Abstracts 11 International Workshop on Dynamical Systems and

Abstracts
11th International Workshop on
Dynamical Systems and Applications
Çankaya University
Ankara, TURKEY
June 26–28, 2012
Preface
The 11th International Workshop on Dynamical Systems and Applications is held at
Çankaya University, Ankara, Turkey, during June 26-28, 2012.
These workshops constitute the annual meetings of the series of dynamical systems seminars traditionally organized at Middle East Technical University throughout each academic year. The theme of this current workshop will be ”Fractional Differential Equations
and Dynamic Equations with Applications”. However, talks are not restricted to these
subjects only.
The workshop brings together about 120 mathematicians from 14 countries.
We would like to express our gratitude to Çankaya University, Turkish Mathematical
Society Ankara Branch, and Casio-Mersa system administration for their support and
sponsorship of the meeting. In addition, the staff at the office of Public Relations at
Çankaya University, the Cultural Affairs office staff and the secretary at the Department
of Mathematics and Computer Science, the Dean’s office secretary at the Faculty of Arts
and Sciences, as well as many faculty and students of the Department of Mathematics
and Computer Science deserve heartfelt thanks.
Organizing committee co-chairs: Billur Kaymakçalan and Ağacık Zafer
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Scientific Committee
Ağacık Zafer, Middle East Technical University
Allaberen Ashyralyev, Fatih University
Aydın Tiryaki, İzmir University
Azer Khanmamedov, Hacettepe University
Etibar Panakhov, Fırat University
Gusein Guseinov, Atılım University
Okay Çelebi, Yeditepe University
Ömer Akın, TOBB Economy and Technology University
Varga Kalantarov, Koç University
Organizing Committee
Ağacık Zafer, Middle East Technical University (co-chair)
Billur Kaymakçalan, Çankaya University (co-chair)
Özlem Defterli, Çankaya University
Dumitru Baleanu, Çankaya University
Fahd Jarad, Çankaya University
Raziye Mert, Çankaya University
Adnan Bilgen, Çankaya University
Feza Güvenilir, Ankara University
Fatma Karakoç, Ankara University
Aytekin Enver, Gazi University
Mustafa Fahri Aktaş, Gazi University
Abdullah Özbekler, Atılım University
Erdal Karapınar, Atılım University
Türker Ertem, Middle East Technical University
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Contents
Preface . . . . . . . . . . . . . . . . . . .
Scientific Committee . . . . . . . . . . .
Organazing Committee . . . . . . . . . .
Plenary Talks . . . . . . . . . . . . . . .
Ravi P. AGARWAL . . . . . . . . .
Ravi P. AGARWAL . . . . . . . .
Delfim F. M. TORRES . . . . . . .
Invited Talks . . . . . . . . . . . . . . .
Murat ADIVAR . . . . . . . . . . .
Alemdar HASANOĞLU . . . . . .
Contributed Talks . . . . . . . . . . . .
Thabet ABDELJAWAD . . . . . .
Nihan ACAR . . . . . . . . . . . .
Ali AKGÜL . . . . . . . . . . . . .
Ömer AKIN . . . . . . . . . . . . .
Elvan AKIN-BOHNER . . . . . . .
Jehad O. ALZABUT . . . . . . . .
Sakri AMINE . . . . . . . . . . . .
Muhammad ARSHAD . . . . . . .
Ola A. ASHOUR . . . . . . . . . .
Allaberen ASHYRALYEV . . . . .
Muzaffer ATEŞ . . . . . . . . . . .
Ferhan M. ATICI . . . . . . . . . .
Akbar AZAM . . . . . . . . . . . .
Dumitru BALEANU . . . . . . . .
Ayşe Hümeyra BİLGE . . . . . . .
Zeyneb BOUDERBALA . . . . . .
Artur M. C. BRITO da CRUZ . . .
Özlem DEFTERLİ . . . . . . . . .
Alireza Khalili GOLMANKHANEH
Gusein Sh. GUSEINOV . . . . . .
Tuba GÜLŞEN . . . . . . . . . . .
Mehmet GÜMÜŞ . . . . . . . . . .
Alaa E. HAMZA . . . . . . . . . .
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ii
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Taher HASSAN . . . . . . . . .
Betül HİÇDURMAZ . . . . . .
Derrardjia ISHAK . . . . . . .
Niyaz İSMAGİLOV . . . . . . .
Fatma KARAKOÇ . . . . . . .
Erdal KARAPINAR . . . . . .
Afshin KHASSEKHAN . . . . .
Atul KUMAR . . . . . . . . . .
Gholam Reza Rokni LAMOOKI
Alaeddin MALEK . . . . . . . .
Raziye MERT . . . . . . . . . .
Karima M. ORABY . . . . . .
Süleyman ÖĞREKÇİ . . . . . .
Abdullah ÖZBEKLER . . . . .
Elif ÖZTÜRK . . . . . . . . . .
M. Mine ÖZYETKİN . . . . . .
Erhan PİŞKİN . . . . . . . . .
Abolhassan RAZMINIA . . . .
Safia SLIMANI . . . . . . . . .
Yeter ŞAHİNER . . . . . . . .
Murat ŞAT . . . . . . . . . . .
Aydın TİRYAKİ . . . . . . . .
Fatma TOKMAK . . . . . . . .
Deniz UÇAR . . . . . . . . . .
Paolo VETTORI . . . . . . . .
Ali YAKAR . . . . . . . . . . .
Ahmet YANTIR . . . . . . . .
Burcu Silindir YANTIR . . . .
Fikriye Nuray YILMAZ . . . .
Tuğba YILMAZ . . . . . . . . .
Uğur YÜKSEL . . . . . . . . .
Ağacık ZAFER . . . . . . . . .
