Search results for: Fractional Derivatives and Integrals

Authors: Manoj Singh, Mumtaz Ahmad Khan, Abdul Hakim Khan

Abstract:

The investigation in the present paper is to obtain certain types of relations for the well known hypergeometric functions by employing the technique of fractional derivative and integral.

Keywords: Fractional Derivatives and Integrals, Hypergeometric functions.

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Authors: Artion Kashuri, Rozana Liko

Abstract:

In this paper, some generalization integral inequalities of Hermite–Hadamard type for functions whose derivatives are s–convex in modulus are given by using k–fractional integrals. Some applications to special means are obtained as well. Some known versions are recovered as special cases from our results. We note that our inequalities can be viewed as new refinements of the previous results. Finally, our results have a deep connection with various fractional integral operators and interested readers can find new interesting results using our idea and technique as well.

Keywords: Hermite–Hadamard’s inequalities, k–Riemann–Liouville fractional integral, H¨older’s inequality, Special means.

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Authors: M. De la Sen

Abstract:

This paper establishes some closed formulas for Riemann- Liouville impulsive fractional integral calculus and also for Riemann- Liouville and Caputo impulsive fractional derivatives.

Keywords: Rimann- Liouville fractional calculus, Caputofractional derivative, Dirac delta, Distributional derivatives, Highorderdistributional derivatives.

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Authors: Kazem Ghanbari, Yousef Gholami

Abstract:

This paper deals with study about fractional order impulsive Hamiltonian systems and fractional impulsive Sturm-Liouville type problems derived from these systems. The main purpose of this paper devotes to obtain so called Lyapunov type inequalities for mentioned problems. Also, in view point on applicability of obtained inequalities, some qualitative properties such as stability, disconjugacy, nonexistence and oscillatory behaviour of fractional Hamiltonian systems and fractional Sturm-Liouville type problems under impulsive conditions will be derived. At the end, we want to point out that for studying fractional order Hamiltonian systems, we will apply recently introduced fractional Conformable operators.

Keywords: Fractional derivatives and integrals, Hamiltonian system, Lyapunov type inequalities, stability, disconjugacy.

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Authors: U.K. Saha, L.K. Arora

Abstract:

The multiindex Mittag-Leffler (M-L) function and the multiindex Dzrbashjan-Gelfond-Leontiev (D-G-L) differentiation and integration play a very pivotal role in the theory and applications of generalized fractional calculus. The object of this paper is to investigate the relations that exist between the Riemann-Liouville fractional calculus and multiindex Dzrbashjan-Gelfond-Leontiev differentiation and integration with multiindex Mittag-Leffler function.

Keywords: Multiindex Mittag-Leffler function, Multiindex Dzrbashjan-Gelfond-Leontiev differentiation and integration, Riemann-Liouville fractional integrals and derivatives.

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Authors: Hussain Al-Qassem

Abstract:

In this paper we study the boundedness properties of certain oscillatory integrals with polynomial phase. We obtain sharp estimates for these oscillatory integrals. By the virtue of these estimates and extrapolation we obtain Lp boundedness for these oscillatory integrals under rather weak size conditions on the kernel function.

Keywords: Fourier transform, oscillatory integrals, Orlicz spaces, Block spaces, Extrapolation, Lp boundedness.

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Authors: Sizhong Zhou, Yang Xu

Abstract:

A graph G is fractional k-covered if for each edge e of G, there exists a fractional k-factor h, such that h(e) = 1. If k = 2, then a fractional k-covered graph is called a fractional 2-covered graph. The binding number bind(G) is defined as follows, bind(G) = min{|NG(X)| |X| : ├ÿ = X Ôèå V (G),NG(X) = V (G)}. In this paper, it is proved that G is fractional 2-covered if δ(G) ≥ 4 and bind(G) > 5 3 .

Keywords: graph, binding number, fractional k-factor, fractional k-covered graph.

