasts and performs better than 5 state-of-the-art systems. Besides, Jaya is more effective in training FNN than BP, MBP, GA, SA, and PSO. Show more

Keywords: fractional Fourier transform, abnormal breast detection, computer-aided diagnosis, mammogram, feedforward neural network, Jaya algorithm

DOI: 10.3233/FI-2017-1487

Citation: Fundamenta Informaticae, vol. 151, no. 1-4, pp. 191-211, 2017

Authors: Kumar, Amit | Kumar, Sunil | Yan, Sheng-Ping

Article Type: Research Article

Abstract: In this article, improved residual power series method (RPSM) is effectively implemented to find the approximate analytical solution of a time fractional diffusion equations. The proposed method is an analytic technique based on the generalized Taylor’s series formula which construct an analytical solution in the form of a convergent series. In order to illustrate the advantages and the accuracy of the RPSM, we have applied the method to two different examples. In case of first example, different cases of initial conditions are considered. Finally, the solutions of the time fractional diffusion equations are investigate through graphical representation, which interpret simplicity, …accuracy and practical usefulness of the present method. Show more

Keywords: Fractional diffusion equation, residual power series, fractional power series, exact solution

DOI: 10.3233/FI-2017-1488

Citation: Fundamenta Informaticae, vol. 151, no. 1-4, pp. 213-230, 2017

Authors: Wu, Shu-Lin | Wu, Guo-Cheng

Article Type: Research Article

Abstract: In this paper, we analyze the convergence properties of the Schwarz waveform relaxation (SWR) algorithm with Robin transmission conditions (TCs) for a class of heat equations with Riemann-Liouville fractional derivative. The Robin TCs contain a free parameter, which has a significant effect on the convergence rate of the SWR algorithm, and optimizing this parameter is an important step for the convergence analysis of the SWR algorithm. By studying the monotonic properties of the convergence factor obtained by applying the Fourier transform to the error functions, we provide a realiable choice of the Robin parameter in the nonoverlapping case. Numerical results …are provided, which show that the analyzed Robin parameter results in satisfactory convergence rate. Show more

Keywords: Schwarz waveform relaxation, fractional heat equations, parameter optimization

DOI: 10.3233/FI-2017-1489

Citation: Fundamenta Informaticae, vol. 151, no. 1-4, pp. 231-240, 2017

Authors: Anastassiou, George A. | Argyros, Ioannis K. | Kumar, Sunil

Article Type: Research Article

Abstract: We present monotone convergence results for general iterative methods in order to approximate a solution of a nonlinear equation defined on a partially ordered linear topological space. The main novelty of the paper is that the operators appearing in the iterative method are not necessarily linear. This way we expand of the applicability of iterative methods. Some applications are also provided from fractional calculus using Caputo and Canavati type fractional derivatives and other areas.

Keywords: Monotone convergence, partially ordered linear topological space, Fractional Calculus, Caputo and Canavati type fractional derivatives

DOI: 10.3233/FI-2017-1490

Citation: Fundamenta Informaticae, vol. 151, no. 1-4, pp. 241-253, 2017

Authors: Heydari, M. H. | Hooshmandasl, M. R. | Cattani, C. | Hariharan, G.

Article Type: Research Article

Abstract: In this paper, a new operational matrix of variable-order fractional derivative (OMV-FD) is derived for the second kind Chebyshev wavelets (SKCWs). Moreover, a new optimization wavelet method based on SKCWs is proposed to solve multi variable-order fractional differential equations (MV-FDEs). In the proposed method, the solution of the problem under consideration is expanded in terms of SKCWs. Then, the residual function and its errors in 2-norm are employed for converting the problem under study to an optimization one, which optimally chooses the unknown coefficients. Finally, the method of constrained extremum is applied, which consists of adjoining the constraint equations obtained …from the given initial conditions to the object function obtained from residual function by a set of unknown Lagrange multipliers. The main advantage of this approach is that it reduces such problems to those optimization problems, which greatly simplifies them and also leads to obtain a good approximate solution for them. Show more

