INTERNATIONAL CONFERENCE OF NUMERICAL ANALYSIS AND APPLIED MATHEMATICS (ICNAAM 2016), 2017
We obtain Sturm Picone type comparison theorems for nonlinear impulsive differential equations. O... more We obtain Sturm Picone type comparison theorems for nonlinear impulsive differential equations. Our results cover the previous results existing in the literature and useful in investigating qualitative behaviour of solutions of such equations.
Hacettepe journal of mathematics and statistics, Feb 14, 2022
In this paper two kinds of dynamic Hardy-Copson type inequalities are derived via diamond alpha i... more In this paper two kinds of dynamic Hardy-Copson type inequalities are derived via diamond alpha integrals. The first kind consists of twelve new integral inequalities which can be considered as mixed type in the sense that these inequalities contain delta, nabla and diamond alpha integrals together. The second kind involves another twelve new inequalities, which are composed of only diamond alpha integrals, unifying delta and nabla Hardy-Copson type inequalities. Our approach is quite new due to the fact that it uses time scale calculus rather than algebra. Therefore both kinds of our results unify some of the known delta and nabla Hardy-Copson type inequalities into one diamond alpha Hardy-Copson type inequalities and offer new Hardy-Copson type inequalities even for the special cases.
Journal of Mathematical Analysis and Applications, Aug 1, 2016
Abstract We present new Lyapunov-type inequalities for Hamiltonian systems, consisting of 2 n -fi... more Abstract We present new Lyapunov-type inequalities for Hamiltonian systems, consisting of 2 n -first-order linear impulsive differential equations, by making use of matrix measure approach. The matrix measure estimates of fundamental matrices of linear impulsive systems are crucial in obtaining sharp inequalities. To illustrate usefulness of the inequalities we have derived new disconjugacy criteria for Hamiltonian systems under impulse effect and obtained new lower bound estimates for eigenvalues of impulsive eigenvalue problems.
Hacettepe Journal of Mathematics and Statistics, 2017
A new and dierent approach to the investigation of the existence and uniqueness of solution of no... more A new and dierent approach to the investigation of the existence and uniqueness of solution of nonhomogenous impulsive boundary value problems involving the Caputo fractional derivative of order α (1 < α ≤ 2) is brought by using Lyapunov type inequality. To express and to analyze the unique solution, Green's function and its bounds are established, respectively. As far as we know, this approach based on the link between fractional boundary value problems and Lyapunov type inequality, has not been revealed even in the absence of impulse eect. Besides, the novel Lyapunov type inequality generalizes the related ones in the literature.
Numerical Heat Transfer Part B-fundamentals, Jun 1, 2013
In this article, we consider a higher-order numerical scheme for the fractional heat equation wit... more In this article, we consider a higher-order numerical scheme for the fractional heat equation with Dirichlet and Neumann boundary conditions. By using a fourth-order compact finitedifference scheme for the spatial variable, we transform the fractional heat equation into a system of ordinary fractional differential equations which can be expressed in integral form. Further, the integral equation is transformed into a difference equation by a modified trapezoidal rule. Numerical results are provided to verify the accuracy and efficiency of the proposed algorithm.
This paper is devoted to the nabla unification of the discrete and continuous Hardy-Copson type i... more This paper is devoted to the nabla unification of the discrete and continuous Hardy-Copson type inequalities. Some of the obtained inequalities are nabla counterparts of their delta versions while the others are new even for the discrete, continuous, and delta cases. Moreover, these dynamic inequalities not only generalize and unify the related ones in the literature but also improve them in the special cases.
Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics
We aim to find the complements of the Bennett-Leindler type inequalities in nabla time scale calc... more We aim to find the complements of the Bennett-Leindler type inequalities in nabla time scale calculus by changing the exponent from $0<\zeta< 1$ to $\zeta>1.$ Different from the literature, the directions of the new inequalities, where $\zeta>1,$ are the same as that of the previous nabla Bennett-Leindler type inequalities obtained for $0<\zeta< 1$. By these settings, we not only complement existing nabla Bennett-Leindler type inequalities but also generalize them by involving more exponents. The dual results for the delta approach and the special cases for the discrete and continuous ones are obtained as well. Some of our results are novel even in the special cases.
