**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**157

# Search results for: Fractional calculus

##### 157 Some Remarks About Riemann-Liouville and Caputo Impulsive Fractional Calculus

**Authors:**
M. De la Sen

**Abstract:**

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

##### 156 Fractional-Order PI Controller Tuning Rules for Cascade Control System

**Authors:**
Truong Nguyen Luan Vu,
Le Hieu Giang,
Le Linh

**Abstract:**

The fractional–order proportional integral (FOPI) controller tuning rules based on the fractional calculus for the cascade control system are systematically proposed in this paper. Accordingly, the ideal controller is obtained by using internal model control (IMC) approach for both the inner and outer loops, which gives the desired closed-loop responses. On the basis of the fractional calculus, the analytical tuning rules of FOPI controller for the inner loop can be established in the frequency domain. Besides, the outer loop is tuned by using any integer PI/PID controller tuning rules in the literature. The simulation study is considered for the stable process model and the results demonstrate the simplicity, flexibility, and effectiveness of the proposed method for the cascade control system in compared with the other methods.

**Keywords:**
Fractional calculus,
fractional–order proportional integral controller,
cascade control system,
internal model control approach.

##### 155 Riemann-Liouville Fractional Calculus and Multiindex Dzrbashjan-Gelfond-Leontiev Differentiation and Integration with Multiindex Mittag-Leffler Function

**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.

##### 154 Existence of Iterative Cauchy Fractional Differential Equation

**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.

##### 153 Fractional Masks Based On Generalized Fractional Differential Operator for Image Denoising

**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.

##### 152 Fractional Order Feedback Control of a Ball and Beam System

**Authors:**
Santosh Kr. Choudhary

**Abstract:**

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.

##### 151 A Mathematical Model Approach Regarding the Children’s Height Development with Fractional Calculus

**Authors:**
Nisa Özge Önal,
Kamil Karaçuha,
Göksu Hazar Erdinç,
Banu Bahar Karaçuha,
Ertuğrul Karaçuha

**Abstract:**

The study aims to use a mathematical approach with the fractional calculus which is developed to have the ability to continuously analyze the factors related to the children’s height development. Until now, tracking the development of the child is getting more important and meaningful. Knowing and determining the factors related to the physical development of the child any desired time would provide better, reliable and accurate results for childcare. In this frame, 7 groups for height percentile curve (3th, 10th, 25th, 50th, 75th, 90th, and 97th) of Turkey are used. By using discrete height data of 0-18 years old children and the least squares method, a continuous curve is developed valid for any time interval. By doing so, in any desired instant, it is possible to find the percentage and location of the child in Percentage Chart. Here, with the help of the fractional calculus theory, a mathematical model is developed. The outcomes of the proposed approach are quite promising compared to the linear and the polynomial method. The approach also yields to predict the expected values of children in the sense of height.

**Keywords:**
Children growth percentile,
children physical development,
fractional calculus,
linear and polynomial model.

##### 150 Observer Based Control of a Class of Nonlinear Fractional Order Systems using LMI

**Authors:**
Elham Amini Boroujeni,
Hamid Reza Momeni

**Abstract:**

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

##### 149 Notes on Fractional k-Covered Graphs

**Authors:**
Sizhong Zhou,
Yang Xu

**Abstract:**

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

##### 148 Stability of Fractional Differential Equation

**Authors:**
Rabha W. Ibrahim

**Abstract:**

We study a Dirichlet boundary value problem for Lane-Emden equation involving two fractional orders. Lane-Emden equation has been widely used to describe a variety of phenomena in physics and astrophysics, including aspects of stellar structure, the thermal history of a spherical cloud of gas, isothermal gas spheres,and thermionic currents. However, ordinary Lane-Emden equation does not provide the correct description of the dynamics for systems in complex media. In order to overcome this problem and describe dynamical processes in a fractalmedium, numerous generalizations of Lane-Emden equation have been proposed. One such generalization replaces the ordinary derivative by a fractional derivative in the Lane-Emden equation. This gives rise to the fractional Lane-Emden equation with a single index. Recently, a new type of Lane-Emden equation with two different fractional orders has been introduced which provides a more flexible model for fractal processes as compared with the usual one characterized by a single index. The contraction mapping principle and Krasnoselskiis fixed point theorem are applied to prove the existence of solutions of the problem in a Banach space. Ulam-Hyers stability for iterative Cauchy fractional differential equation is defined and studied.

**Keywords:**
Fractional calculus,
fractional differential equation,
Lane-Emden equation,
Riemann-Liouville fractional operators,
Volterra integral equation.

##### 147 On Fractional (k,m)-Deleted Graphs with Constrains Conditions

**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.

##### 146 A Neighborhood Condition for Fractional k-deleted Graphs

**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.

##### 145 Application of a SubIval Numerical Solver for Fractional Circuits

**Authors:**
Marcin Sowa

**Abstract:**

**Keywords:**
Numerical method,
SubIval,
fractional calculus,
numerical solver,
circuit analysis.

##### 144 Derivation of Fractional Black-Scholes Equations Driven by Fractional G-Brownian Motion and Their Application in European Option Pricing

**Authors:**
Changhong Guo,
Shaomei Fang,
Yong He

**Abstract:**

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

##### 143 Stability of Interval Fractional-order Systems with Order 0 < α < 1

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

**Abstract:**

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.

##### 142 Relation between Roots and Tangent Lines of Function in Fractional Dimensions: A Method for Optimization Problems

**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.

##### 141 Lyapunov Type Inequalities for Fractional Impulsive Hamiltonian Systems

**Authors:**
Kazem Ghanbari,
Yousef Gholami

**Abstract:**

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

##### 140 Application of Fractional Model Predictive Control to Thermal System

**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.

##### 139 Realization of Fractional-Order Capacitors with Field-Effect Transistors

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

**Abstract:**

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

##### 138 Calculus-based Runtime Verification

**Authors:**
Xuan Qi,
Changzhi Zhao

**Abstract:**

**Keywords:**
calculus,
eagle logic,
monitor synthesis,
runtime verification

##### 137 A Design of Fractional-Order PI Controller with Error Compensation

**Authors:**
Mazidah Tajjudin,
Norhashim Mohd Arshad,
Ramli Adnan

**Abstract:**

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.

##### 136 Oil Displacement by Water in Hauterivian Sandstone Reservoir of Kashkari Oil Field

**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.

##### 135 An Efficient Hamiltonian for Discrete Fractional Fourier Transform

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

**Abstract:**

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

##### 134 Lower Bound of Time Span Product for a General Class of Signals in Fractional Fourier Domain

**Authors:**
Sukrit Shankar,
Chetana Shanta Patsa,
Jaydev Sharma

**Abstract:**

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.

##### 133 Perturbation in the Fractional Fourier Span due to Erroneous Transform Order and Window Function

**Authors:**
Sukrit Shankar,
Chetana Shanta Patsa,
Jaydev Sharma

**Abstract:**

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

##### 132 Quality Factor Variation with Transform Order in Fractional Fourier Domain

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

**Abstract:**

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

##### 131 Operational Representation of Certain Hypergeometric Functions by Means of 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.

##### 130 Effect of Fractional Flow Curves on the Heavy Oil and Light Oil Recoveries in Petroleum Reservoirs

**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.

##### 129 Group Invariant Solutions of Nonlinear Time-Fractional Hyperbolic Partial Differential Equation

**Authors:**
Anupma Bansal,
Rajeev Budhiraja,
Manoj Pandey

**Abstract:**

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

##### 128 Stability Analysis of Fractional Order Systems with Time Delay

**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.