Showing papers in "Abstract and Applied Analysis in 2014"

Application of Local Fractional Series Expansion Method to Solve Klein-Gordon Equations on Cantor Sets

Abstract: We use the local fractional series expansion method to solve the Klein-Gordon equations on Cantor sets within the local fractional derivatives. The analytical solutions within the nondifferential terms are discussed. The obtained results show the simplicity and efficiency of the present technique with application to the problems of the liner differential equations on Cantor sets.

A Survey of Modelling and Identification of Quadrotor Robot

Abstract: A quadrotor is a rotorcraft capable of hover, forward flight, and VTOL and is emerging as a fundamental research and application platform at present with flexibility, adaptability, and ease of construction. Since a quadrotor is basically considered an unstable system with the characteristics of dynamics such as being intensively nonlinear, multivariable, strongly coupled, and underactuated, a precise and practical model is critical to control the vehicle which seems to be simple to operate. As a rotorcraft, the dynamics of a quadrotor is mainly dominated by the complicated aerodynamic effects of the rotors. This paper gives a tutorial of the platform configuration, methodology of modeling, comprehensive nonlinear model, the aerodynamic effects, and model identification for a quadrotor.

Comparison of the Finite Volume and Lattice Boltzmann Methods for Solving Natural Convection Heat Transfer Problems inside Cavities and Enclosures

Abstract: Different numerical methods have been implemented to simulate internal natural convection heat transfer and also to identify the most accurate and efficient one. A laterally heated square enclosure, filled with air, was studied. A FORTRAN code based on the lattice Boltzmann method (LBM) was developed for this purpose. The finite difference method was applied to discretize the LBM equations. Furthermore, for comparison purpose, the commercially available CFD package FLUENT, which uses finite volume Method (FVM), was also used to simulate the same problem. Different discretization schemes, being the first order upwind, second order upwind, power law, and QUICK, were used with the finite volume solver where the SIMPLE and SIMPLEC algorithms linked the velocity-pressure terms. The results were also compared with existing experimental and numerical data. It was observed that the finite volume method requires less CPU usage time and yields more accurate results compared to the LBM. It has been noted that the 1st order upwind/SIMPLEC combination converges comparatively quickly with a very high accuracy especially at the boundaries. Interestingly, all variants of FVM discretization/pressure-velocity linking methods lead to almost the same number of iterations to converge but higher-order schemes ask for longer iterations.

Fixed Point Theory in -Complete Metric Spaces with Applications

Abstract: The aim of this paper is to introduce new concepts of --complete metric space and --continuous function and establish fixed point results for modified ---rational contraction mappings in --complete metric spaces. As an application, we derive some Suzuki type fixed point theorems and new fixed point theorems for -graphic-rational contractions. Moreover, some examples and an application to integral equations are given here to illustrate the usability of the obtained results.

Generalized Kudryashov Method for Time-Fractional Differential Equations

Abstract: In this study, the generalized Kudryashov method (GKM) is handled to find exact solutions of time-fractional Burgers equation, time-fractional Cahn-Hilliard equation, and time-fractional generalized third-order KdV equation. These time-fractional equations can be turned into another nonlinear ordinary differantial equation by travelling wave transformation. Then, GKM has been implemented to attain exact solutions of time-fractional Burgers equation, time-fractional Cahn-Hilliard equation, and time-fractional generalized third-order KdV equation. Also, some new hyperbolic function solutions have been obtained by using this method. It can be said that this method is a generalized form of the classical Kudryashov method.

Local Fractional Sumudu Transform with Application to IVPs on Cantor Sets

Abstract: Local fractional derivatives were investigated intensively during the last few years. The coupling method of Sumudu transform and local fractional calculus (called as the local fractional Sumudu transform) was suggested in this paper. The presented method is applied to find the nondifferentiable analytical solutions for initial value problems with local fractional derivative. The obtained results are given to show the advantages.

