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