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26
Plenary Talks
Lidstone and Complementary Lidstone Polynomials and Interpolation I
Ravi P. AGARWAL
Texas A&M University - Kingsville, TX, USA
[email protected]
In this lecture we shall:
• Define Lidstone polynomials and provide its several different representations which
involve Bernoulli polynomials, Bernoulli numbers, Euler polynomials and Euler
numbers.
• Establish several equalities and inequalities (most of these are the best possible).
• Present an explicit representation of the Lidstone interpolating polynomial, and
then for the error function give Peano’s and Cauchy’s representations.
• Provide best possible error inequalities, best possible criterion for the convergence
of Lidstone series, and a quadrature formula with best possible error bound.
• Construct complementary Lidstone interpolating polynomial, provide explicit representation of the error function, best possible error inequalities, best possible criterion for the convergence of complementary Lidstone series, and a quadrature formula
with best possible error bound.
Lidstone and Complementary Lidstone Boundary Value Problems II
Ravi P. AGARWAL
Texas A&M University - Kingsville, TX, USA
[email protected]
The equalities and inequalities established in Lecture 1 will be used here to study
Lidstone and Complementary Lidstone boundary value problems. We shall:
• Provide necessary and sufficient conditions for the existence and uniqueness of solutions.
• Establish sufficient conditions for the convergence of Picard’s and Approximate
Picard’s iterative methods.
• Develop sufficient conditions for the convergence of Quasilinearization and Approximate Quasilinearization.
• Define Lidstone Disconjugacy and give a best possible criterion.
• Obtain sufficient conditions for the existence of positive solutions.
1
Combined Derivatives on Time Scales and Fractional Variational Calculus
Delfim F. M. TORRES
University of Aveiro, PORTUGAL
[email protected]
Combined derivatives appear naturally in the context of mechanics and the calculus
of variations, both on time scale and fractional settings. In this talk we give a personal
view to the subject, and review some recent results on delta-nabla time scale derivatives
and right-left fractional derivatives.
2
Invited Talks
Convex Analysis on Multidimensional Mixed Domains
Murat ADIVAR
İzmir University of Economics, İzmir, TURKEY
[email protected]
Coauthors: Shu-Cherng FANG
In this study, we establish a platform which enables us to study convexity and duality
properties of the sets, cones and functions on multidimensional discrete, continuous and
mixed domains.
Nonlinear Diffusion and Monotonicity of Differential Operators
Alemdar HASANOĞLU
İzmir University, İzmir, TURKEY
[email protected]
This study paper deals with nonlinear differential operators
Au := −div(D(|∇u|2 )∇u)
in Ω ∈ R, arising in PDE-based image processing (J. Weickert, in: Scale-Space Theory in
Computer Vision, B. ter Haar Romeny, L. Florack, J. Koenderink, M. Viergever (eds.),
Lecture Notes in Computer Science, Springer, Berlin, 1997.), computational material science (A. Hasanov, Int. J. Non-Linear Mechanics, 46(5)(2011)), diffusion and nonlinear
heat transfer (E. Zeidler, Nonlinear Functional Analysis and Its Applications, II/B Nonlinear Monotone Operators, New York: Springer, 1990). We show that, the in all these
(quite different) physical models, defined to be as nonlinear isotropic diffusion models, the
properties of nonlinear differential operators are almost the same. Using these properties
we formulate some new coefficient inverse problems, with nonlocal measured output data
(i.e. in the form of integral operator) for the steady state (Au := −div(D(|∇u|2 )∇u))
and evolution (Au := −div(D(|∇u|2 )∇u)) equations. Then we derive a quasisolution
approach based on weak solution theory for PDEs, which permits one to prove existence
of a solution of the considered coefficient inverse problems. Important questions related
to instability of a solution are also discussed.
3
Contributed Talks
On Delta and Nabla Riemann and Caputo Fractional Differences
Thabet ABDELJAWAD
Çankaya University, Ankara, TURKEY
[email protected]
We investigate two types of dual identities for Riemann and Caputo fractional sums
and differences. The first type relates nabla and delta type fractional sums and differences.
The second type, represented by the Q-operator, relates left and right fractional sums and
differences. These dual identities insist that in the definition of right fractional differences
we have to mix both the nabla and delta operators.
The Sequential Fractional Difference Equations
Nihan ACAR
Western Kentucky University, Kentucky, USA
[email protected]
Coauthors: Ferhan M. ATICI
In this talk, the extended table for discrete Laplace transform known as N -transform
will be presented for some basic functions of discrete fractional calculus with nabla operator. N -transform is a great tool to solve α-th order nabla fractional difference equations
as demonstrated in the paper [F. M. Atici and P. W. Eloe, The Rocky Mountain Journal
of Mathematics, Special issue honoring Prof. Lloyd Jackson 41(2011), 353-370]. We will
also demonstrate how to use N -transform to solve some classes of fractional difference
equations with initial conditions. Next, we give the definition of Casoration for the set
of solutions up to n-th order nabla fractional difference equations. By calculating the
Casoration of the solutions we will classify the solutions of the sequential nabla fractional
difference equations as linearly independent or linearly dependent. Finally, we concentrate on the solutions of up to second order nabla fractional difference equations. We will
examine characteristic roots in three cases, namely real and distinct, real and same, and
complex. Riemann-Liouville definition of fractional difference will be used throughout our
work.
4
Numerical Solution of the Second-Order One-Dimensional Telegraph Equation
Based on Reproducing Kernel Hilbert Space Method
Ali AKGÜL
Dicle University, Diyarbakır, TURKEY
[email protected]
Coauthors: Mustafa İNÇ
In this paper, we proposed a reproducing kernel method for solving the telegraph
equation with initial and boundary conditions based on the reproducing kernel theory.