In this thesis, Lyapunov type inequalities and their applications for impulsive systems of linear... more In this thesis, Lyapunov type inequalities and their applications for impulsive systems of linear/nonlinear differential equations are studied. Since systems under impulse effect are one of the fundamental problems in most branches of applied mathematics, science and technology, investigation of their theory has developed rapidly in the last three decades. In addition, Lyapunov type inequalities have become a popular research area in recent years due to the fact that they provide not only better understanding of the qualitative nature of the solutions of ordinary and impulsive systems, for instance oscillation, disconjugacy, stability and asymptotic behavior of solutions, but also deeper analysis for boundary and eigenvalue problems. This thesis consists of 7 chapters. Chapter 1 is introductory and contains detailed literature review, and brief information about the linear systems of impulsive differential equations and Hamiltonian systems. The main contributions of the thesis, whic...
In this paper two kinds of dynamic Hardy-Copson type inequalities are derived via diamond alpha i... more In this paper two kinds of dynamic Hardy-Copson type inequalities are derived via diamond alpha integrals. The first kind consists of twelve new integral inequalities which can be considered as mixed type in the sense that these inequalities contain delta, nabla and diamond alpha integrals together. The second kind involves another twelve new inequalities, which are composed of only diamond alpha integrals, unifying delta and nabla Hardy-Copson type inequalities. Our approach is quite new due to the fact that it uses time scale calculus rather than algebra. Therefore both kinds of our results unify some of the known delta and nabla Hardy-Copson type inequalities into one diamond alpha Hardy-Copson type inequalities and offer new Hardy-Copson type inequalities even for the special cases.
This paper is devoted to novel diamond alpha Hardy–Copson type dynamic inequalities, which are co... more This paper is devoted to novel diamond alpha Hardy–Copson type dynamic inequalities, which are complements of the classical ones obtained for and their applications to difference equations. We obtain two kinds of diamond alpha Hardy–Copson type inequalities for , one of which is mixed type and established by the convex linear combinations of the related delta and nabla inequalities while the other one is new and is obtained by using time scale calculus rather than algebra. In contrast to the works existing in the literature, these complements are derived by preserving the directions of the classical inequalities. Therefore both kinds of our results unify some of the known delta and nabla Hardy–Copson type inequalities obtained for into one diamond alpha Hardy–Copson type inequalities and offer new types of diamond alpha Hardy–Copson type inequalities which have the same directions as the classical ones and can be considered as complementary inequalities. Moreover the application of these inequalities in the oscillation theory of half linear difference equations provides several nonoscillation criteria for such equations.
Mathematical Methods in the Applied Sciences, 2021
In this paper, two kinds of dynamic Bennett‐Leindler type inequalities via the diamond alpha inte... more In this paper, two kinds of dynamic Bennett‐Leindler type inequalities via the diamond alpha integrals are derived. The first kind consists of eight new integral inequalities which can be considered as mixed type in the sense that these inequalities contain delta, nabla and diamond alpha integrals together due to the fact that convex linear combinations of delta and nabla Bennett‐Leindler type inequalities give diamond alpha Bennett‐Leindler type inequalities. The second kind involves four new inequalities, which are composed of only diamond alpha integrals, unifying delta and nabla Bennett‐Leindler type inequalities. For the second type, choosing α=1 or α=0 not only yields the same results as the ones obtained for delta and nabla cases but also provides novel results for them. Therefore, both kinds of our results expand some of the known delta and nabla Bennett‐Leindler type inequalities, offer new types of these inequalities, and bind and unify them into one diamond alpha Bennett‐Leindler type inequalities. Moreover, an application of dynamic Bennett‐Leindler type inequalities to the oscillation theory of the second‐order half linear dynamic equation is developed and presented for the first time ever.
