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Optimal Control of a Class of Parabolic Partial Fractional Differential Equations
Mahmoud M. El-borai
,
Mohamed A. Abdou
,
Mai Taha Elsayed
Issue: Volume 6, Issue 5-1, September 2017
Pages: 66-70
Received: 28 July 2017
Accepted: 31 July 2017
Published: 9 August 2017
DOI:
10.11648/j.ajtas.s.2017060501.20
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Abstract: In this paper, the existence of the solution of the parabolic partial fractional differential equation is studied and the solution bound estimate which are used to prove the existence of the solution of the optimal control problem in a Banach space is also studied, then apply the classical control theory to parabolic partial differential equation in a bounded domain with boundary problem. An expansion formula for fractional derivative, optimal conditions and a new solution scheme is proposed.
Abstract: In this paper, the existence of the solution of the parabolic partial fractional differential equation is studied and the solution bound estimate which are used to prove the existence of the solution of the optimal control problem in a Banach space is also studied, then apply the classical control theory to parabolic partial differential equation i...
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Statistical Analysis of Strength of W/S Test of Normality Against Non-normal Distribution Using Monte Carlo Simulation
Ukponmwan H. Nosakhare
,
Ajibade F. Bright
Issue: Volume 6, Issue 5-1, September 2017
Pages: 62-65
Received: 24 February 2017
Accepted: 1 March 2017
Published: 11 July 2017
DOI:
10.11648/j.ajtas.s.2017060501.19
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Abstract: Among the test of normality in existence is the W/S which has standard table as the interval for critical region with both lower and upper bound. The test is suitable for sample size ranging from 3 as displayed in the W/S Critical table. But the sensitivity of the test can be determined by computation of power of the test which would show how sensitive the test is to non-normal distribution. The paper addressed the sensitivity of the test using some selected distributions which are from asymmetric and symmetric in nature. Monte Carlo Simulation technique was used with 100 replicates for sample sizes of 5 to 100 with regular interval of 5. Distributions considered include; Weibull, Chi-Square, t and Cauchy distributions. The result shows inconsistency of the test as it has weak power for distribution used except Cauchy distribution. The findings shows that the test should be used with caution has it has weak or low power which could lead to statistical error, thereby call for proper modification of the test to improve its power.
Abstract: Among the test of normality in existence is the W/S which has standard table as the interval for critical region with both lower and upper bound. The test is suitable for sample size ranging from 3 as displayed in the W/S Critical table. But the sensitivity of the test can be determined by computation of power of the test which would show how sensi...
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Evaluation of Techniques for Univariate Normality Test Using Monte Carlo Simulation
Ukponmwan H. Nosakhare
,
Ajibade F. Bright
Issue: Volume 6, Issue 5-1, September 2017
Pages: 51-61
Received: 24 February 2017
Accepted: 1 March 2017
Published: 9 June 2017
DOI:
10.11648/j.ajtas.s.2017060501.18
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Abstract: This paper examines the sensitivity of nine normality test statistics; W/S, Jaque-Bera, Adjusted Jaque-Bera, D’Agostino, Shapiro-Wilk, Shapiro-Francia, Ryan-Joiner, Lilliefors’and Anderson Darlings test statistics, with a view to determining the effectiveness of the techniques to accurately determine whether a set of data is from normal distribution or not. Simulated data of sizes 5, 10, …, 100 is used for the study and each test is repeated 100 times for increased reliability. Data from normal distributions (N (2, 1) and N (0, 1)) and non-normal distributions (asymmetric and symmetric distributions: Weibull, Chi-Square, Cauchy and t-distributions) are simulated and tested for normality using the nine normality test statistics. To ensure uniformity of results, one statistical software is used in all the data computations to eliminate variations due to statistical software. The error rate of each of the test statistic is computed; the error rate for the normal distribution is the type I error and that for non-normal distribution is type II error. Power of test is computed for the non-normal distributions and use to determine the strength of the methods. The ranking of the nine normality test statistics in order of superiority for small sample sizes is; Adjusted Jarque-Bera, Lilliefor’s, D’Agostino, Ryan-Joiner, Shapiro-Francia, Shapiro-Wilk, W/S, Jarque-Bera and Anderson-Darling test statistics while for large sample sizes, we have; D’Agostino, Ryan-Joiner, Shapiro-Francia, Jarque-Bera, Anderson-Darling, Lilliefor’s, Adjusted Jarque-Bera, Shapiro-Wilk and W/S test statistics. Hence, only D’Agostino test statistic is classified as Uniformly Most Powerful since it is effective for both small and large sample sizes. Other methods are Locally Most Powerful. Shapiro-Francia, an improvement of Shapiro-Wilk is more sensitive for both small and large samples, hence should replace Shapiro-Wilk while the Adjsted Jarque-Bera and the Jarque-Bera should both be retained for small and large samples respectively.
