Difference Equations

In this study, we use a compartmental nonlinear deterministic mathematical model to investigate the effect of different optimal control strategies in controlling Tuberculosis (TB) disease transmission in the community. We employ stability theory of differential equations to investigate the qualitative behavior of the model by obtaining the basic reproduction number and determining the local stability conditions for the disease-free and endemic equilibria. We consider three control strategies representing distancing, case finding, and treatment efforts and numerically compare the levels of exposed and infectious populations with and without control strategies. The results suggest that combination of all controls is the best strategy to eradicate TB disease from the community at an optimal level with minimum cost of interventions.

The present paper deals with a fractional-order mathematical epidemic model of malaria transmission accompanied by temporary immunity and relapse. The model is revised by using Caputo fractional operator for the index of memory. We also recommend the utilization of temporary immunity and the possibility of relapse. The theory of locally bounded and Lipschitz is employed to inspect the existence and uniqueness of the solution of the malaria model. It is shown that temporary immunity has a great effect on the dynamical transmission of host and vector populations. The stability analysis of these equilibrium points for fractional-order derivative α and basic reproduction number R 0 $\mathcal{R}_{0}$ is discussed. The model will exhibit a Hopf-type bifurcation. The two control variables are introduced in this model to decrease the number of populations. Mandatory conditions for the control problem are produced. Two types of numerical method via Laplace Adomian decomposition and Runge–Kut...

In this paper we have proposed a stochastic model for studying the dynamics of tuberculosis (TB) by incorporating vaccination of newly born babies. The total population in this model is subdivided in to four compartments, namely susceptible $S ( t )$ S ( t ) , infected $I ( t )$ I ( t ) , vaccinated newborns $V ( t ) $ V ( t ) , and recovered $R ( t ) $ R ( t ) . First, the developed model is expressed and analyzed by the deterministic approach. Since this approach neglects the randomness of the dynamics of the process, it has great limitations in the modeling process. To avoid this kind of issues, we transform the deterministic approach into a stochastic one, which is known to play a significant role by providing additional degree of realism compared to the deterministic approach. The analysis of the model is done employing both approaches. The invariant region, positivity of the solution, equilibrium points and their stability are checked. According to the analysis, we came to rea...

The present study deals with the analysis of a Lotka-Volterra model describing competition between tumor and immune cells. The model consists of differential equations with piecewise constant arguments and based on metamodel constructed by Stepanova. Using the method of reduction to discrete equations, it is obtained a system of difference equations from the system of differential equations. In order to get local and global stability conditions of the positive equilibrium point of the system, we use Schur-Cohn criterion and Lyapunov function that is constructed. Moreover, it is shown that periodic solutions occur as a consequence of Neimark-Sacker bifurcation.

In this paper, by using the Lie symmetry analysis, all of the geometric vector fields of the ( 3 + 1 ) $(3+1)$ -Burgers system are obtained. We find the 1, 2, and 3-dimensional optimal system of the Burger system and then by applying the 3-dimensional optimal system reduce the order of the system. Also the nonclassical symmetries of the ( 3 + 1 ) $(3+1)$ -Burgers system will be found by employing nonclassical methods. Finally, the ansatz solutions of BS equations with the aid of the tanh method has been presented. The achieved solutions are investigated through two- and three-dimensional plots for different values of parameters. The analytical simulations are presented to ensure the efficiency of the considered technique. The behavior of the obtained results for multiple cases of symmetries is captured in the present framework. The outcomes of the present investigation show that the considered scheme is efficient and powerful to solve nonlinear differential equations that arise in t...

Coloring of fuzzy graphs has many real life applications in combinatorial optimization problems like traffic light system, exam scheduling, register allocation, etc. In this paper, the concept of fuzzy chromatic polynomial of fuzzy graph is introduced and defined based on í µí»¼-cuts of fuzzy graph. Two different types of fuzziness to fuzzy graph are considered in the paper. The first type was fuzzy graph with crisp vertex set and fuzzy edge set and the second type was fuzzy graph with fuzzy vertex set and fuzzy edge set. Depending on this, the fuzzy chromatic polynomials for some fuzzy graphs are discussed. Some interesting remarks on fuzzy chromatic polynomial of fuzzy graphs have been derived. Further, some results related to the concept are proved. Lastly, fuzzy chromatic polynomials for complete fuzzy graphs and fuzzy cycles are studied and some results are obtained.