Survey of Direct Transcription for Low-Thrust Space Trajectory Optimization with Applications

Abstract: Space trajectory design is usually addressed as an optimal control problem. Although it relies on the classic theory of optimal control, this branch possesses some peculiarities that led to the development of ad hoc techniques, which can be grouped into two categories: direct and indirect methods. This paper gives an overview of the principal techniques belonging to the direct methods. The technique known as “direct transcription and collocation” is illustrated by considering Hermite-Simpson, high-order Gauss-Lobatto, and pseudospectral methods. Practical examples are given, and several hints to improve efficiency and robustness are implemented.

Some Subordination Results on -Analogue of Ruscheweyh Differential Operator

Abstract: We derive the -analogue of the well-known Ruscheweyh differential operator using the concept of -derivative. Here, we investigate several interesting properties of this -operator by making use of the method of differential subordination.

A Weighted Voting Classifier Based on Differential Evolution

Abstract: Ensemble learning is to employ multiple individual classifiers and combine their predictions, which could achieve better performance than a single classifier. Considering that different base classifier gives different contribution to the final classification result, this paper assigns greater weights to the classifiers with better performance and proposes a weighted voting approach based on differential evolution. After optimizing the weights of the base classifiers by differential evolution, the proposed method combines the results of each classifier according to the weighted voting combination rule. Experimental results show that the proposed method not only improves the classification accuracy, but also has a strong generalization ability and universality.

Robust Η∞Control for a Class of Discrete Time-Delay Stochastic Systems with Randomly Occurring Nonlinearities

Abstract: Copyright © 2014 Yamin Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Local Fractional Variational Iteration and Decomposition Methods for Wave Equation on Cantor Sets within Local Fractional Operators

Abstract: We perform a comparison between the fractional iteration and decomposition methods applied to the wave equation on Cantor set. The operators are taken in the local sense. The results illustrate the significant features of the two methods which are both very effective and straightforward for solving the differential equations with local fractional derivative.

Nonlinear Methodologies for Identifying Seismic Event and Nuclear Explosion Using Random Forest, Support Vector Machine, and Naive Bayes Classification

Abstract: The discrimination of seismic event and nuclear explosion is a complex and nonlinear system. The nonlinear methodologies including Random Forests (RF), Support Vector Machines (SVM), and Naive Bayes Classifier (NBC) were applied to discriminant seismic events. Twenty earthquakes and twenty-seven explosions with nine ratios of the energies contained within predetermined “velocity windows” and calculated distance are used in discriminators. Based on the one out cross-validation, ROC curve, calculated accuracy of training and test samples, and discriminating performances of RF, SVM, and NBC were discussed and compared. The result of RF method clearly shows the best predictive power with a maximum area of 0.975 under the ROC among RF, SVM, and NBC. The discriminant accuracies of RF, SVM, and NBC for test samples are 92.86%, 85.71%, and 92.86%, respectively. It has been demonstrated that the presented RF model can not only identify seismic event automatically with high accuracy, but also can sort the discriminant indicators according to calculated values of weights.

On Fractional SIRC Model with Salmonella Bacterial Infection

Abstract: We propose a fractional order SIRC epidemic model to describe the dynamics of Salmonella bacterial infection in animal herds. The infection-free and endemic steady sates, of such model, are asymptotically stable under some conditions. The basic reproduction number is calculated, using next-generation matrix method, in terms of contact rate, recovery rate, and other parameters in the model. The numerical simulations of the fractional order SIRC model are performed by Caputo’s derivative and using unconditionally stable implicit scheme. The obtained results give insight to the modelers and infectious disease specialists.