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Authors: Sizhong Zhou, Hongxia Liu

Abstract:

Let G be a graph of order n, and let k  2 and m  0 be two integers. Let h : E(G)  [0, 1] be a function. If e∋x h(e) = k holds for each x  V (G), then we call G[Fh] a fractional k-factor of G with indicator function h where Fh = {e  E(G) : h(e) > 0}. A graph G is called a fractional (k,m)-deleted graph if there exists a fractional k-factor G[Fh] of G with indicator function h such that h(e) = 0 for any e  E(H), where H is any subgraph of G with m edges. In this paper, it is proved that G is a fractional (k,m)-deleted graph if (G)  k + m + m k+1 , n  4k2 + 2k − 6 + (4k 2 +6k−2)m−2 k−1 and max{dG(x), dG(y)}  n 2 for any vertices x and y of G with dG(x, y) = 2. Furthermore, it is shown that the result in this paper is best possible in some sense.

Keywords: Graph, degree condition, fractional k-factor, fractional (k, m)-deleted graph.

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Authors: Sizhong Zhou, Hongxia Liu

Abstract:

Abstract–Let k ≥ 3 be an integer, and let G be a graph of order n with n ≥ 9k +3- 42(k - 1)2 + 2. Then a spanning subgraph F of G is called a k-factor if dF (x) = k for each x ∈ V (G). A fractional k-factor is a way of assigning weights to the edges of a graph G (with all weights between 0 and 1) such that for each vertex the sum of the weights of the edges incident with that vertex is k. A graph G is a fractional k-deleted graph if there exists a fractional k-factor after deleting any edge of G. In this paper, it is proved that G is a fractional k-deleted graph if G satisfies δ(G) ≥ k + 1 and |NG(x) ∪ NG(y)| ≥ 1 2 (n + k - 2) for each pair of nonadjacent vertices x, y of G.

Keywords: Graph, minimum degree, neighborhood union, fractional k-factor, fractional k-deleted graph.

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Authors: M. Reni Sagayaraj, S. Anand Gnana Selvam, R. Reynald Susainathan

Abstract:

We analyze stochastic integrals associated with a mutation process. To be specific, we describe the cell population process and derive the differential equations for the joint generating functions for the number of mutants and their integrals in generating functions and their applications. We obtain first-order moments of the processes of the two-way mutation process in first-order moment structure of X (t) and Y (t) and the second-order moments of a one-way mutation process. In this paper, we obtain the limiting behaviour of the integrals in limiting distributions of X (t) and Y (t).

Keywords: Stochastic integrals, single–server queue model, catastrophes, busy period.

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Authors: Hamid A. Jalab, Rabha W. Ibrahim

Abstract:

This paper introduces an image denoising algorithm based on generalized Srivastava-Owa fractional differential operator for removing Gaussian noise in digital images. The structures of nxn fractional masks are constructed by this algorithm. Experiments show that, the capability of the denoising algorithm by fractional differential-based approach appears efficient to smooth the Gaussian noisy images for different noisy levels. The denoising performance is measured by using peak signal to noise ratio (PSNR) for the denoising images. The results showed an improved performance (higher PSNR values) when compared with standard Gaussian smoothing filter.

Keywords: Fractional calculus, fractional differential operator, fractional mask, fractional filter.

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Authors: Santosh Kr. Choudhary

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In this paper, fractional order feedback control of a ball beam model is investigated. The ball beam model is a particular example of the double Integrator system having strongly nonlinear characteristics and unstable dynamics which make the control of such system a challenging task. Most of the work in fractional order control systems are in theoretical nature and controller design and its implementation in practice is very small. In this work, a successful attempt has been made to design a fractional order PIλDμcontroller for a benchmark laboratory ball and beam model. Better performance can be achieved using a fractional order PID controller and it is demonstrated through simulations results with a comparison to the classic PID controller.

Keywords: Fractional order calculus, fractional order controller, fractional order system, ball and beam system, PIλDμ controller, modelling, simulation.

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Authors: Changhong Guo, Shaomei Fang, Yong He

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In this paper, fractional Black-Scholes models for the European option pricing were established based on the fractional G-Brownian motion (fGBm), which generalizes the concepts of the classical Brownian motion, fractional Brownian motion and the G-Brownian motion, and that can be used to be a tool for considering the long range dependence and uncertain volatility for the financial markets simultaneously. A generalized fractional Black-Scholes equation (FBSE) was derived by using the Taylor’s series of fractional order and the theory of absence of arbitrage. Finally, some explicit option pricing formulas for the European call option and put option under the FBSE were also solved, which extended the classical option pricing formulas given by F. Black and M. Scholes.

