Showing papers in "Abstract and Applied Analysis in 2013"
TL;DR: In this paper, a table of fractional order derivatives of some functions in Riemann-Liouville sense is presented and some advantages and disadvantages of these fractional derivatives are discussed.
Abstract: The purpose of this note is to present the different fractional order derivatives definition that are commonly used in the literature on one hand and to present a table of fractional order derivatives of some functions in Riemann-Liouville sense On other the hand. We present some advantages and disadvantages of these fractional derivatives. And finally we propose alternative fractional derivative definition.
200 citations
TL;DR: A class of fractional-order differential models of biological systems with memory, such as dynamics of tumor-immune system and dynamics of HIV infection of CD4 are provided.
Abstract: We provide a class of fractional-order differential models of biological systems with memory, such as dynamics of tumor-immune system and dynamics of HIV infection of CD4
147 citations
TL;DR: In this paper, the modified trial equation method (MTEM) was applied to the one-dimensional nonlinear fractional wave equation (FWE) and time fractional generalized Burgers equation.
Abstract: The fractional partial differential equations stand for natural phenomena all over the world from science to engineering. When it comes to obtaining the solutions of these equations, there are many various techniques in the literature. Some of these give to us approximate solutions; others give to us analytical solutions. In this paper, we applied the modified trial equation method (MTEM) to the one-dimensional nonlinear fractional wave equation (FWE) and time fractional generalized Burgers equation. Then, we submitted 3D graphics for different value of .
130 citations
TL;DR: In this article, the exp-function method is used for finding the exact solutions of nonlinear fractional equations and new exact solutions are constructed in fractional complex transform to convert fractional differential equations into ordinary differential equations.
Abstract: The exp-function method is presented for finding the exact solutions of nonlinear fractional equations. New solutions are constructed in fractional complex transform to convert fractional differential equations into ordinary differential equations. The fractional derivatives are described in Jumarie's modified Riemann-Liouville sense. We apply the exp-function method to both the nonlinear time and space fractional differential equations. As a result, some new exact solutions for them are successfully established.
119 citations
TL;DR: In this paper, a new analytic method is applied to singular initial-value Lane-Emden type problems, and the effectiveness and performance of the method is studied, which obtains a Taylor expansion of the solution, and when the solution is polynomial, their method reproduces the exact solution.
Abstract: A new analytic method is applied to singular initial-value Lane-Emden-type problems, and the effectiveness and performance of the method is studied. The proposed method obtains a Taylor expansion of the solution, and when the solution is polynomial, our method reproduces the exact solution. It is observed that the method is easy to implement, valuable for handling singular phenomena, yields excellent results at a minimum computational cost, and requires less time. Computational results of several test problems are presented to demonstrate the viability and practical usefulness of the method. The results reveal that the method is very effective, straightforward, and simple.
117 citations
TL;DR: A survey paper concerning the notions of hyperstability and superstability, which are connected to the issue of Ulam's type stability, is presented in this article, where the authors present the recent results on those subjects.
Abstract: This is a survey paper concerning the notions of hyperstability and
superstability, which are connected to the issue of Ulam’s type stability. We
present the recent results on those subjects.
110 citations
TL;DR: In this paper, the Sumudu transform was used to solve nonhomogeneous fractional ordinary differential equations (FODEs) and then the solutions were used to form two-dimensional (2D) graphs.
Abstract: We introduce the rudiments of fractional calculus and the consequent applications of the Sumudu transform on fractional derivatives. Once this connection is firmly established in the general setting, we turn to the application of the Sumudu transform method (STM) to some interesting nonhomogeneous fractional ordinary differential equations (FODEs). Finally, we use the solutions to form two-dimensional (2D) graphs, by using the symbolic algebra package Mathematica Program 7.
105 citations
TL;DR: In this paper, a modified Runge-Kutta-Nystrom method of fourth algebraic order is developed, which is based on the fitting of the coefficients, due to the nullification not only of the phase lag and of the amplification error, but also of their derivatives.