Complex Differences and Difference Equations

Abstract: In more recent years, activity in the area of the complex differences and the complex difference equations has fleetly increased. This journal has set up a column of this special issue. We were pleased to invite the interested authors to contribute their original research papers as well as good expository papers to this special issue that will make better improvement on the theory of complex differences and difference equations. In this special issue, many good results are obtained. Difference equations are widely applied to mathematical physics, economics, and chemistry. In this special issue, Z.B. Huang and R.-R. Zhang, J. Li et al., and L. Gao and Y. Wang investigate the growth, a Borel exceptional value of meromorphic solutions to different types of higher order nonliear difference equations, respectively; B. Chen and S. Li investigate the Schwarzian type difference equation. D. Liu et al., Z. Mao and H. Liu, and S. Li and B. Chen investigate unicity of meromorphic functions concerning different types of difference operators. Recently, many difference analogues of the classic Nevanlinna theory are obtained. In this special issue, X.-M. Zheng and H. Y. Xu obtain a differential difference analogue of Valiron-Mohonko theorem. Related topics with complex difference, J. E. Kim and K. H. Shon investigate the regularity of functions on dual split quaternions in Clifford analysis and the tensor product representation of polynomials of weak type in a DF-space; Q. Zhang and Z. Liu et al. investigate different types of real difference equations, respectively; L. Shen and Q. Xu investigate stochastic differential equations. This special issue stimulates the continuing efforts to the complex differences and the complex difference equations.

Stability for Caputo Fractional Differential Systems

Abstract: We introduce the notion of h-stability for fractional differential systems. Then we investigate the boundedness and h-stability of solutions of Caputo fractional differential systems by using fractional comparison principle and fractional Lyapunov direct method. Furthermore, we give examples to illustrate our results.

Modelling Fractal Waves on Shallow Water Surfaces via Local Fractional Korteweg-de Vries Equation

Abstract: A mathematical model of fractal waves on shallow water surfaces is developed by using the concepts of local fractional calculus. The derivations of linear and nonlinear local fractional versions of the Korteweg-de Vries equation describing fractal waves on shallow water surfaces are obtained.

Extended Auxiliary Equation Method and Its Applications to Three Generalized NLS Equations

Abstract: The auxiliary equation method proposed by Sirendaoreji is extended to construct new types of elliptic function solutions of nonlinear evolution equations. The effectiveness of the extended method is demonstrated by applications to the RKL model, the generalized derivative NLS equation and the Kundu-Eckhaus equation. Not only are the Jacobian elliptic function solutions are derived, but also the solitary wave solutions and trigonometric function solutions are obtained in a unified way.

Asymptotic Behavior of Higher-Order Quasilinear Neutral Differential Equations

Abstract: We study asymptotic behavior of solutions to a class of higher-order quasilinear neutral differential equations under the assumptions that allow applications to even- and odd-order differential equations with delayed and advanced arguments, as well as to functional differential equations with more complex arguments that may, for instance, alternate indefinitely between delayed and advanced types. New theorems extend a number of results reported in the literature. Illustrative examples are presented.

Analytical Study of Fractional-Order Multiple Chaotic FitzHugh-Nagumo Neurons Model Using Multistep Generalized Differential Transform Method

Abstract: The multistep generalized differential transform method is applied to solve the fractional-order multiple chaotic FitzHugh-Nagumo (FHN) neurons model The algorithm is illustrated by studying the dynamics of three coupled chaotic FHN neurons equations with different gap junctions under external electrical stimulation The fractional derivatives are described in the Caputo sense Furthermore, we present figurative comparisons between the proposed scheme and the classical fourth-order Runge-Kutta method to demonstrate the accuracy and applicability of this method The graphical results reveal that only few terms are required to deduce the approximate solutions which are found to be accurate and efficient

On Fractional Order Hybrid Differential Equations

Abstract: We develop the theory of fractional hybrid differential equations with linear and nonlinear perturbations involving the Caputo fractional derivative of order . Using some fixed point theorems we prove the existence of mild solutions for two types of hybrid equations. Examples are given to illustrate the obtained results.

Generalized Metric Spaces Do Not Have the Compatible Topology

Abstract: We study generalized metric spaces, which were introduced by Branciari (2000). In particular, generalized metric spaces do not necessarily have the compatible topology. Also we prove a generalization of the Banach contraction principle in complete generalized metric spaces.