Keywords: European option pricing, fractional Black-Scholes equations, fractional G-Brownian motion, Taylor’s series of fractional order, uncertain volatility.

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Authors: Rabha W. Ibrahim

Abstract:

Our main aim in this paper is to use the technique of non expansive operators to more general iterative and non iterative fractional differential equations (Cauchy type ). The non integer case is taken in sense of Riemann-Liouville fractional operators. Applications are illustrated.

Keywords: Fractional calculus, fractional differential equation, Cauchy equation, Riemann-Liouville fractional operators, Volterra integral equation, non-expansive mapping, iterative differential equation.

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Authors: Hong Li, Shou-ming Zhong, Hou-biao Li

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In this paper, some brief sufficient conditions for the stability of FO-LTI systems dαx(t) dtα = Ax(t) with the fractional order are investigated when the matrix A and the fractional order α are uncertain or both α and A are uncertain, respectively. In addition, we also relate the stability of a fractional-order system with order 0 < α ≤ 1 to the stability of its equivalent fractional-order system with order 1 ≤ β < 2, the relationship between α and β is presented. Finally, a numeric experiment is given to demonstrate the effectiveness of our results.

Keywords: Interval fractional-order systems, linear matrix inequality (LMI), asymptotical stability.

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Authors: Ali Dorostkar

Abstract:

In this paper, a basic schematic of fractional dimensional optimization problem is presented. As will be shown, a method is performed based on a relation between roots and tangent lines of function in fractional dimensions for an arbitrary initial point. It is shown that for each polynomial function with order N at least N tangent lines must be existed in fractional dimensions of 0 < α < N+1 which pass exactly through the all roots of the proposed function. Geometrical analysis of tangent lines in fractional dimensions is also presented to clarify more intuitively the proposed method. Results show that with an appropriate selection of fractional dimensions, we can directly find the roots. Method is presented for giving a different direction of optimization problems by the use of fractional dimensions.

Keywords: Tangent line, fractional dimension, root, optimization problem.

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Authors: Aymen Rhouma, Khaled Hcheichi, Sami Hafsi

Abstract:

The article presents an application of Fractional Model Predictive Control (FMPC) to a fractional order thermal system using Controlled Auto Regressive Integrated Moving Average (CARIMA) model obtained by discretization of a continuous fractional differential equation. Moreover, the output deviation approach is exploited to design the K -step ahead output predictor, and the corresponding control law is obtained by solving a quadratic cost function. Experiment results onto a thermal system are presented to emphasize the performances and the effectiveness of the proposed predictive controller.

Keywords: Fractional model predictive control, fractional order systems, thermal system.

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Authors: Steve Hung-Lung Tu, Yu-Hsuan Cheng

Abstract:

A novel and efficient approach to realize fractional-order capacitors is investigated in this paper. Meanwhile, a new approach which is more efficient for semiconductor implementation of fractional-order capacitors is proposed. The feasibility of the approach has been verified with the preliminary measured results.

Keywords: Fractional-order, field-effect transistors, RC transmission lines.

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Authors: Elham Amini Boroujeni, Hamid Reza Momeni

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Design of an observer based controller for a class of fractional order systems has been done. Fractional order mathematics is used to express the system and the proposed observer. Fractional order Lyapunov theorem is used to derive the closed-loop asymptotic stability. The gains of the observer and observer based controller are derived systematically using the linear matrix inequality approach. Finally, the simulation results demonstrate validity and effectiveness of the proposed observer based controller.

Keywords: Fractional order calculus, Fractional order observer, Linear matrix inequality, Nonlinear Systems, Observer based Controller.