Abstract: A new modified Runge-Kutta-Nystrom method of fourth algebraic order is developed. The new modified RKN method is based on the fitting of the coefficients, due to the nullification not only of the phase lag and of the amplification error, but also of their derivatives. Numerical results indicate that the new modified method is much more efficient than other methods derived for solving numerically the Schrodinger equation.
103 citations
TL;DR: In this article, a local fractional series expansion method was proposed to solve wave and diffusion equations on Cantor sets, and the results showed the efficiency and accuracy of the proposed method to obtain analytical solutions to differential equations.
Abstract: We proposed a local fractional series expansion method to solve the wave and diffusion equations on Cantor sets. Some examples are given to illustrate the efficiency and accuracy of the proposed method to obtain analytical solutions to differential equations within the local fractional derivatives.
91 citations
TL;DR: The purpose of the present paper is to review the RCIP method in a simple setting, to show how easily the method can be implemented in MATLAB, and to present new applications of RCIP to integral equations of scattering theory on planar curves with corners.
Abstract: Recursively compressed inverse preconditioning (RCIP) is a numerical method for obtaining highly accurate solutions to integral equations on piecewise smooth surfaces. The method originated in 2008 as a technique within a scheme for solving Laplace’s equation in two-dimensional domains with corners. In a series of subsequent papers, the technique was then refined and extended as to apply to integral equation formulations of a broad range of boundary value problems in physics and engineering. The purpose of the present paper is threefold: first, to review the RCIP method in a simple setting; second, to show how easily the method can be implemented in MATLAB; third, to present new applications of RCIP to integral equations of scattering theory on planar curves with corners.
87 citations
TL;DR: In this article, the homotopy decomposition method (HPM) was used for solving the nonlinear fractional coupled Korteweg-de-Vries equations.
Abstract: We put into practice a relatively new analytical technique, the homotopy decomposition method, for solving the nonlinear fractional coupled-Korteweg-de-Vries equations. Numerical solutions are given, and some properties exhibit reasonable dependence on the fractional-order derivatives’ values. The fractional derivatives are described in the Caputo sense. The reliability of HDM and the reduction in computations give HDM a wider applicability. In addition, the calculations involved in HDM are very simple and straightforward. It is demonstrated that HDM is a powerful and efficient tool for FPDEs. It was also demonstrated that HDM is more efficient than the adomian decomposition method (ADM), variational iteration method (VIM), homotopy analysis method (HAM), and homotopy perturbation method (HPM).
TL;DR: In this paper, a lower bound on the product of the effective widths of quaternion-valued signals in the spatial and frequency domains is established for the quaternionic linear canonical transform (QLCT).
Abstract: We generalize the linear canonical transform (LCT) to quaternion-valued signals, known as the quaternionic linear canonical transform (QLCT). Using the properties of the LCT we establish an uncertainty principle for the QLCT. This uncertainty principle prescribes a lower bound on the product of the effective widths of quaternion-valued signals in the spatial and frequency domains. It is shown that only a 2D Gaussian signal minimizes the uncertainty.
TL;DR: To prove strong convergence of the algorithms, a new lemma is established, which can be used as a fundamental tool for solving some nonlinear problems.
Abstract: Consider the variational inequality of finding a point satisfying the property , for all , where is the intersection of finite level sets of convex functions defined on a real Hilbert space and is an -Lipschitzian and -strongly monotone operator. Relaxed and self-adaptive iterative algorithms are devised for computing the unique solution of . Since our algorithm avoids calculating the projection (calculating by computing several sequences of projections onto half-spaces containing the original domain ) directly and has no need to know any information of the constants and , the implementation of our algorithm is very easy. To prove strong convergence of our algorithms, a new lemma is established, which can be used as a fundamental tool for solving some nonlinear problems.