An analytical delay model for BiCMOS driver circuits is presented. The model is based on physical device parameters and can be used to estimate both the pull-up and the pull-down times for a variety of circuit configurations. The intrinsic delay associated with the bipolar transistors is taken into consideration by using a charge control model that incorporates the high-injection effects

Although recently there has been a great interest in studying of the behaviour of the solutions of rational difference equations, there are only a few papers devoted to systems of the rational difference equations. The aim of this study is to investigate the periodic character of the some systems of difference equations and to contribute the studies on this subject. This study consists of four sections.

In the first section; the definitions of difference equations are given.

In the second section; the studies on the systems of difference equations are summarized.
In the third section;  the periodic character of the some systems of difference equations are investigated.

In the fourth section; the systems of difference equations that investigated in third section are generalized.

Son zamanlarda rasyonel fark denklemlerinin çözümlerinin davranışları ile ilgili önemli çalışmalar yapılmasına rağmen , rasyonel fark denklem sistemleri ile ilgili az sayıda çalışma vardır. Bu çalışmanın amacı, bazı fark denklem sistemlerinin çözümlerinin periyodikliğini incelemek ve bu konudaki çalışmalara katkıda bulunmaktır. Bu amaçla hazırlanan çalışma dört bölümden oluşmuştur.

Birinci bölümde fark denklemleri ile ilgili tanımlar verilmiştir.

İkinci bölümde fark denklem sistemleri ile ilgili yapılan çalışmalar özetlenmiştir.

Üçüncü bölümde bazı fark denklem sistemlerinin çözümlerinin periyodikliği incelenmiştir.

Dördüncü bölümde, üçüncü bölümde incelenen fark denklem sistemleri genelleştirilmiştir.

Physiologically-based pharmacokinetic (PBPK) models have been used to describe the distribution and elimination characteristics of intravenous ethanol administration. Further, these models have been used to estimate the ethanol infusion profile required to prescribe a specific breath ethanol concentration time course in a specific human being, providing a platform upon which other pharmacokinetic and pharmacodynamic investigations are based. In these PBPK models, the equivalence of two different peripheral tissue models are shown and issues concerning the mass flow into the liver in comparison with ethanol metabolism in the liver are explained.

A new family of p-Bernoulli numbers and polynomials was introduced by Rahmani (J. Number Theory 157:350–366, 2015) with the help of the Gauss hypergeometric function. Motivated by that paper and in the light of the recent interests in finding degenerate versions, we construct the generalized degenerate Bernoulli numbers and polynomials by means of the Gauss hypergeometric function. In addition, we construct the degenerate type Eulerian numbers as a degenerate version of Eulerian numbers. For the generalized degenerate Bernoulli numbers, we express them in terms of the degenerate Stirling numbers of the second kind, of the degenerate type Eulerian numbers, of the degenerate p-Stirling numbers of the second kind and of an integral on the unit interval. As to the generalized degenerate Bernoulli polynomials, we represent them in terms of the degenerate Stirling polynomials of the second kind.

The aim of this paper is to study Jindalrae and Gaenari numbers and polynomials in connection with Jindalrae-Stirling numbers of the first and second kinds. For this purpose, we first introduce Jindalrae-Stirling numbers of the first and second kinds as extensions of the notions of the degenerate Stirling numbers of the first and second kinds, and deduce several relations involving those special numbers. Then we introduce Jindalrae and Gaenari numbers and polynomials and obtain some explicit expressions and identities associated with those numbers and polynomials. In addition, we interpret our results by using umbral calculus.

A stand growth model for Mediterranean maritime pine in the Iberian Peninsula (Pinus pinaster Ait.) is presented. The model consist of a site index submodel developed from 281 stem analysis and validated with the data from 92 permanent sample plots. Three height growth equations in difference form are tested and the Bailey and Clutter (1974) function is considered appropriate due to its good performance with both fitting and validation data. A compatible growth and yield submodel is then elaborated with the data from the Forest Research Centre (CIFOR-INIA) permanent sample plots network. The future state of the stand is determined by the current state, characterized by basal area, age and density. The model is completed with a control function that predicts the diameter after thinning and a mortality rate for non-thinned intervals. The global model was validated with data of 13 independent permanent thinning plots from two experimental sites. The simulation of the long-term projection of volume, basal area and DBH after thinning shows certain overestimation in diameter after thinning and in volume. However, the results show errors lower than 5% and little bias.