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Authors: Mazidah Tajjudin, Norhashim Mohd Arshad, Ramli Adnan

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Fractional-order controller was proven to perform better than the integer-order controller. However, the absence of a pole at origin produced marginal error in fractional-order control system. This study demonstrated the enhancement of the fractionalorder PI over the integer-order PI in a steam temperature control. The fractional-order controller was cascaded with an error compensator comprised of a very small zero and a pole at origin to produce a zero steady-state error for the closed-loop system. Some modification on the error compensator was suggested for different order fractional integrator that can improve the overall phase margin.

Keywords: Fractional-order PI, Ziegler-Nichols tuning, Oustaloup's Recursive Approximation, steam temperature control.

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Authors: B. I. Yun

Abstract:

A dual-reciprocity boundary element method is presented for the numerical solution of a class of axisymmetric elastodynamic problems. The domain integrals that arise in the integrodifferential formulation are converted to line integrals by using the dual-reciprocity method together suitably constructed interpolating functions. The second order time derivatives of the displacement in the governing partial differential equations are suppressed by using Laplace transformation. In the Laplace transform domain, the problem under consideration is eventually reduced to solving a system of linear algebraic equations. Once the linear algebraic equations are solved, the displacement and stress fields in the physical domain can be recovered by using a numerical technique for inverting Laplace transforms.

Keywords: Axisymmetric elasticity, boundary element method, dual-reciprocity method, Laplace transform.

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Authors: A. J. Nazari, S. Honma

Abstract:

This paper evaluates oil displacement by water in Hauterivian sandstone reservoir of Kashkari oil field in North of Afghanistan. The core samples of this oil field were taken out from well No-21^st, and the relative permeability and fractional flow are analyzed. Steady state flow laboratory experiments are performed to empirically obtain the fractional flow curves and relative permeability in different water saturation ratio. The relative permeability represents the simultaneous flow behavior in the reservoir. The fractional flow approach describes the individual phases as fractional of the total flow. The fractional flow curve interprets oil displacement by water, and from the tangent of fractional flow curve can find out the average saturation behind the water front flow saturation. Therefore, relative permeability and fractional flow curves are suitable for describing the displacement of oil by water in a petroleum reservoir. The effects of irreducible water saturation, residual oil saturation on the displaceable amount of oil are investigated through Buckley-Leveret analysis.

Keywords: Fractional flow, oil displacement, relative permeability, simultaneously flow.

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Authors: Sukrit Shankar, Pardha Saradhi K., Chetana Shanta Patsa, Jaydev Sharma

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Fractional Fourier Transform, which is a generalization of the classical Fourier Transform, is a powerful tool for the analysis of transient signals. The discrete Fractional Fourier Transform Hamiltonians have been proposed in the past with varying degrees of correlation between their eigenvectors and Hermite Gaussian functions. In this paper, we propose a new Hamiltonian for the discrete Fractional Fourier Transform and show that the eigenvectors of the proposed matrix has a higher degree of correlation with the Hermite Gaussian functions. Also, the proposed matrix is shown to give better Fractional Fourier responses with various transform orders for different signals.

Keywords: Fractional Fourier Transform, Hamiltonian, Eigen Vectors, Discrete Hermite Gaussians.

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Authors: Sukrit Shankar, Chetana Shanta Patsa, Jaydev Sharma

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Fractional Fourier Transform is a generalization of the classical Fourier Transform which is often symbolized as the rotation in time- frequency plane. Similar to the product of time and frequency span which provides the Uncertainty Principle for the classical Fourier domain, there has not been till date an Uncertainty Principle for the Fractional Fourier domain for a generalized class of finite energy signals. Though the lower bound for the product of time and Fractional Fourier span is derived for the real signals, a tighter lower bound for a general class of signals is of practical importance, especially for the analysis of signals containing chirps. We hence formulate a mathematical derivation that gives the lower bound of time and Fractional Fourier span product. The relation proves to be utmost importance in taking the Fractional Fourier Transform with adaptive time and Fractional span resolutions for a varied class of complex signals.

Keywords: Fractional Fourier Transform, uncertainty principle, Fractional Fourier Span, amplitude, phase.