Its exact solution is represented in the form of series in the reproducing kernel Hilbert
space. Some numerical examples have been studied to demonstrate the accuracy of the
present method. Results of numerical examples show that the presented method is simple
and effective.
Possible Fuzzy Solutions For Second Order Initial Value Problems
Ömer AKIN
TOBB University of Economics and Technology, Ankara, TURKEY
[email protected]
Coauthors: Burhan TÜRKŞEN
In this study, we state a fuzzy initial value problem of the second order fuzzy differential
equations. Here we investigate problems with fuzzy coefficients, fuzzy initial values and
fuzzy forcing functions. We propose an algorithm based on alpha-cut of a fuzzy set.
Finally we present some examples by using our proposed algorithm. Following these we
try to extend the results for low as well as high fuzzy sets.
Oscillation Criteria for Second Order Strongly Superlinear and Strongly Sublinear Dynamic Inclusions
Elvan AKIN-BOHNER
Missouri University S&T, Rolla, MS, USA
[email protected]
Coauthors: Shurong SUN
In this paper, we establish some oscillation criteria for strongly superlinear and strongly
sublinear dynamic inclusions. Oscillation problems are in differential and difference equations have become very attractive recently. These areas have started to be unified and
extended for more powerful general theory, so called dynamic equations on time scales.
Results in this paper even are new in continuous case.
5
Oscillation of Solutions for Third–Order Half-Linear Neutral Difference Equations
Jehad O. ALZABUT
Prince Sultan University, Riyadh, SAUDI ARABIA
[email protected]
Coauthors: Ömer AKIN, Yaşar BOLAT, N. DOĞAN
In this article, we study the oscillation of solutions for third order neutral difference
equations of the form
α ∆ a (n) ∆2 [x (n) ± p (n) x (δ (n))]
+ q (n) xα (τ (n)) = 0.
(1)
Sufficient conditions are established to prove that every solution of (1) either oscillates or
converges to zero. We support the main results by numerical examples.
Asymptotic Dynamics of the Slow-Fast Hindmarsh-Rose Neuronal System
Sakri AMINE
Cheffia Eltarf, ALGERIA
[email protected]
Coauthors: Benchetteh AZDINE
This work addresses the asymptotic dynamics of a neuronal mathematical model. The
first aim is the understanding of the biological meaning of existing mathematical systems
concerning neurons such as Hodgkin-Huxley or Hindmarsh-Rose models. The local stability and the numerical asymptotic analysis of Hindmarsh-Rose model are then developed
in order to comprehend bifurcations and dynamics evolution of a single Hindmarsh-Rose
neuron. This has been performed using numerical tools borrowed from the nonlinear
dynamical system theory.
Fixed points of Kannan and Chatterjea mappings on a closed ball without the
assumption of continuity
Muhammad ARSHAD
International Islamic University, Islamabad, PAKISTAN
marshad [email protected]
Coauthors: Saqib HUSSAIN
After Banach contraction principal, a variety of generalizations in the setting of point
to point mappings have been obtained. In 1969, Kannan [R. Kannan, Some results on
fixed points, Bull. Calcutta Math. Soc., 60(1968), 71-76] a premier Indian Mathematician
proved a contraction theorem for a complete metric space proved that a mapping T : X →
X satisfying a contraction condition d(T x, T y) < k[d(x, T x) + d(y, T y)] for all x, y ∈ X
where 0 < k < (1/2) has a unique fixed point in X. Chatterjea [S. K. Chatterjea, Fixed
6
point theorems, C.R. Acad. Bulgare Sci., 25(1972), 727–730] followed Kannan and in 1972
proved a fixed point theorem for a complete metric space (X, d), states that a mapping
T : X → X satisfying a contraction condition d(T x, T y) < k[d(x, T y) + d(y, T x)] for all
x, y ∈ X where 0 < k < (1/2) has a unique fixed point in X. It is impartant to note that
these three theorems are independent of each other and have laid down the foundation
of modern fixed points theory. In this study we establish some fixed point theorems
for mappings satisfying Kannan and Chatterjea locally contractive conditions on a closed
ball in a complete metric space. Our results generalize/improve some well-known classical
results of the literature.
On q-dual Integral Equations
Ola A. ASHOUR
Cairo University, Cairo, EGYPT
oa [email protected]
Our aim in this talk is to introduce q-analogues of some known Dual Integral Equations
and to obtain their solutions in different techniques.
Finite Difference Method for Stochastic Parabolic and Hyperbolic Equations
Allaberen ASHYRALYEV
Fatih University, İstanbul, TURKEY
[email protected]
It is known that most problems in heat flow, fusion process, model financial instruments like options, bonds and interest rates and other areas which are involved with
uncertainty lead to stochastic differential equation with parabolic type. These equations
can be derived as models of indeterministic systems and consider as methods for solving
boundary value problems. The method of operators as a tool for investigation of the solution to stochastic partial differential equations in Hilbert and Banach spaces, has been
systematically developed by several authors. Numerical analytic methods for stochastic
differential equations have been studied extensively by many researchers. However, finite
difference method for stochastic partial differential equations was not well-investigated.
The main goal of this study is to construct and investigate the difference schemes for
stochastic parabolic and hyperbolic equations. The single step and two step difference
schemes for the numerical solution of stochastic parabolic and hyperbolic equations are
presented. The convergence estimates for the solution of these difference schemes are
established. For the numerical study, procedure of modified Gauss elimination method is
used to solve these difference schemes.
7
Global Asymptotic Stability and Ultimate Boundednes of a Class of Third
Order Nonlinear Systems
Muzaffer ATEŞ
Yüzüncü Yıl University, Van, TURKEY
[email protected]
In this paper, we study the problems of the global stability and the boundedness
results of the second order vector differential equations,
...
.
..
..
.
.