TL;DR: In this paper, a user friendly algorithm based on new homotopy perturbation Sumudu transform method (HPSTM) is proposed to solve nonlinear fractional gas dynamics equation.
Abstract: A user friendly algorithm based on new homotopy perturbation Sumudu transform method (HPSTM) is proposed to solve nonlinear fractional gas dynamics equation. The fractional derivative is considered in the Caputo sense. Further, the same problem is solved by Adomian decomposition method (ADM). The results obtained by the two methods are in agreement and hence this technique may be considered an alternative and efficient method for finding approximate solutions of both linear and nonlinear fractional differential equations. The HPSTM is a combined form of Sumudu transform, homotopy perturbation method, and He’s polynomials. The nonlinear terms can be easily handled by the use of He’s polynomials. The numerical solutions obtained by the proposed method show that the approach is easy to implement and computationally very attractive.
TL;DR: A new structure of relation between two CAIFSs, called complex Atanassov's intuitionistic fuzzy relation (CAIFR), is obtained, in which the ranges of values of CAIFR are extended to the unit circle in complex plane for both membership and nonmembership functions instead of [0, 1].
Abstract: This paper presents distance measure between two complex Atanassov's intuitionistic fuzzy sets (CAIFSs). This distance measure is used to illustrate an application of CAIFSs in solving one of the most core application areas of fuzzy set theory, which is multiattributes decision-making (MADM) problems, in complex Atanassov's intuitionistic fuzzy realm. A new structure of relation between two CAIFSs, called complex Atanassov's intuitionistic fuzzy relation (CAIFR), is obtained. This relation is formally generalised from a conventional Atanassov's intuitionistic fuzzy relation, based on complex Atanassov's intuitionistic fuzzy sets, in which the ranges of values of CAIFR are extended to the unit circle in complex plane for both membership and nonmembership functions instead of [0, 1] as in the conventional Atanassov's intuitionistic fuzzy functions. Definition and some mathematical concepts of CAIFS, which serve as a foundation for the creation of complex Atanassov's intuitionistic fuzzy relation, are recalled. We also introduce the Cartesian product of CAIFSs and derive two properties of the product space. The concept of projection and cylindric extension of CAIFRs are also introduced. An example of CAIFR in real-life situation is illustrated in this paper. Finally, we introduce the concept of composition of CAIFRs.
TL;DR: In this paper, the authors deal with the put option pricing problems based on the time-fractional Black-Scholes equation, where the fractional derivative is a so-called modified Riemann-Liouville fractional derivatives.
Abstract: This work deals with the put option pricing problems based on the time-fractional Black-Scholes equation, where the fractional derivative is a so-called modified Riemann-Liouville fractional derivative. With the aid of symbolic calculation software, European and American put option pricing models that combine the time-fractional Black-Scholes equation with the conditions satisfied by the standard put options are numerically solved using the implicit
scheme of the finite difference method.
TL;DR: In this paper, a new proof of this result is found under new conditions which are much weaker than Han and Yuan's assumptions, and in order to accelerate the ADMM with three blocks, they also propose a relaxed ADMM involving an additional computation of optimal step size and establish its global convergence under mild conditions.
Abstract: We consider a class of linearly constrained separable convex programming problems whose objective functions are the sum of three convex functions without coupled variables. For those problems, Han and Yuan (2012) have shown that the sequence generated by the alternating direction method of multipliers (ADMM) with three blocks converges globally to their KKT points under some technical conditions. In this paper, a new proof of this result is found under new conditions which are much weaker than Han and Yuan’s assumptions. Moreover, in order to accelerate the ADMM with three blocks, we also propose a relaxed ADMM involving an additional computation of optimal step size and establish its global convergence under mild conditions.
TL;DR: In this paper, a nonlinear filter for image noise reduction based on the diffusion flow generated by the porous media equation is proposed, where is a non-linear continuous function of the form,.