Keywords: Second kind Chebyshev wavelets (SKCWs), Optimization method, Operational matrix of variable-order fractional derivative (OMV-FD), Multi variable-order fractional differential equation (MV-FDE), Caputo’s variable-order fractional derivative

DOI: 10.3233/FI-2017-1491

Citation: Fundamenta Informaticae, vol. 151, no. 1-4, pp. 255-273, 2017

Authors: Wang, Shuihua | Li, Peng | Chen, Peng | Phillips, Preetha | Liu, Ge | Du, Sidan | Zhang, Yudong

Article Type: Research Article

Abstract: (Aim) In order to detect pathological brains in a more efficient way, (Method) we proposed a novel system of pathological brain detection (PBD) that combined wavelet packet Tsallis entropy (WPTE), feedforward neural network (FNN), and real-coded biogeography-based optimization (RCBBO). (Results) The experiments showed the proposed WPTE + FNN + RCBBO approach yielded an average accuracy of 99.49% over a 255-image dataset. (Conclusions) The WPTE + FNN + RCBBO performed better than 10 state-of-the-art approaches.

Keywords: pathological brain detection, feed-forward neural network, wavelet packet Tsallis entropy, real-coded biogeography-based optimization

DOI: 10.3233/FI-2017-1492

Citation: Fundamenta Informaticae, vol. 151, no. 1-4, pp. 275-291, 2017

Authors: Moghaddam, B. P. | Machado, J. A. T.

Article Type: Research Article

Abstract: In this paper we discuss different definitions of variable-order derivatives of high order and we propose accurate and robust algorithms for their approximate calculation. The proposed algorithms are based on finite difference approximations and B-spline interpolation. We compare the performance of the algorithms by experimental convergence order. Numerical examples are presented demonstrating the efficiency and accuracy of the proposed algorithms.

Keywords: Fractional calculus, Variable-order derivative, Spline approximation, Finite difference approximation, Convergence order, Numerical method

DOI: 10.3233/FI-2017-1493

Citation: Fundamenta Informaticae, vol. 151, no. 1-4, pp. 293-311, 2017

Authors: Atıcı, Ferhan M. | Atıcı, Mustafa | Belcher, Michael | Marshall, Dana

Article Type: Research Article

Abstract: In this paper, we introduce a new class of nonlinear discrete fractional equations to model tumor growth rates in mice. For the data fitting purpose, we develop a new method which can be considered as an improved version of the partial sum method for parameter estimations. We demonstrate the goodness of fit by comparing the models with three statistical measures.

Keywords: Discrete Fractional Calculus, Parameter Estimations, Data Fitting, Method of Partial Sums

DOI: 10.3233/FI-2017-1494

Citation: Fundamenta Informaticae, vol. 151, no. 1-4, pp. 313-324, 2017

Authors: Wang, Shuihua | Phillips, Preetha | Liu, Aijun | Du, Sidan

Article Type: Research Article

Abstract: (Objective) In order to increase classification accuracy of tea-category identification (TCI) system, this paper proposed a novel approach. (Method) The proposed methods first extracted 64 color histogram to obtain color information, and 16 wavelet packet entropy to obtain the texture information. With the aim of reducing the 80 features, principal component analysis was harnessed. The reduced features were used as input to generalized eigenvalue proximal support vector machine (GEPSVM). Winner-takes-all (WTA) was used to handle the multiclass problem. Two kernels were tested, linear kernel and Radial basis function (RBF) kernel. Ten repetitions of 10-fold stratified cross validation technique were used …to estimate the out-of-sample errors. We named our method as GEPSVM + RBF + WTA and GEPSVM + WTA. (Result) The results showed that PCA reduced the 80 features to merely five with explaining 99.90% of total variance. The recall rate of GEPSVM + RBF + WTA achieved the highest overall recall rate of 97.9%. (Conclusion) This was higher than the result of GEPSVM + WTA and other five state-of-the-art algorithms: back propagation neural network, RBF support vector machine, genetic neural-network, linear discriminant analysis, and fitness-scaling chaotic artificial bee colony artificial neural network. Show more

Keywords: Tea category identification, computer vision, color histogram, wavelet packet entropy, winner-takes-all, radial basis function, artificial neural network, pattern recognition, support vector machine

DOI: 10.3233/FI-2017-1495

Citation: Fundamenta Informaticae, vol. 151, no. 1-4, pp. 325-339, 2017

Authors: Lenzi, M. K. | Lenzi, E. K. | Novatski, A. | Menechini Neto, R. | da Silva, L. R.