.
..
X +F (X, X , X ) X +G(X, X ) X +H(X) = P (t, X, X , , X )
(∗)
in two cases (i) P = 0 and (ii) P (6= 0), ||P (t, X, Y )|| ≤ (A + ||Y ||)q(t), where δ0 , δ1 are
constants. For case (i) we obtain some sufficient conditions which ensure that the solution
x = 0 of Eq. (*) is globally asymptotically stable. For case (ii) the ultimate boundedness
results of Eq. (*) is obtained. Our results include a well-known result in the literature.
Finally, a concrete example is given to check our results.
Models of Inventory with Deteriorating Items on Non-periodic Time Domains
Ferhan M. ATICI
Western Kentucky University, Kentucky, USA
[email protected]
Coauthors: Alex LEBEDINSKY, Fahriye M. UYSAL
In this talk, we first introduce deterministic model of inventory with deteriorating
items on complex time domains which may be periodic as discrete or non periodic as
unevenly distributed collection of time. Then we show how the dynamic model can be
optimized using techniques of time scale calculus. After we demonstrate that both the
conventional discrete-time and time scale calculus model yield the same results, we show
an example based on this model that could not be solved using conventional discrete-time
techniques because the points on time scales may be spaced at uneven intervals.
8
Coincidence Points of Fuzzy Mappings
Akbar AZAM
COMSATS Institute of Information Technology, Islamabad, PAKISTAN
[email protected]
A large variety of the most important problems of applied mathematics reduced to
finding solutions of nonlinear functional equation, which can be formulated in terms of
finding the fixed points of a nonlinear operator. Since the appearance of celebrated Banach
contraction principle in 1932, several generalizations and improvements of this theorem
have been obtained. Heilpern [S. Heilpern, Fuzzy mappings and fixed point theorems,
J. Math. Anal.Appl. 83(1981), 566-569.] generalized the Banach Contraction Principle
by introducing a contraction condition for fuzzy mappings and established a fixed point
theorem for fuzzy mappings in complete metric linear spaces. Subsequently several originators studied the existence of fixed points and common fixed points of fuzzy mappings
satisfying a Banach type contractive condition. In the present paper, we establish a coincidence point theorem for a pair of fuzzy mappings under a contractive type condition
and obtain a significant extension of Heilpern result.
About a Nonlinear Fractional Differential Equation
Dumitru BALEANU
Çankaya University, Ankara, TURKEY and Institute of Space Sciences, Magurele-Bucharest,
ROMANIA
[email protected]
In this paper we study the existence and uniqueness of a nonlinear fractional differential equation with periodic boundary condition.
The Classification of Integrable Evolution Equation in 1+1 Dimensions
Ayşe Hümeyra BİLGE
Kadir Has University, İstanbul, TURKEY
[email protected]
All integrable polynomial evolution equations in 1+1 dimensions are known to be
symmetries of the Korteweg deVries, the Sawada-Kotera and the Kaup equations. We
classify the non-polynomial equations by the existence of a formal symmetry and we show
that there are non-polynomial hierarchies which are nevertheless expected to be related
to the equations above.
9
Periodic Solutions for a class of Autonomous Newton Differential Equations
Zeyneb BOUDERBALA
Badji Mokhtar University, Annaba, ALGERIA
[email protected]
Coauthors: Amar MAKHLOUF
In this work, we provide sufficient conditions for the existence of periodic solutions of
the second order autonomous differential equation
x00 = M (x)
with M(x) a 2π periodic function (with a small parameter of perturbation ). Note
that this is a particular class of autonomous Newton differential equations. Moreover we
provide some applications.
A Symmetric Dynamic Calculus on Time Scales
Artur M. C. BRITO da CRUZ
The Polytechnic Institute of Setubal and University of Aveiro, PORTUGAL
[email protected]
Coauthors: Natalia MARTINS and Delfim F. M. TORRES
We define a symmetric derivative on time scales and derive some of its properties. We
introduce partial symmetric derivatives for two-variable functions. A generalized diamond
integral, which is a refined version of the diamond-α integral, is also introduced. A mean
value theorem is proved for the generalized diamond integral as well as versions of Holder’s,
Cauchy-Schwarz’s and Minkowski’s inequalities.
10
Anticipation of the Dynamics of Genetic Regulatory Networks
Özlem DEFTERLİ
Çankaya University, Ankara, TURKEY.
[email protected]
Coauthors: Armin FUGENSCHUH, Gerhard-Wilhelm WEBER
Inferring and anticipation of genetic networks based on experimental data and environmental measurements is a challenging research problem of mathematical modeling. In this study, we discuss the models of genetic regulatory systems, so-called geneenvironment networks. The dynamics of such kind of systems are described by a class
of time-continuous systems of ordinary differential equations containing unknown parameters to be optimized. Accordingly, time-discrete version of that model class is studied
and improved by using higher-order numerical methods. The presented time-continuous
and time-discrete dynamical models are identified based on given data, as an illustrative
example, by solving the constrained and regularized nonlinear mixed-integer problem. By
using this solution and applying both the new and existing discretization schemes, we
generate corresponding time-series of gene-expressions for each numerical method with a
comparative study with respect to various criteas.
Chaos On New System With Fractional Order
Alireza Khalili GOLMANKHANEH
Islamic Azad University, Urmia, IRAN
[email protected]
Coauthors: R. AREFI, D. BALEANU
In this paper, the fractional version of a new system which is similar to Liu system has
been studied. We have shown that this chaotic system again will be chaotic when the order
of system is less than 3. We use the Adams-Bash forth algorithm to solve the system and
show strange attractors in trajectories of solutions. Fixed points and Lyapunov exponent
of this system have been found and their stability have been investigated for existence of
chaos.