Abstract: This paper examines the sensitivity of nine normality test statistics; W/S, Jaque-Bera, Adjusted Jaque-Bera, D’Agostino, Shapiro-Wilk, Shapiro-Francia, Ryan-Joiner, Lilliefors’and Anderson Darlings test statistics, with a view to determining the effectiveness of the techniques to accurately determine whether a set of data is from normal distributio...
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On The Fractional Optimal Control Problem with Free End Point
Mahmoud M. El-borai
,
Wagdy G. ElSayed
,
M. A. Abdou
,
M. Taha E.
Issue: Volume 6, Issue 5-1, September 2017
Pages: 46-50
Received: 15 March 2017
Accepted: 16 March 2017
Published: 11 April 2017
DOI:
10.11648/j.ajtas.s.2017060501.17
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Abstract: We present a necessary optimality conditions for a class of optimal control problems. The dynamical control system involves integer and fractional order derivatives and the final time is free. Optimality conditions are obtained. Feedback control laws for linear dynamic system are obtained.
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Approximate Solutions for Mathematical Model of Carcinogenesis Using Adomian Decomposition Method
Mahmoud M. El-borai
,
M. A. Abdou
,
E. M. Youssef
Issue: Volume 6, Issue 5-1, September 2017
Pages: 40-45
Received: 20 March 2017
Accepted: 21 March 2017
Published: 5 April 2017
DOI:
10.11648/j.ajtas.s.2017060501.16
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Abstract: In this paper, the Adomian decomposition method (ADM) is applied to obtain the approximate solution of a mathematical model of carcinogenesis which is a Riccati differential equation derived by Moolgavkar and Venzon (see [9]). The numerical solution obtained by this way have been compared with the exact solution which obtained by Moolgavkar and Venzon (see [11]). This comparison show that the (ADM) is a powerful method for solving this differential equations. The method does not need weak nonlinearity assumptions or perturbation theory, the decomposition procedure of Adomian will be obtained easily without linearization the problem by implementing the decomposition method rather than the standard methods for the exact solutions.
Abstract: In this paper, the Adomian decomposition method (ADM) is applied to obtain the approximate solution of a mathematical model of carcinogenesis which is a Riccati differential equation derived by Moolgavkar and Venzon (see [9]). The numerical solution obtained by this way have been compared with the exact solution which obtained by Moolgavkar and Ven...
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Synchronization and Impulsive Control of Some Parabolic Partial Differential Equations
Mahmoud M. El-Borai
,
Wagdy G. Elsayed
,
Turkiya Alhadi Aljamal
Issue: Volume 6, Issue 5-1, September 2017
Pages: 30-39
Received: 7 March 2017
Accepted: 8 March 2017
Published: 5 April 2017
DOI:
10.11648/j.ajtas.s.2017060501.15
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Abstract: Novel equi-attractivity in large generalized non-linear partial differential equations were performed for the impulsive control of spatiotemporal chaotic. Attractive solutions of these general partial differential equations were determined. A proof for existence of a certain kind of impulses for synchronization such that the small error dynamics that is equi-attractive in the large is established. A comparative study between these general non-linear partial differential equations and the existent reported numerical theoretical models was developed. Several boundary conditions were given to confirm the theoretical results of the general non-linear partial differential equations. Moreover, the equations were applied to Kuramoto–Sivashinsky PDE′s equation; Grey–Scott models, and Lyapunov exponents for stabilization of the large chaotic systems with elimination of the dynamic error.
Abstract: Novel equi-attractivity in large generalized non-linear partial differential equations were performed for the impulsive control of spatiotemporal chaotic. Attractive solutions of these general partial differential equations were determined. A proof for existence of a certain kind of impulses for synchronization such that the small error dynamics th...