Article Type: Research Article

Abstract: We analyze the behavior of a system governed by a fractional diffusion equation with spherical symmetry and subjected to integro–differential boundary conditions which can simulate sorption, desorption and reaction processes. We consider the processes defined in terms of kinetic equations that couple the surface processes with the bulk dynamic enable us to describe scenarios where the surface modifies the bulk dynamics and this may change the behavior on surface. This problem is presented in terms of a general formulation satisfying the mass balance and a particular application characterized by a reversible process on the surface is analyzed. For this application, …we obtain exact solutions in terms of the Green function approach and evaluate the concentrations on the spherical surface and in the bulk for different processes. These results lead to a rich class of scenarios which can be related to an anomalous diffusion. Show more

Keywords: Fractional Time Derivative, Memory Effect, Kinetic Equation, Sorption

DOI: 10.3233/FI-2017-1496

Citation: Fundamenta Informaticae, vol. 151, no. 1-4, pp. 341-354, 2017

Authors: Zhang, Xianmin | Zhang, Xianzhen | Cao, Hui

Article Type: Research Article

Abstract: In this paper we mainly study a kind of fractional differential equations with not instantaneous impulses, and find the equivalent equations of the impulsive system. The obtained result discovers that there exist general solution for the impulsive system. Next, an example is given to illustrate the obtained result.

Keywords: fractional differential equations, impulsive fractional differential equations, not instantaneous impulse, impulse, general solution

DOI: 10.3233/FI-2017-1497

Citation: Fundamenta Informaticae, vol. 151, no. 1-4, pp. 355-369, 2017

Authors: Hristov, Jordan

Article Type: Research Article

Abstract: This paper focuses on an approximate analytical solution of an initial-boundary value problem of spatial-fractional partial differential diffusion equation with RiemannLiouville fractional derivative in space. The spatial correlation of the superdiffusion coefficient as a power-law has been discussed in cases of fast and slow spatial superdiffusion. Approximate closed form solutions in terms of non-linear similarity variable are based on the integral-balance method and series expansion of the assumed parabolic profile with undefined exponent. The law of the spatial and temporal propagation of the solution was the primary issue and discussed in two cases: fast and slow superdiffussion.

Keywords: approximate analytical solution, integral-balance method, spatial-fractional diffusion equation

DOI: 10.3233/FI-2017-1498

Citation: Fundamenta Informaticae, vol. 151, no. 1-4, pp. 371-388, 2017

Authors: David, Sergio A. | Machado, J. A. Tenreiro | Trevisan, Lucas R. | Inácio Jr., Cláudio M.C. | Lopes, António M.

Article Type: Research Article

Abstract: This paper examines the time series of four important agricultural commodities, namely the soybean, corn, coffee and sugar prices. Time series can exhibit long-range dependence and persistence in their observation. The long memory feature of data is a documented fact and there has been an increasing interest in studying such concepts in the perspective of economics and finance. In this work, we start by analyzing the time series of the four commodities by means of the Fractional Fourier Transform (FrFT) to unveil time-frequency patterns in the data. In a second phase, we apply Auto Regressive Integrated Moving Average (ARIMA) and …Auto Regressive Fractionally Integrated Moving Average (ARFIMA) models for obtaining the spot price composition and predict future price. The ARFIMA process is a known class of long memory model, representing a generalization of the ARIMA algorithm. We compare the performances of the ARIMA and the ARFIMA models and we show that the ARFIMA has a superior performance for future price forecasting. Show more

Keywords: Fractional statistic systems, business and management, Fractional Fourier Transform, computer modeling and simulation

DOI: 10.3233/FI-2017-1499

Citation: Fundamenta Informaticae, vol. 151, no. 1-4, pp. 389-408, 2017

Authors: Yang, Xiao-Jun | Srivastava, H. M. | Torres, Delfim F. M. | Zhang, Yudong

Article Type: Research Article

Abstract: We investigate local fractional nonlinear Riccati differential equations (LFNRDE) by transforming them into local fractional linear ordinary differential equations. The case of LFNRDE with constant coefficients is considered and non-differentiable solutions for special cases obtained.