INTERNATIONAL CONFERENCE OF NUMERICAL ANALYSIS AND APPLIED MATHEMATICS (ICNAAM 2016), 2017
We obtain Sturm Picone type comparison theorems for nonlinear impulsive differential equations. O... more We obtain Sturm Picone type comparison theorems for nonlinear impulsive differential equations. Our results cover the previous results existing in the literature and useful in investigating qualitative behaviour of solutions of such equations.
Hacettepe journal of mathematics and statistics, Feb 14, 2022
In this paper two kinds of dynamic Hardy-Copson type inequalities are derived via diamond alpha i... more In this paper two kinds of dynamic Hardy-Copson type inequalities are derived via diamond alpha integrals. The first kind consists of twelve new integral inequalities which can be considered as mixed type in the sense that these inequalities contain delta, nabla and diamond alpha integrals together. The second kind involves another twelve new inequalities, which are composed of only diamond alpha integrals, unifying delta and nabla Hardy-Copson type inequalities. Our approach is quite new due to the fact that it uses time scale calculus rather than algebra. Therefore both kinds of our results unify some of the known delta and nabla Hardy-Copson type inequalities into one diamond alpha Hardy-Copson type inequalities and offer new Hardy-Copson type inequalities even for the special cases.
Journal of Mathematical Analysis and Applications, Aug 1, 2016
Abstract We present new Lyapunov-type inequalities for Hamiltonian systems, consisting of 2 n -fi... more Abstract We present new Lyapunov-type inequalities for Hamiltonian systems, consisting of 2 n -first-order linear impulsive differential equations, by making use of matrix measure approach. The matrix measure estimates of fundamental matrices of linear impulsive systems are crucial in obtaining sharp inequalities. To illustrate usefulness of the inequalities we have derived new disconjugacy criteria for Hamiltonian systems under impulse effect and obtained new lower bound estimates for eigenvalues of impulsive eigenvalue problems.
Hacettepe Journal of Mathematics and Statistics, 2017
A new and dierent approach to the investigation of the existence and uniqueness of solution of no... more A new and dierent approach to the investigation of the existence and uniqueness of solution of nonhomogenous impulsive boundary value problems involving the Caputo fractional derivative of order α (1 < α ≤ 2) is brought by using Lyapunov type inequality. To express and to analyze the unique solution, Green's function and its bounds are established, respectively. As far as we know, this approach based on the link between fractional boundary value problems and Lyapunov type inequality, has not been revealed even in the absence of impulse eect. Besides, the novel Lyapunov type inequality generalizes the related ones in the literature.
Numerical Heat Transfer Part B-fundamentals, Jun 1, 2013
In this article, we consider a higher-order numerical scheme for the fractional heat equation wit... more In this article, we consider a higher-order numerical scheme for the fractional heat equation with Dirichlet and Neumann boundary conditions. By using a fourth-order compact finitedifference scheme for the spatial variable, we transform the fractional heat equation into a system of ordinary fractional differential equations which can be expressed in integral form. Further, the integral equation is transformed into a difference equation by a modified trapezoidal rule. Numerical results are provided to verify the accuracy and efficiency of the proposed algorithm.
This paper is devoted to the nabla unification of the discrete and continuous Hardy-Copson type i... more This paper is devoted to the nabla unification of the discrete and continuous Hardy-Copson type inequalities. Some of the obtained inequalities are nabla counterparts of their delta versions while the others are new even for the discrete, continuous, and delta cases. Moreover, these dynamic inequalities not only generalize and unify the related ones in the literature but also improve them in the special cases.
Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics
We aim to find the complements of the Bennett-Leindler type inequalities in nabla time scale calc... more We aim to find the complements of the Bennett-Leindler type inequalities in nabla time scale calculus by changing the exponent from $0<\zeta< 1$ to $\zeta>1.$ Different from the literature, the directions of the new inequalities, where $\zeta>1,$ are the same as that of the previous nabla Bennett-Leindler type inequalities obtained for $0<\zeta< 1$. By these settings, we not only complement existing nabla Bennett-Leindler type inequalities but also generalize them by involving more exponents. The dual results for the delta approach and the special cases for the discrete and continuous ones are obtained as well. Some of our results are novel even in the special cases.