Abstract: A novel PDE-based image denoising approach is proposed in this paper. One designs here a nonlinear filter for image noise reduction based on the diffusion flow generated by the porous media equation , where is a nonlinear continuous function of the form , . With respect to standard 2D Gaussian smoothing and some nonlinear PDE-based filters, this one is more efficient to remove noise from degraded images and also to reduce “staircasing” effects and preserve the image edges.
TL;DR: In this article, the authors define modified weak contractive mappings and establish fixed point results for such mappings defined on partial metric spaces using the notion of triangular-admissibility.
Abstract: The aim of this paper is to define modified weak --contractive mappings and to establish fixed point results for such mappings defined on partial metric spaces using the notion of triangular -admissibility. As an application, we prove new fixed point results for graphic weak -contractive mappings. Moreover, some examples and an application to integral equation are given here to illustrate the usability of the obtained results.
TL;DR: In this article, the existence, uniqueness, and Ulam-Hyers stability results for fixed point problems via contractive mapping of type-() in the framework of -metric spaces are investigated.
Abstract: We will investigate some existence, uniqueness, and Ulam-Hyers stability results for fixed point problems via --contractive mapping of type-() in the framework of -metric spaces. The presented theorems extend, generalize, and unify several results in the literature, involving the results of Samet et al. (2012).
TL;DR: In this paper, the local fractional variational iteration (LVAE) method for the Laplace equation was investigated and the obtained results reveal that the method is very effective.
Abstract: The local fractional variational iteration method for local fractional Laplace equation is investigated in this paper. The operators are described in the sense of local fractional operators. The obtained results reveal that the method is very effective.
TL;DR: In this paper, the authors consider a new type of fractional derivative (which interpolates the Hadamard derivative and its Caputo counterpart) and prove the well-posedness for a basic Cauchy type fractional differential equation involving this kind of derivative.
Abstract: Motivated by the Hilfer fractional derivative (which interpolates the Riemann-Liouville derivative and the Caputo derivative), we consider a new type of fractional derivative (which interpolates the Hadamard derivative and its Caputo counterpart). We prove the well-posedness for a basic Cauchy type fractional differential equation involving this kind of derivative. This is established in an appropriate underlying space after proving the equivalence of this problem with a certain corresponding Volterra integral equation.
TL;DR: The generalized Laguerre pseudo-spectral approximation based on the generalized Lagueerre operational matrix is investigated to reduce the nonlinear multiterm FDEs and its initial conditions to nonlinear algebraic system, thus greatly simplifying the problem.
Abstract: We present a direct solution technique for approximating linear multiterm fractional differential equations (FDEs) on semi-infinite interval, using generalized Laguerre polynomials. We derive the operational matrix of Caputo fractional derivative of the generalized Laguerre polynomials which is applied together with generalized Laguerre tau approximation for implementing a spectral solution of linear multiterm FDEs on semi-infinite interval subject to initial conditions. The generalized Laguerre pseudo-spectral approximation based on the generalized Laguerre operational matrix is investigated to reduce the nonlinear multiterm FDEs and its initial conditions to nonlinear algebraic system, thus greatly simplifying the problem. Through several numerical examples, we confirm the accuracy and performance of the proposed spectral algorithms. Indeed, the methods yield accurate results, and the exact solutions are achieved for some tested problems.
TL;DR: This work presents an analysis methodology based on the use of the Finite Element Method (FEM) nowadays considered one of the main numerical tools for solving Boundary Value Problems (BVPs) showing robustness and computational efficiency in comparison with FEM solutions obtained with a commercial code, despite the fact that cg-FEM has been fully implemented in MATLAB.