Keywords: nonlinear Riccati equations, non-differentiable functions, local fractional derivatives

DOI: 10.3233/FI-2017-1500

Citation: Fundamenta Informaticae, vol. 151, no. 1-4, pp. 409-417, 2017

Authors: Zhao, Xiao-Hu | Zhang, Yudong | Zhao, Duan | Yang, Xiaojun

Article Type: Research Article

Abstract: A non-differentiable resistor-capacitor circuit comprised of the capacitor and resistor in the fractal-time domain is first proposed in this article. The solution behavior of the corresponding local fractional ordinary differential equation is presented for the Mittag-Leffler decay defined on Cantor sets. The obtained results reveal the sufficiency of the local fractional calculus in the analysis of the fractal electrical systems.

Keywords: ordinary differential equations, Mittag-Leffler decay, electric circuit, local fractional derivative

DOI: 10.3233/FI-2017-1501

Citation: Fundamenta Informaticae, vol. 151, no. 1-4, pp. 419-429, 2017

Authors: Tarasov, Vasily E.

Article Type: Research Article

Abstract: In this paper, we propose an “informatic” interpretation of the Riemann-Liouville and Caputo derivatives of non-integer orders as reconstruction from infinite sequence of standard derivatives of integer orders. The reconstruction is considered with respect to orders of derivatives.

Keywords: Fractional derivative, Riemann-Liouville derivative, Caputo derivative, derivations of Hadamard type, sampling theorem

DOI: 10.3233/FI-2017-1502

Citation: Fundamenta Informaticae, vol. 151, no. 1-4, pp. 431-442, 2017

Authors: Hassani, Hossein | Dahaghin, Mohammad Shafie | Heydari, Mohammad Hossein | Eshkaftaki, Ali Bayati

Article Type: Research Article

Abstract: In this paper, we present a new optimization method based on a new class of functions, namely generalized polynomials (GPs) for solving linear and nonlinear fractional differential equations (FDEs). In the proposed method, the solution of the problem under study is expanded in terms of the GPs with fixed coefficients , free coefficients and control parameters . The initial conditions are employed to compute the fixed coefficients . The residual function and its ‖.‖2 are employed for converting the problem under consideration to an optimization one and then choosing the unknown free coefficients and control parameters …optimally. As a useful result, the necessary conditions of optimality are derived as a system of nonlinear algebraic equations with unknown free coefficients and control parameters . The validity and accuracy of the approach are illustrated by some numerical examples. The obtained results show that the proposed method is very efficient and accurate. Show more

Keywords: Optimization method, Generalized polynomials (GPs), Free coefficients, Fixed coefficients, Control parameters, Fractional differential equations (FDEs)

DOI: 10.3233/FI-2017-1503

Citation: Fundamenta Informaticae, vol. 151, no. 1-4, pp. 443-457, 2017

Authors: Cattani, C. | Guariglia, E. | Wang, S.

Article Type: Research Article

Abstract: The α -order fractional derivative of the Dirichlet η function is computed in order to investigate the behavior of the fractional derivative of the Riemann zeta function ζ (α ) on the critical strip. The convergence of η (α ) is studied. In particular, its half-plane of convergence gives the possibility to better understand the ζ (α ) and its critical strip. As an application, two signal processing networks, corresponding to η (α ) and to its Fourier transform respectively, are shortly described.