In this thesis, Lyapunov type inequalities and their applications for impulsive systems of linear... more In this thesis, Lyapunov type inequalities and their applications for impulsive systems of linear/nonlinear differential equations are studied. Since systems under impulse effect are one of the fundamental problems in most branches of applied mathematics, science and technology, investigation of their theory has developed rapidly in the last three decades. In addition, Lyapunov type inequalities have become a popular research area in recent years due to the fact that they provide not only better understanding of the qualitative nature of the solutions of ordinary and impulsive systems, for instance oscillation, disconjugacy, stability and asymptotic behavior of solutions, but also deeper analysis for boundary and eigenvalue problems. This thesis consists of 7 chapters. Chapter 1 is introductory and contains detailed literature review, and brief information about the linear systems of impulsive differential equations and Hamiltonian systems. The main contributions of the thesis, whic...
In this paper two kinds of dynamic Hardy-Copson type inequalities are derived via diamond alpha i... more In this paper two kinds of dynamic Hardy-Copson type inequalities are derived via diamond alpha integrals. The first kind consists of twelve new integral inequalities which can be considered as mixed type in the sense that these inequalities contain delta, nabla and diamond alpha integrals together. The second kind involves another twelve new inequalities, which are composed of only diamond alpha integrals, unifying delta and nabla Hardy-Copson type inequalities. Our approach is quite new due to the fact that it uses time scale calculus rather than algebra. Therefore both kinds of our results unify some of the known delta and nabla Hardy-Copson type inequalities into one diamond alpha Hardy-Copson type inequalities and offer new Hardy-Copson type inequalities even for the special cases.
This paper is devoted to novel diamond alpha Hardy–Copson type dynamic inequalities, which are co... more This paper is devoted to novel diamond alpha Hardy–Copson type dynamic inequalities, which are complements of the classical ones obtained for and their applications to difference equations. We obtain two kinds of diamond alpha Hardy–Copson type inequalities for , one of which is mixed type and established by the convex linear combinations of the related delta and nabla inequalities while the other one is new and is obtained by using time scale calculus rather than algebra. In contrast to the works existing in the literature, these complements are derived by preserving the directions of the classical inequalities. Therefore both kinds of our results unify some of the known delta and nabla Hardy–Copson type inequalities obtained for into one diamond alpha Hardy–Copson type inequalities and offer new types of diamond alpha Hardy–Copson type inequalities which have the same directions as the classical ones and can be considered as complementary inequalities. Moreover the application of these inequalities in the oscillation theory of half linear difference equations provides several nonoscillation criteria for such equations.
Mathematical Methods in the Applied Sciences, 2021
In this paper, two kinds of dynamic Bennett‐Leindler type inequalities via the diamond alpha inte... more In this paper, two kinds of dynamic Bennett‐Leindler type inequalities via the diamond alpha integrals are derived. The first kind consists of eight new integral inequalities which can be considered as mixed type in the sense that these inequalities contain delta, nabla and diamond alpha integrals together due to the fact that convex linear combinations of delta and nabla Bennett‐Leindler type inequalities give diamond alpha Bennett‐Leindler type inequalities. The second kind involves four new inequalities, which are composed of only diamond alpha integrals, unifying delta and nabla Bennett‐Leindler type inequalities. For the second type, choosing α=1 or α=0 not only yields the same results as the ones obtained for delta and nabla cases but also provides novel results for them. Therefore, both kinds of our results expand some of the known delta and nabla Bennett‐Leindler type inequalities, offer new types of these inequalities, and bind and unify them into one diamond alpha Bennett‐Leindler type inequalities. Moreover, an application of dynamic Bennett‐Leindler type inequalities to the oscillation theory of the second‐order half linear dynamic equation is developed and presented for the first time ever.
Uploads
Papers by Zeynep Kayar