11
Description of the structure of arbitrary functions of the Laplace-Beltrami
operator
Gusein Sh. GUSEINOV
Atılım University, Ankara, TURKEY
[email protected]
One of the fundamental methods of investigation in the theory of operators is to
examine the functions of the operator and get in this way informations about the operator
itself. In this work, we describe the structure of arbitrary rapidly decreasing function of the
Laplace-Beltrami operator in n-dimensional hyperbolic space showing that the function
of the Laplace-Beltrami operator is an integral operator and giving an explicit formula
for its kernel.
Isospectral Problems for Differential Operators
Tuba GÜLŞEN
Fırat University, Elazığ, TURKEY
[email protected]
Coauthors: Etibar PENAHLI
In this talk, we give a relatively proof of the Gelfand-Levitan equation and a proof of
the existence of transmutation operator and investigate isospectral problem of Dirac operator. In particular, research of inverse problems for Sturm-Liouville and Dirac operators
have been investigated by many mathematicians: J.Poschel and E.Trubowitz, JR.Max
Jodeit and B.M. Levitan, H-H.Chern, etc.
On The Dynamics of the recursive sequences xn+1 = a + xpn−k /xqn
Mehmet GÜMÜŞ
Bülent Ecevit University, Zonguldak, TURKEY
[email protected]
Coauthors: Özkan ÖCALAN, Nilüfer B. FELAH
In this paper, we investigate the boundedness character, the oscillatory and the periodic character of positive solutions of the difference equation xn+1 = a + ((xpn−k )/(xqn )),
n = 0, 1, ... where k = 2, 3, ... and a, p, q are positive constans and the initial conditions
x−k , ..., x0 are arbitrary positive numbers. We investigate the existence of a prime two
periodic solution for k is odd and we find solutions which converge to this periodic solution. Moreover, when k is even, we prove that there are no prime two periodic solutions
of the equation above.
12
Semigroups of Operators on Time Scales and Applications
Alaa E. HAMZA
Cairo University, Giza, EGYPT
[email protected]
Coauthors: Karima ORABY
In this paper we define the generator A of a strongly continuous semigroup (C0 semigroup) {T (t) : t ∈ T} of bounded linear operators from a Banach space X into
itself, where T ⊆ R≥0 is a time scale, which is an additive semigroup. Many properties
of {T (t) : t ∈ T} and its generator A are established. We also prove that the dynamic
equations of the form
x∆ (t) = Ax(t),
x(0) = x0 ∈ D(A), t ∈ T,
has the unique solution
x(t) = T (t)x0 , t ∈ T,
where D(A) is the domain of A. Finally, some of well-known results, like Hille-Yosida
Theorem, are generalized.
Oscillation Criteria for Second Order Nonlinear Dynamic Equations with pLaplacian and Damping
Taher HASSAN
Mansoura University, Mansoura, EGYPT (Present Address: Hail University, KSA)
[email protected]
Coauthors: Qingkai KONG
This paper concerns the oscillation of solutions of the second order nonlinear dynamic
equation with p-Laplacian and damping
∆
r(t) ϕα x∆ (t)
+ p (t) ϕα x∆σ (t) + q(t)f (xσ (t)) = 0
on a time scale T which is unbounded above. Sign changes are allowed for the coefficient
functions r (t), p (t), and q (t). Several examples are given to illustrate the main results.
13
A Study on Difference Schemes of a Fractional Schrodinger Differential Equation
Betül HİÇDURMAZ
İstanbul Medeniyet University, İstanbul, TURKEY
[email protected]
Coauthors: Allaberen ASHYRALYEV
The study is on difference schemes of a special type of fractional Schrodinger differential equation. The first and second orders of accuracy difference schemes for numerical
solution of the fractional Schrodinger differential equation are considered. Then, stability
estimates for solutions of these difference schemes are obtained. Some applications of
these stability theorems on different problems are presented.
Fixed Points and Stability in Neutral Nonlinear Differential Equations with
Variable Delays
Derrardjia ISHAK
University of Annaba, ALGERIA
[email protected]
Coauthors: Ahcene DJOUDI
By means of Krasnoselskii’s fixed point theorem we obtain boundedness and stability
results of a neutral nonlinear differential equation with variable delays. A stability theorem with a necessary and sufficient condition is given. The results obtained here extend
and improve the work of C.H. Jin and J.W. Luo [Nonlinear Anal. 68 (2008), 3307–3315],
and also those of T.A. Burton [Fixed Point Theory 4 (2003), 15-32; Dynam. Systems
Appl. 11 (2002), 499–519] and B. Zhang [Nonlinear Anal. 63 (2005), e233–e242]. In the
end we provide an example to illustrate our claim.
14
On Pathwise Optimality for Controlled Diffusion type Processes
Niyaz ISMAGILOV
Ufa State Aviation Technical University, Ufa, RUSSIA
[email protected]
In the work, we consider a stochastic optimal control problem of diffusion type processes with pathwise cost functional, that is, the problem of finding a control function
such that it minimizes cost for every single trajectory of state variable. More precisely,
consider a stochastic differential equation describing dynamics of some system
dxt = b(t, xt , ut ) dt + σ(t, xt , ut )dWt ,
(1)
with initial value x(0) = x0 . In the above equation xt is state variable function, ut is
control function and Wt is a standard Wiener process. To measure performance of control
we introduce a cost functional of the following form
Z T
f (t, xt , ut )dt.
(2)
J=
0
The problem is to find control that minimizes the functional (2) subject to dynamics
equation (1). Previous work on optimal control of diffusion processes has mainly been
concerned with problem of finding control that minimizes “mean” value of cost
Z T
¯
f (t, xt , ut )dt → min,
J =E
0
where E denotes expectation (see [Krylov, N.V., Controlled Diffusion Processes. SpringerVerlag, Berlin, 2009], [Gichman, I.I., Skorochod,A.V., Controlled stochastic processes.