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On Some Lag Synchronization and Higher Order Parabolic Systems
Khairia El-Said El-Nadi
,
Wagdy G. Elsayed
,
Mabroka F. Bader
Issue: Volume 6, Issue 5-1, September 2017
Pages: 23-29
Received: 7 March 2017
Accepted: 8 March 2017
Published: 20 March 2017
DOI:
10.11648/j.ajtas.s.2017060501.14
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Abstract: Chaos synchronization is a topic of great interest, due to its observation in a huge variety of phenomena of different nature. We study synchronization of two chaotic oscillators in a Master- Slave configuration. The two dynamic systems are coupled via a directed feedback that randomly switches among a finite set of given constant function at a prescribed time rate. And we use Lyapunov stability theory. This paper discussed the using of lag synchronization approach, and provided the equilibrium solutions of a new class of higher order parabolic partial differential equations to be applicable for Lorenz chaotic system in order to minimize the dynamical error of large Lorenz chaotic system
Abstract: Chaos synchronization is a topic of great interest, due to its observation in a huge variety of phenomena of different nature. We study synchronization of two chaotic oscillators in a Master- Slave configuration. The two dynamic systems are coupled via a directed feedback that randomly switches among a finite set of given constant function at a pre...
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Solvability of Some Nonlinear Integral Functional Equations
Mahmoud M. El-Borai
,
Wagdy G. El-Sayed
,
Noura N. Khalefa
Issue: Volume 6, Issue 5-1, September 2017
Pages: 13-22
Received: 11 February 2017
Accepted: 15 February 2017
Published: 28 February 2017
DOI:
10.11648/j.ajtas.s.2017060501.13
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Abstract: This paper discussed some existence theorems for nonlinear functional integral equations in the space L^1 of Lebesgue integrable functions,by using the Darbo fixed point theorem associated with the Hausdorff measure of noncompactness. Also, as an application, we discuss the existence of solutions for some nonlinear integral equations with fractional order.
Abstract: This paper discussed some existence theorems for nonlinear functional integral equations in the space L^1 of Lebesgue integrable functions,by using the Darbo fixed point theorem associated with the Hausdorff measure of noncompactness. Also, as an application, we discuss the existence of solutions for some nonlinear integral equations with fractiona...
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A Generalization of Some Lag Synchronization of System with Parabolic Partial Differential Equation
Mahmoud M. El-Borai
,
Wagdy G. El-sayed.
,
Aafaf E. Abduelhafid
Issue: Volume 6, Issue 5-1, September 2017
Pages: 8-12
Received: 27 January 2017
Accepted: 3 February 2017
Published: 18 February 2017
DOI:
10.11648/j.ajtas.s.2017060501.12
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Abstract: In this paper, we study generalized adaptive synchronization of Lorenz chaotic system with parabolic partial differential equation. Systems with three uncertain parameters and the non-linear adaptive feedback control technique are considered. Moreover, a systematic design process of parameters identification and Lag synchronization of chaotic system is considered. Finally, a sufficient condition is given for Lyapunov stability.
Abstract: In this paper, we study generalized adaptive synchronization of Lorenz chaotic system with parabolic partial differential equation. Systems with three uncertain parameters and the non-linear adaptive feedback control technique are considered. Moreover, a systematic design process of parameters identification and Lag synchronization of chaotic syste...
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Marshall-Olkin Exponential Pareto Distribution with Application on Cancer Stem Cells
Khairia El-Said El-Nadi
,
L. M. Fatehy
,
Nourhan Hamdy Ahmed
Issue: Volume 6, Issue 5-1, September 2017
Pages: 1-7
Received: 8 December 2016
Accepted: 27 December 2016
Published: 24 January 2017
DOI:
10.11648/j.ajtas.s.2017060501.11
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Abstract: A Marshall–Olkin variant of exponential Pareto distribution is being introduced in this paper. Some of its statistical functions and numerical characteristics among others characteristics function, moment generalizing function, central moments of real order are derived in the computational series expansion form and various illustrative special cases are discussed. This density function is utilized to model a real data set of cancer stem cells patients. The new distribution provides a better fit than related distributions. The proposed distribution could find applications for instance in the physical and biological sciences, hydrology, medicine, meteorology and engineering.
Abstract: A Marshall–Olkin variant of exponential Pareto distribution is being introduced in this paper. Some of its statistical functions and numerical characteristics among others characteristics function, moment generalizing function, central moments of real order are derived in the computational series expansion form and various illustrative special case...
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