Abstract: This work presents an analysis methodology based on the use of the Finite Element Method (FEM) nowadays considered one of the main numerical tools for solving Boundary Value Problems (BVPs). The proposed methodology, so-called cg-FEM (Cartesian grid FEM), has been implemented for fast and accurate numerical analysis of 2D linear elasticity problems. The traditional FEM uses geometry-conforming meshes; however, in cg-FEM the analysis mesh is not conformal to the geometry. This allows for defining very efficient mesh generation techniques and using a robust integration procedure, to accurately integrate the domain’s geometry. The hierarchical data structure used in cg-FEM together with the Cartesian meshes allow for trivial data sharing between similar entities. The cg-FEM methodology uses advanced recovery techniques to obtain an improved solution of the displacement and stress fields (for which a discretization error estimator in energy norm is available) that will be the output of the analysis. All this results in a substantial increase in accuracy and computational efficiency with respect to the standard FEM. cg-FEM has been applied in structural shape optimization showing robustness and computational efficiency in comparison with FEM solutions obtained with a commercial code, despite the fact that cg-FEM has been fully implemented in MATLAB.
TL;DR: In this paper, the strong consistency of estimator of fixed design regression model under widely dependent random variables, which generalizes the corresponding one of independent random variables was studied. And the authors used the Bernstein-type inequality and the truncated method.
Abstract: We present the Bernstein-type inequality for widely dependent random variables. By using the Bernstein-type inequality and the truncated method, we further study the strong consistency of estimator of fixed design regression model under widely dependent random variables, which generalizes the corresponding one of independent random variables. As an application, the strong consistency for the nearest neighbor estimator is obtained.
TL;DR: In this article, a mathematical model of the switched interval drive-response error system is established and synchronization criteria are derived for switched interval networks under the arbitrary switching rule, which are easy to verify in practice.
Abstract: This paper investigates synchronization problem of switched delay networks with interval parameters uncertainty, based on the theories of the switched systems and drive-response technique, a mathematical model of the switched interval drive-response error system is established. Without constructing Lyapunov-Krasovskii functions, introducing matrix measure method for the first time to switched time-varying delay networks, combining Halanay inequality technique, synchronization criteria are derived for switched interval networks under the arbitrary switching rule, which are easy to verify in practice. Moreover, as an application, the proposed scheme is then applied to chaotic neural networks. Finally, numerical simulations are provided to illustrate the effectiveness of the theoretical results.
TL;DR: In this paper, two new subclasses of the function class Σ of bi-univalent functions defined in the open unit disc are introduced, and estimates on the coefficients and for functions in these subclasses are found.
Abstract: We introduce two new subclasses of the function class Σ of bi-univalent functions defined in the open unit disc. Furthermore, we find estimates on the coefficients and for functions in these new subclasses. Also consequences of the results are pointed out.
TL;DR: The fractional subequation method was applied to solve Cahn-Hilliard and Klein-Gordon equations of fractional order as discussed by the authors, and the accuracy and efficiency of the scheme were discussed for these illustrative examples.
Abstract: The fractional subequation method is applied to solve Cahn-Hilliard and Klein-Gordon equations of fractional order. The accuracy and efficiency of the scheme are discussed for these illustrative examples.
TL;DR: In this paper, the authors give sufficient conditions for the parameters of the normalized form of the generalized Struve functions to be convex and star-like in the open unit disk.
Abstract: We give sufficient conditions for the parameters of the normalized form of the generalized Struve functions to be convex and starlike in the open unit disk.
TL;DR: In this article, a new set of sufficient conditions for approximate controllability of fractional Sobolev-type differential equations, formulated and proved using Schauder fixed point theorem, fractional calculus and methods of controllable theory, is presented.
Abstract: We discuss the approximate controllability of semilinear fractional Sobolev-type differential system under the assumption that the corresponding linear system is approximately controllable. Using Schauder fixed point theorem, fractional calculus and methods of controllability theory, a new set of sufficient conditions for approximate controllability of fractional Sobolev-type differential equations, are formulated and proved. We show that our result has no analogue for the concept of complete controllability. The results of the paper are generalization and continuation of the recent results on this issue.