Springer Verlag, New York, 1979.]). In contrast, stated above problem is concerned with
minimization of functional (2), which represents cost for every single path of state variable.
Hence pathwise optimality is the distinguishing feature of this work. In the work we
introduce new method of solving problems of pathwise cost minimization. The main idea
of the method is that original stochastic control problem can be reduced to deterministic
control problem (Nasyrov, F.S., Local times, symmetric integrals and stochastic analysis.
Fizmatlit, Moscow, 2011 (in Russian).). Solution to the latter gives pathwise optimal
solution to the original problem.
15
Oscillatory and Periodic Solutions of Impulsive Differential Equations with
Piecewise Constant Argument
Fatma KARAKOÇ
Ankara University, Ankara, TURKEY
[email protected]
Coauthors: Hüseyin Bereketoğlu, Gizem Seyhan
In this talk we give some results for the existence of oscillatory and periodic solutions
of a class of impulsive differential equations with piecewise constant argument.
Remarks on Coupled Fixed Point Theorems
Erdal KARAPINAR
Atılım University, Ankara, TURKEY
[email protected]
In this talk, we prove new coupled fixed point theorems extending some recent results
in the literature on this topic. We also present applications of these new results through
a number of examples.
Approximate Solution of a class of Fredholm Integral Equation of second kind
with hypersingular kernel
Afshin KHASSEKHAN
Tarbiat Moallem Center, Salmas, IRAN
[email protected]
In this work a method is offered for solving a class of hyper singular integral equation of
the second kind. Chebyshev polynomials of first kind are used to approximate the kernel
function. The integrals are computed in terms of chebyshev polynomials too. In addition,
numerical examples that illustrate the pertinent features of the method are presented.
16
One-Dimensional Solute Transport For Uniform And Varying Pulse Type Input Point Source Through Inhomgeneous Medium
Atul KUMAR
Lucknow University, Uttar Pradesh, INDIA
[email protected]
Coauthors: Dilip Kumar JAISLAW and R. R. YADAV
To solves the analytically, conservative solute transport equation for a solute undergoing convection, dispersion, retardation in a one-dimensional inhomogeneous porous
medium. In the present study, the solute dispersion parameter is considered uniform
while the velocity of the flow is considered spatially dependent. Retardation factor is also
considered. The velocity of the flow is considered inversely proportional to the spatially
dependent function while retardation factor is considered inversely proportional to square
of the velocity of flow. Analytical approaches introduced for two cases: former one is for
uniform input point source and latter case is for varying input point source where the
solute transport is considered initially solute free from the domain. The variable coefficients in the advection-diffusion equation are reduced into constant coefficients with the
help of the transformations which introduce by new space variables, respectively. The
Laplace transformation technique is used to get the analytical solutions. Figures are presented illustrating the dependence of the solute transport upon velocity, dispersion and
adsorption coefficient.
A Dynamical Systems Approach in Mathematical Modeling of Thyroid
Gholam Reza Rokni LAMOOKI
University of Tehran and Institute for Research in Fundamental Sciences (IPM), Tehran,
IRAN
[email protected]
Coauthors: Alireza MANI, Amirhossein SHIRAZI
In this paper we utilize the reaction kinetics of chemicals involved in thyroid gland to
obtain a system of differential equations. The system has a large scale with one input
and is highly nonlinear. We specifically focus on relaxation oscillations of the system.
See Berman (1961) and Danziger (1956) for early applications of differential equations
to study thyroid, Rosenfeld (2000) for a historic review, and Leow (2007) for underlying
feedback mechanism. Mentioned relaxation oscillations is under the influence of the input
and the negative feedback regulator and can be destroyed by various phenomena including
input bandwidth, input amplitude and the rates of reactions. Various thyroid malfunctioning will be discussed including inhibition of reactions via minerals. The results will
be illustrated by a series of analysis as well as graphs describing the normal behavior
and malfunctioning. The analysis above opens an aperture into the understanding of the
largest gland in body.
17
Novel Formulation for the Two Phase Immiscible Flow in Petroleum Reservoir
Alaeddin MALEK
Tarbiat Modares University, Tehran, IRAN
[email protected]
Coauthors: Samaneh KHODAYARI-SAMGHABADI
The cognition of the fluid flow in porous media is an important task for engineers and
mathematicians. One of the complicated kinds of this phenomenon is the fluid of two
phase flow in porous media. In this case, two immiscible fluids that are supposed to be
incompressible, contact with each other through a porous media. The problem is modeled
with a set of governing equations (PDE initial boundary value problem); involving the
Darcy law and mass conservation equation for each phase that pressure and saturation
are unknown. In this paper we propose new formulation for the two phase immiscible
flow in petroleum reservoir, when we have one injection and one production well that
are existed in the Five-spot pattern. We split the coupled system into pressure and
saturation equations. We solve pressure equation with second order implicit method and
for the saturation if there is not capillary pressure equation, we can get closed analytical
form of the solution for the water saturation equation; otherwise saturation equation is
numerically solved by using the Runge-Kutta method. A control volume finite element
method on unstructured grid is applied for spatial discretization of the corresponding
non-linear partial differential system. Numerical results for pressure, saturation after
2000 days for a reservoir containing injection and production wells are presented.
A Halanay-type Inequality on Time Scales in Higher Dimensional Spaces
Raziye MERT
Çankaya University, Ankara, TURKEY
[email protected]
Coauthors: Jia BAOGUO, Lynn ERBE
In this study, we investigate a certain class of Halanay-type inequalities on time scales
in higher dimensional spaces. By means of the obtained inequality, we derive some new
global stability conditions for linear delay dynamic systems on time scales. An example
is given to illustrate the results.