Transmission dynamics of swine influenza pandemic is analysed through a deterministic model. Qualitative analysis of the model includes global asymptotic stability of disease-free and endemic equilibria under a certain condition based on the reproduction number. Sensitivity analysis to ponder the effect of model parameters on the reproduction number is performed and control strategies are designed. It is also verified that the obtained numerical results are in good agreement with the analytical ones.

A stand growth model for Mediterranean maritime pine in the Iberian Peninsula (Pinus pinaster Ait.) is presented. The model consist of a site index submodel developed from 281 stem analysis and validated with the data from 92 permanent sample plots. Three height growth equations in difference form are tested and the Bailey and Clutter (1974) function is considered appropriate due to its good performance with both fitting and validation data. A compatible growth and yield submodel is then elaborated with the data from the Forest Research Centre (CIFOR-INIA) permanent sample plots network. The future state of the stand is determined by the current state, characterized by basal area, age and density. The model is completed with a control function that predicts the diameter after thinning and a mortality rate for non-thinned intervals. The global model was validated with data of 13 independent permanent thinning plots from two experimental sites. The simulation of the long-term projection of volume, basal area and DBH after thinning shows certain overestimation in diameter after thinning and in volume. However, the results show errors lower than 5% and little bias.

ÖZ Bu çalışmada Celal Bayar Üniversitesi'nde 2014-2015 eğitim-öğretim döneminde formasyon eğitimine katılan öğretmen adaylarının Bilgisayar Destekli Öğretime ilişkin tutumları ile demografik bazı değişkenler arasındaki ilişkiler incelenmiştir. Bu amaçla, 478 öğretmen adayına Arslan (2006) tarafından geliştirilen Bilgisayar Destekli Eğitim Yapmaya İlişkin Tutum Ölçeği uygulanmıştır. Sonuç olarak, formasyon eğitimine katılan öğretmen adaylarının bilgisayar destekli öğretime yönelik tutumlarının olumlu yönde olduğu, adayların bilgisayar destekli öğretim yapmaya yönelik tutumlarında cinsiyete veya öğretmenlik deneyimlerine bağlı anlamlı farklılık olmadığı; ancak, yaşa bağlı olarak bilgisayar destekli öğretim yapmaya ilişkin tutumlar incelendiğinde 30 yaş üstü öğretmen adaylarının 25 yaş altı adaylara göre pozitif yönde anlamlı derecede olumlu tutumda oldukları görülmüştür. Ayrıca, İlahiyat alanından mezun öğretmen adaylarının tutumlarında Fizik, Kimya, Matematik ve Türk Dili ve Edebiyatı alanlarından mezun adaylara göre pozitif yönde anlamlı farklılık gözlenmiştir. Son olarak Bilgisayar Destekli Öğretim/Eğitim derslerinin uygulamalı olarak yapılması gerektiği önerilmiştir. Anahtar Kelimeler: BDE tutum ölçeği, formasyon öğretmen adayları, bilgisayar destekli öğretim.

Integral transform methods are widely used to solve the several dynamic equations with initial values or boundary conditions which are represented by integral equations. With this purpose, the Sumudu transform is introduced in this article as a new integral transform on a time scale "Equation missing" to solve a system of dynamic equations. The Sumudu transform on time scale "Equation missing" has not been presented before. The results in this article not only can be applied on ordinary differential equations when T = ℝ , difference equations when T = ℕ 0 , but also, can be applied for q-difference equations when T = q ℕ 0 , where q ℕ 0 : = { q t : t ∈ ℕ 0  for  q > 1 } or T = q ℤ ¯ : = q ℤ ∪ { 0 } for q > 1 (which has important applications in quantum theory) and on different types of time scales like T = h ℕ 0 , T = ℕ 0 2 and T = T n the space of the harmonic numbers. Finally, we give some applications to illustrate our main results. 2010 Mathematics Subj...