Keywords: Fractional derivatives, Riemann ζ functions, Dirichlet η functions, signal processings, Fourier transform, critical strip

DOI: 10.3233/FI-2017-1504

Citation: Fundamenta Informaticae, vol. 151, no. 1-4, pp. 459-472, 2017

Authors: Derdar, Nedjemeddine | Debbouche, Amar

Article Type: Research Article

Abstract: We investigate the unique solvability of a class of nonlinear nonlocal differential equations associated with degenerate linear operator at the fractional Caputo derivative. For the main results, we use the theory of fractional calculus and (L , p )-boundedness technique that based on the analysis of both strongly (L , p )-sectorial operators and strongly (L , p )-radial operators. The obtained results are applicable to degenerate fractional Cauchy and Showalter–Sidorov problems in Banach spaces. Finally, we give an application described by time-fractional order Oskolkov system.

Keywords: Degenerate nonlinear equation, fractional calculus, (L, p)-bounded operator, non-local condition

DOI: 10.3233/FI-2017-1505

Citation: Fundamenta Informaticae, vol. 151, no. 1-4, pp. 473-485, 2017

Authors: Kalamani, P. | Mallika Arjunan, M. | Mallika, D. | Baleanu, D.

Article Type: Research Article

Abstract: Based on concepts for semigroup theory, fractional calculus, Banach contraction principle and Krasnoselskii fixed point theorem (FPT), this manuscript is principally involved with existence results of Riemann-Liouville (RL) fractional neutral integro-differential systems (FNIDS) with nonlocal conditions (NLCs) in Banach spaces. An example is offered to demonstrate the theoretical concepts.

Keywords: Fractional order integro-differential equations, Riemann-Liouville fractional derivatives, fixed point, semigroup theory

DOI: 10.3233/FI-2017-1506

Citation: Fundamenta Informaticae, vol. 151, no. 1-4, pp. 487-504, 2017

Authors: Wang, Shuihua | Yang, Ming | Li, Jianwu | Wu, Xueyan | Wang, Hainan | Liu, Bin | Dong, Zhengchao | Zhang, Yudong

Article Type: Research Article

Abstract: To detect the sensorineural hearing loss (SNHL) from healthy people accurately, we used magnetic resonance imaging (MRI) to obtain the imaging data, and then proposed a new computer-aided diagnosis (CAD) system, on the basis of texture analysis method. In the first, we extracted 12-element feature from each brain image via fractional Fourier entropy (FRFE). Afterwards, multilayer perceptron (MLP) was employed as the classifier, which was trained by a novel fitness-scaling adaptive genetic algorithm (FSAGA). The statistical analysis over 49 subjects showed the overall accuracy of our method yielded 95.51%. Experimental results performed better than four state-of-the-art weight optimization methods, and …this CAD system give significantly better performance than manual interpretation. Show more

Keywords: texture analysis, fractional Fourier entropy, power-rank fitness scaling, sensorineural hearing loss, genetic algorithm

DOI: 10.3233/FI-2017-1507

Citation: Fundamenta Informaticae, vol. 151, no. 1-4, pp. 505-521, 2017

Authors: Sharmaa, J. B. | Sharma, K.K. | Purohit, S.D. | Atangana, Abdon

Article Type: Research Article

Abstract: Watermarking is proposed as solution to authentication, copyright protection and security requirements of multimedia objects (speech, image and video). In this paper a watermarking scheme based on finite radon transform (FRAT), fractional Fourier Transform (FRFT) and singular value decomposition is proposed. In the proposed scheme, image to be watermarked is first transformed by finite radon transform, the radon transformed image is further transformed by FRFT, and singular values of FRFT transformed image are modified to embed the watermark. Inverse transformation is applied to obtain watermarked image. Simulations are performed under various test conditions with different FRFT transform angles for improved …robustness and visual transparence of watermarked image. Results of the proposed scheme are better in comparison to the existing schemes for most of the attacks. Proposed scheme provide additional degree of freedom in security, robustness, payload capacity and visual transparence. Proposed scheme can also be used to communicate or store the watermarked image as erasure code, to reduce communication errors over a network, due to the use of FRAT. Show more

Keywords: Image watermarking, Fractional Fourier transform, Finite Radon transforms, Singular Value Decomposition

DOI: 10.3233/FI-2017-1508

Citation: Fundamenta Informaticae, vol. 151, no. 1-4, pp. 523-543, 2017

Article Type: Other

Citation: Fundamenta Informaticae, vol. 151, no. 1-4, pp. 545-548, 2017