18
Stability of Abstract Dynamic Equations on Time Scales
Karima M. ORABY
Suez Canal University, EGYPT
[email protected]
Coauthors: Alaa E. HAMZA
In this paper, we investigate many types of stability (uniform stability, asymptotic
stability, uniform asymptotic stability, global stability, global asymptotic stability, exponential stability, and uniform exponential stability) of the homogeneous linear dynamic
equations. Finally, we give an illustrative example for a non-regressive homogeneous first
order linear dynamic equation and we investigate its stability.
Oscillation Theorems For Second Order Nonlinear Differential Equations
Süleyman ÖĞREKÇİ
Gazi University, Ankara, TURKEY
[email protected]
Coauthors: Adil MISIR
In this study, we give some oscillation criterions for non-linear differential equations
of second-order by using Riccati technique. Our results improve the theorems given in
[Fanwei Meng, Yan Huang, Interval oscillation criteria for a forced second-order nonlinear
differential equations with damping, Applied Mathematics and Computation 218 (2011)
1857–1861] and some known results inthe literature.
Oscillation Criterion for Half-Linear Differential Equations with Periodic Coefficients
Abdullah ÖZBEKLER
Atılım University
[email protected]
Coauthors: O. DOSLY, R. Simon HILSCHER
We present an oscillation criterion for second order half-linear differential equations
with periodic coefficients. The method is based on the nonexistence of a proper solution
of the related modified Riccati equation. Our result can be regarded as an oscillatory
counterpart to the nonoscillation criterion by Sugie and Matsumura (2008). These two
theorems provide a complete half-linear extension of the oscillation criterion of Kwong
and Wong (2003) dealing with the Hill’s equation.
19
Difference Schemes for Elliptic Equations
Elif ÖZTÜRK
Uludağ University, Bursa, TURKEY
[email protected]
Coauthors: Allaberen ASHYRALYEV
The nonlocal Bitsadze-Samarskii type nonlocal boundary value problems

d2 u(t)

−
+ Au(t) = f (t), 0 < t < 1,

dt2


J

P

u(0) = ϕ, u(1) =
αj u(λj ) + ψ,
j=1

 P
J



|αj | ≤ 1, 0 < λ1 < λ2 < · · · < λJ < 1

j=1
for the differential equations in a Hilbert space H with the self-adjoint positive definite
operator A with a domain D(A) ⊂ H is considered. The second and fourth orders of accuracy difference schemes are presented. The stability and almost coercive stability of these
difference schemes are established. A procedure of modified Gauss elimination method
is used for solving these difference schemes for the two-dimensional elliptic differential
equation. The method is illustrated by numerical examples.
Integer Order Approximation of Uncertain Fractional Order Differentiations
and Integrations
M. Mine ÖZYETKİN
İnönü University, Malatya, TURKEY
[email protected]
Coauthors: Nusret TAN
In this talk we deal with the robust stability of the fractional order system having
interval order uncertainty. For this aim, integer order approximations of s[α,α] are computed. Using interval arithmetic rules, integer order equivalences of s[α,α] are obtained in
terms of first, second, third and fourth order integer approximations dependent on α. It is
shown that integer order equivalences of s[α,α] have interval transfer function (coefficients
of equivalence transfer function have interval structure). Kharitonov stability criterion
is applied to check the stability of the system. The proposed idea is supported through
numerical examples. The details of the proposed method will be given in the final version
of the presentation.
20
On the decay and blow up of solutions for coupled wave equations of Kirchhoff
type with nonlinear damping and source terms
Erhan PİŞKİN
Dicle University, Diyarbakır, TURKEY
[email protected]
Coauthors: Necat POLAT
In this work, we consider an initial boundary value problem for coupled wave equations
of Kirchhoff type. For some restrictions on the initial data, we establish the exponential
and polynomial decay. After that, we obtain the blow up of the solution with negative
initial energy.
Fractional Hyperchaotic Telecommunication Systems: A New Paradigm
Abolhassan RAZMINIA
Persian Gulf University, Bushehr, IRAN
[email protected]
Coauthors: Dumitru BALEANU
The dynamics of hyperchaotic and fractional-order systems have increasingly attracted
attention in recent years. In this paper, we mix two complex dynamics to construct a new
telecommunication system. Using a hyperchaotic fractional order system, we propose
a novel synchronization scheme between receiver and transmitter which increases the
security of data transmission and communication. Indeed this is first work that can open
a new way in secure communication system.
Analysis of a predator-prey model with modified Leslie-Gower and Hollingtype II schemes with term refuge
Safia SLIMANI
Annaba, ALGERIA
slimani [email protected]
Coauthors: Azzedine BENCHATTEH
In this study we present a two-dimensional continuous time dynamical system modelling a predator-prey food chain based on a modified version of the Leslie-Gower scheme
and on the Holling-type II scheme incorporating a term refuge. We show the boundedness
of solutions, existence of an attracting set and global stability of the coexisting interior
equilibrium
21
On Oscillation of Elliptic Inequalities
Yeter ŞAHİNER
Hacettepe University, Ankara, TURKEY
[email protected]
Coauthors: Ağacık ZAFER
Sufficient conditions are obtained for the oscillation of solutions of half-linear elliptic
inequalities with p(x)-Laplacian. The results obtained are new even for one-dimensional
case.
Inverse Problem For Interior Spectral Data of The Hydrogen Atom Equation
Murat ŞAT
Erzincan University, Erzincan, TURKEY
murat [email protected]
Coauthors: Etibar S. PENAHLI
We consider an inverse problem for the second order differential operators with a
regular singularity, and show that the potential function can be uniquely determined by
a set of values of eigenfunctions at some interior point and parts of two spectra.