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Authors: Sukrit Shankar, Chetana Shanta Patsa, Jaydev Sharma

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Fractional Fourier Transform is a generalization of the classical Fourier Transform. The Fractional Fourier span in general depends on the amplitude and phase functions of the signal and varies with the transform order. However, with the development of the Fractional Fourier filter banks, it is advantageous in some cases to have different transform orders for different filter banks to achieve better decorrelation of the windowed and overlapped time signal. We present an expression that is useful for finding the perturbation in the Fractional Fourier span due to the erroneous transform order and the possible variation in the window shape and length. The expression is based on the dependency of the time-Fractional Fourier span Uncertainty on the amplitude and phase function of the signal. We also show with the help of the developed expression that the perturbation of span has a varying degree of sensitivity for varying degree of transform order and the window coefficients.

Keywords: Fractional Fourier Transform, Perturbation, Fractional Fourier span, amplitude, phase, transform order, filterbanks.

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Authors: H. M. Al-Qassem

Abstract:

We obtain appropriate sharp estimates for rough oscillatory integrals. Our results represent significant improvements as well as natural extensions of what was known previously.

Keywords: Oscillatory singular integral, Rough kernel, Singular integral, L^{p} boundedness.

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Authors: Sukrit Shankar, Chetana Shanta Patsa, K. Pardha Saradhi, Jaydev Sharma

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Fractional Fourier Transform is a powerful tool, which is a generalization of the classical Fourier Transform. This paper provides a mathematical relation relating the span in Fractional Fourier domain with the amplitude and phase functions of the signal, which is further used to study the variation of quality factor with different values of the transform order. It is seen that with the increase in the number of transients in the signal, the deviation of average Fractional Fourier span from the frequency bandwidth increases. Also, with the increase in the transient nature of the signal, the optimum value of transform order can be estimated based on the quality factor variation, and this value is found to be very close to that for which one can obtain the most compact representation. With the entire mathematical analysis and experimentation, we consolidate the fact that Fractional Fourier Transform gives more optimal representations for a number of transform orders than Fourier transform.

Keywords: Fractional Fourier Transform, Quality Factor, Fractional Fourier span, transient signals.

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Authors: Abdul Jamil Nazari, Shigeo Honma

Abstract:

This paper evaluates and compares the effect of fractional flow curves on the heavy oil and light oil recoveries in a petroleum reservoir. Fingering of flowing water is one of the serious problems of the oil displacement by water and another problem is the estimation of the amount of recover oil from a petroleum reservoir. To address these problems, the fractional flow of heavy oil and light oil are investigated. The fractional flow approach treats the multi-phases flow rate as a total mixed fluid and then describes the individual phases as fractional of the total flow. Laboratory experiments are implemented for two different types of oils, heavy oil, and light oil, to experimentally obtain relative permeability and fractional flow curves. Application of the light oil fractional curve, which exhibits a regular S-shape, to the water flooding method showed that a large amount of mobile oil in the reservoir is displaced by water injection. In contrast, the fractional flow curve of heavy oil does not display an S-shape because of its high viscosity. Although the advance of the injected waterfront is faster than in light oil reservoirs, a significant amount of mobile oil remains behind the waterfront.

Keywords: Fractional flow curve, oil recovery, relative permeability, water fingering.

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Authors: Anupma Bansal, Rajeev Budhiraja, Manoj Pandey

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In this paper, we have investigated the nonlinear time-fractional hyperbolic partial differential equation (PDE) for its symmetries and invariance properties. With the application of this method, we have tried to reduce it to time-fractional ordinary differential equation (ODE) which has been further studied for exact solutions.

Keywords: Nonlinear time-fractional hyperbolic PDE, Lie Classical method, exact solutions.

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Authors: Hong Li, Shou-Ming Zhong, Hou-Biao Li

Abstract:

In this paper, we mainly study the stability of linear and interval linear fractional systems with time delay. By applying the characteristic equations, a necessary and sufficient stability condition is obtained firstly, and then some sufficient conditions are deserved. In addition, according to the equivalent relationship of fractional order systems with order 0 < α ≤ 1 and with order 1 ≤ β < 2, one may get more relevant theorems. Finally, two examples are provided to demonstrate the effectiveness of our results.

Keywords: Fractional order systems, Time delay, Characteristic equation.

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