High-Order Algorithms for Riesz Derivative and Their Applications $(I)$

Abstract: We firstly develop the high-order numerical algorithms for the left and right Riemann-Liouville derivatives. Using these derived schemes, we can get high-order algorithms for the Riesz fractional derivative. Based on the approximate algorithm, we construct the numerical scheme for the space Riesz fractional diffusion equation, where a fourth-order scheme is proposed for the spacial Riesz derivative, and where a compact difference scheme is applied to approximating the first-order time derivative. It is shown that the difference scheme is unconditionally stable and convergent. Finally, numerical examples are provided which are in line with the theoretical analysis.

Study on Support Vector Machine-Based Fault Detection in Tennessee Eastman Process

Abstract: This paper investigates the proficiency of support vector machine (SVM) using datasets generated by Tennessee Eastman process simulation for fault detection. Due to its excellent performance in generalization, the classification performance of SVM is satisfactory. SVM algorithm combined with kernel function has the nonlinear attribute and can better handle the case where samples and attributes are massive. In addition, with forehand optimizing the parameters using the cross-validation technique, SVM can produce high accuracy in fault detection. Therefore, there is no need to deal with original data or refer to other algorithms, making the classification problem simple to handle. In order to further illustrate the efficiency, an industrial benchmark of Tennessee Eastman (TE) process is utilized with the SVM algorithm and PLS algorithm, respectively. By comparing the indices of detection performance, the SVM technique shows superior fault detection ability to the PLS algorithm.

Numerical Solutions of Nonlinear Fractional Partial Differential Equations Arising in Spatial Diffusion of Biological Populations

Abstract: The main aim of this work is to present a user friendly numerical algorithm based on homotopy perturbation Sumudu transform method for nonlinear fractional partial differential arising in spatial diffusion of biological populations in animals. The movements are made generally either by mature animals driven out by invaders or by young animals just reaching maturity moving out of their parental territory to establish breeding territory of their own. The homotopy perturbation Sumudu transform method is a combined form of the Sumudu transform method and homotopy perturbation method. The obtained results are compared with Sumudu decomposition method. The numerical solutions obtained by the proposed method indicate that the approach is easy to implement and accurate. These results reveal that the proposed method is computationally very attractive.

Generalized Convex Functions on Fractal Sets and Two Related Inequalities

Abstract: We introduce the generalized convex function on fractal sets of real line numbers and study the properties of the generalized convex function. Based on these properties, we establish the generalized Jensen’s inequality and generalized Hermite-Hadamard's inequality. Furthermore, some applications are given.

Local Fractional Function Decomposition Method for Solving Inhomogeneous Wave Equations with Local Fractional Derivative

Abstract: We propose the local fractional function decomposition method, which is derived from the coupling method of local fractional Fourier series and Yang-Laplace transform. The forms of solutions for local fractional differential equations are established. Some examples for inhomogeneous wave equations are given to show the accuracy and efficiency of the presented technique.

Convolution Properties for Certain Classes of Analytic Functions Defined by $q$-Derivative Operator

Abstract: We investigate convolution properties and coefficients estimates for two classes of analytic functions involving the -derivative operator defined in the open unit disc. Some of our results improve previously known results.

An Existence Theorem for Fractional Hybrid Differential Inclusions of Hadamard Type with Dirichlet Boundary Conditions

Abstract: This paper studies the existence of solutions for a boundary value problem of nonlinear fractional hybrid differential inclusions by using a fixed point theorem due to Dhage (2006). The main result is illustrated with the aid of an example.

A Local Fractional Integral Inequality on Fractal Space Analogous to Anderson’s Inequality

Abstract: Anderson's inequality (Anderson, 1958) as well as its improved version given by Fink (2003) is known to provide interesting examples of integral inequalities. In this paper, we establish local fractional integral analogue of Anderson's inequality on fractal space under some suitable conditions. Moreover, we also show that the local fractional integral inequality on fractal space, which we have proved in this paper, is a new generalization of the classical Anderson's inequality.

A Linear Functional Equation of Third Order Associated with the Fibonacci Numbers

Abstract: Given a vector space , we investigate the solutions of the linear functional equation of third order , which is strongly associated with a well-known identity for the Fibonacci numbers. Moreover, we prove the Hyers-Ulam stability of that equation.