Boundedness Results of Certain Type Second Order Nonlinear Differential
Equations
Aydın TİRYAKİ
İzmir University, İzmir, TURKEY
[email protected]
In this paper, by using standard methods we present some boundedness results for
certain type second order nonlinear differential equations. The special cases of these
results are valid for the well-known Emden-Fowler equation and half-linear equations.
Also some examples illustrating the theory will be given.
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Positive Solutions For Second-Order Impulsive Boundary Value Problems on
Time Scales
Fatma TOKMAK
Ege University, İzmir, TURKEY
[email protected]
Coauthors: İlkay Y. KARACA
In this study, we consider second-order impulsive boundary value problems and eigenvalue problem on time scales. By using some fixed point theorems, we investigate the
existence of positive solutions.
Oscillatory Behaviour of a Higher Order Nonlinear Neutral Type Functional
Dynamic Equation with Oscillating Coefficients
Deniz UÇAR
Uşak University, Uşak, TURKEY
[email protected]
Coauthors: Yaşar BOLAT
In this paper we are concerned with the oscillation of solutions of a certain more
general higher order nonlinear neutral type functional dynamic equation with oscillating
coefficients. We obtain some sufficient criteria for oscillatory behaviour of its solutions.
Stability Conditions for Linear Fractional Difference Systems
Paolo VETTORI
University of Aveiro, PORTUGAL
[email protected]
The well-known condition for the stability of fractional differential systems published
by Matignon in 1996 [Stability results for fractional differential equations with applications
to control processing. In: Computational Engineering in Systems Applications, pp. 963–
968] will be extended to discrete-time systems defined by fractional difference equations
(nabla calculus). This will be accomplished using Laplace transforms on time scales,
unifying the continuous and discrete-time cases.
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On the Existence of Solutions for Fractional Differential Equations
Ali YAKAR
Gaziosmanpaşa University, Tokat, TURKEY
[email protected]
In this work, existence of solutions for fractional differential equations (FDEs) involving standart Riemann Liouville fractional derivative is established. Our main tool is the
method of upper and lower solutions which is one of the effective ways to reveal the existence of solutions of nonlinear FDEs. Also we use functions satisfying Cp continuity
assumption instead of imposing Holder continuity. In addition, it is shown that these results can be generalized to the existence of solutions for systems of fractional differential
equations.
Caratheodory solutions of Sturm-Liouville dynamic equation with a measure
of noncompactness in Banach spaces
Ahmet YANTIR
Yaşar University, İzmir, TURKEY
[email protected]
Coauthors: Ireneusz Kubiaczyk, Aneta Sikorska-Nowak
We prove the existence result for Carathéodory type solutions of the nonlinear SturmLiouville boundary value problem on time scales in Banach spaces. We obtain the sufficient conditions for the existence of Carathéodory solutions in terms of Kuratowski measure of noncompactness. We express the problem of finding the Carathéodory solutions of
Sturm- Liouville boundary value problem as an integral operator on an appropriate set.
The existence of fixed points this integral operator is proved by using Mönch’s fixed point
theorem. We also remark that Kuratowski measure of noncompactness can be replaced
by any axiomatic measure of noncompactness.
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q-discrete Generalized Toda Equation
Burcu Silindir YANTIR
İzmir University of Economics, İzmir, TURKEY
[email protected]
We present the q-analogue of generalized Toda equation. We develop its three-qsoliton solutions, which are expressed in the form of polynomials in power functions. The
q-analogue of generalized Toda equation is a unified equation as proper reductions of
parameters give rise to various types of q-difference analogues of soliton equations such
as Toda equation, KdV equation a nd sine-Gordon equation. Therefore, it comprises
three-q-soliton solutions of these various types of q-difference soliton equations.
All-at-once approach for the Optimal Control of Burgers Equation
Fikriye Nuray YILMAZ
Gazi University, Ankara, TURKEY
[email protected]
Coauthors: Bülent KARASÖZEN
In this work, we apply the all-at-once method for the optimal control of unsteady
Burgers equation. The all-at-once methods were applied in recent years for optimal control problems governed by linear elliptic and parabolic equations. The state equation is
discretized and then the optimality system for the finite dimensional optimization problem
is derived. This approach is also referred to as the black-box approach. In other words, an
existing algorithm for the solution of the state equation is embedded into an optimization
loop. For space discretization, we use the Galerkin finite element method. The nonlinear
Burgers equation is discretized in time using the semi implicit discretization which results
an effective linearization of the optimal control problem. Numerical results for distributed
unconstrained and control constrained problems illustrate the performance of all-at-once
approach with semi-implicit time discretization
Fractional Davey-Stewartson Equations within Variational Iteration Method
Tuğba YILMAZ
Çankaya University, Ankara, TURKEY
[email protected]
Coauthors: Dumitru BALEANU
The variational iteration method is applied for the fractional Davey-Stewartson equations in the Caputo sense and the approximate analytical solutions are obtained.
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Initial Value Problems in Clifford Algebras Depending on Parameters
Uğur YÜKSEL
Atılım University, Ankara, TURKEY
[email protected]
This study deals with initial value problems of type
∂t u(t, x) = Lu(t, x),
u(0, x) = u0 (x)
in Clifford algebras depending on parameters.
Lyapunov-type Inequalities for Planar Linear Dynamic Hamiltonian Systems
Ağacık ZAFER
Middle East Technical University, Ankara, TURKEY
[email protected]
Coauthors: Martin Bohner
Lyapunov-type inequalities are useful in studying the qualitative behavior of solutions
such as oscillation, disconjugacy, and eigenvalue problems for differential and difference
equations. In this talk we give new Lyapunov-type inequalities for linear Hamiltonian
systems on arbitrary time scales, which improve recently published results and hence all
the related ones in the literature.
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