A simulation model based on **nonlinear** **ordinary** **differential** **equations** to in- terpret the transmission dynamics of Zika Virus (ZIKV), is formulated and analyzed, integrating the asymptomatic human population and **coupled** to the Aedes aegypti dynamics, the epidemic threshold Basic Reproduction Number R 0 is determined, as the spectral radius of Next-Generation Matrix and the

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Most of the existing literature deals with static networks, in which neither the set of nodes nor the set of edges varies over time. By contrast, in this paper we introduce a (simple) evolution model of social networks which is motivated by the work of Hoff, et al., [13]. We represent the dynamics of relationships between individuals by a **coupled** system of **nonlinear** stochastic **ordinary** **differential** **equations** in which the pairwise relationships are driven by both observable and unobservable (latent) characteristics.

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-7- In Chapter IV an optimal filter is derived for a completely general class of stochastic systems governed by coupled nonlinear ordinary and partial differential equations of either fi[r]

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Though the theory of positive solutions (PSs) for **ordinary** diﬀerential **equations** with parameters is mature, not much has been done for FDEs with parameters [12, 13, 17, 20]. By using the Guo–Krasnosel’skii ﬁxed point theorem on cones, some suﬃcient conditions for the existence of multiple PSs and eigenvalue intervals are established in [17] for the following FDEs with parameter:

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By substituting y in 3.1, we obtain a system of **equations**, and, with attention to Gaussian nodes, we must solve the set of **nonlinear** **equations**. The boundary conditions are imposed on the system of **equations** with elimination of two **equations**, and we add these boundary conditions to the system as follows:

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The **differential** **equations** of fractional order arise in many scientific disciplines, such as physics, chemistry, control theory, signal processing and biophysics. For more details, we refer the reader to [7, 9, 12] and the references therein. Recently, there has been an important progress in the investigation of these **equations**, (see [3, 4, 13]). On the other hand, the study of **coupled** systems of fractional **differential** **equations** is also of a great importance. These systems occur in various problems of applied science and engineering. For some recent results, we refer the interested reader to ([1, 2, 5, 6, 11].

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The study on periodic solutions for **ordinary** diﬀerential **equations** is a very important branch in the diﬀerential equation theory. Many results about the existence of periodic solutions for second-order diﬀerential **equations** have been obtained by combining the classical method of lower and upper solutions and the method of alternative problems The Lyapunov-Schmidt method as discussed by many authors 1–10. In 11, the author gives a simple method to discuss the existence and uniqueness of **nonlinear** two-point boundary value problems. In this paper, we will extend this method to the periodic problem.

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[10] Ibanez J, Hernandez V., Arias E. and Ruiz P.A., 2009. Solving initial value problems for **ordinary** **differential** **equations** by two approaches: BDF and piecewise- linearized methods, Comput. Phys. Commun. 180(5), 712--723.

level of cell concentration in the body [2, 5]. In this paper, a new kind of analytical approach for a non-linear system of **ordinary** **differential** **equations** called **Differential** transformation method (DTM) is addressed and used to approximate solutions for a well-known non-linear system. The **differential** transformation method is a kind of analytical technique based on the Taylor series expansion. This method constructs an analytic approximation to the solution, polynomial form. The concept of **differential** transform method was first proposed by Zhou and was applied to solve linear and **nonlinear** initial value problems in electric circuit analysis [6]. Chen and Liu applied this method to solve two-boundary-value problems [7]. Jang, Chen and Liu used two-dimensional **differential** transform method to solve partial **differential** **equations** [8]. Yu and Chen applied the **differential** transformation method for optimization of the rectangular fins with variable thermal parameters [9, 10]. Unlike the traditional high order Taylor series method that requires many symbolic computations, the **differential** transform method is an iterative procedure for obtaining Taylor series solutions. This method will not consume too much computer time when applying to non-linear or parameter varying systems.

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Abstract— Synthesis of biomolecular circuits for controlling molecular-scale processes is an important goal of synthetic biology with a wide range of in vitro and in vivo applications, including biomass maximization, nanoscale drug delivery, and many others. In this paper, we present new results on how abstract chemical reactions can be used to implement com- monly used system theoretic operators such as the polynomial functions, rational functions and Hill-type nonlinearity. We first describe how idealised versions of multi-molecular reactions, catalysis, annihilation, and degradation can be combined to implement these operators. We then show how such chemical reactions can be implemented using enzyme-free, entropy- driven DNA reactions. Our results are illustrated through three applications: (1) implementation of a Stan-Sepulchre oscillator, (2) the computation of the ratio of two signals, and (3) a PI+antiwindup controller for regulating the output of a static **nonlinear** plant.

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only with Hall current absent (m = 0); for m > 0, values are always positive. Therefore the Hall current assists the secondary magnetic induction but opposes the primary magnetic induction. We note that the Figures 11(a) and (b) correspond to the inclined applied magnetic field case of θ = π/4. In Tables 1 to 6 the critical Grashof number as computed in **Equations** (38) and (40) has been evalu- ated for various values of the thermophysical parameters for both the primary and secondary flow fields. We note that the critical Grashof number for the primary flow, as defined by

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Three-point boundary value problems for ordinary differential equations by matching solutions, Nonlinear Anal.. Exlstence-unlqueness theorems for three-polnt.[r]

At higher values of Ra, due to instability to asymmetric disturbances arising from machine round-off, the bottom half of this vector becomes nonzero and the **nonlinear** coupling between the two sub-blocks becomes so strong that the symmetry of the solution is broken. It is noted that the splitting of the original problem makes the linear stability analysis of the solutions very natural and straightforward, although it will not be discussed here.

The assumption that a disturbance can be represented by wave components, according to the method of normal modes, serves to separate the variables and reduces the linearized **equations** of motion from partial to **ordinary** **differential** **equations**. The final process consists of solving the set of **coupled**, homogeneous, **ordinary** linearized **differential** **equations** governing the amplitude A (z), subject to appropriate boundary conditions of the problem under investigation. Indeed, the requirement that the equation allows a non-trivial solution satisfying the various boundary conditions leads directly to a characteristic value problem for c. In general, the characteristic value for c will be complex, whose real and imaginary parts will apart from various modes, depend on the physically significant parameters involved in the system.

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In this section, the modified mathematical model of Ngwa and Shu’s model for the evolution of the malaria in a population formulated in [1] is presented in (1) – (9). The **equations** are based on both human and mosquito population, infection rates of human and mosquito per unit time, rescaling of critical population of each class by total species population and bifurcation analysis. Following the basic ideas and structure of mathematical modeling in epidemiology, the model for the malaria disease was developed under the terms in tables 1 and 2 (see [1] for detailed analysis).

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Fractional order **differential** **equations** are available in various disciplines of sci- ence and technology, we refer to [4, 6, 15, 19, 22, 25] for some of the applications. These **equations** have solutions under certain conditions. The area devoted to in- vestigate sufficient conditions for existence of positive solutions to these **equations** is well studied and plenty of research papers are available in literature, we refer to some of them [5, 16, 18, 20, 23, 7, 21] and the references therein. Recently, existence of solutions to boundary value problems for **coupled** systems of fractional order dif- ferential **equations** have also attracted some attentions, we refer to [1, 3, 24, 26, 27]. The area of **differential** **equations** devoted to quadratic perturbations of **nonlinear** problems, also known as hybrid **differential** **equations**, is one of the most important area and have attracted considerable attention from researchers. It is because, the class of hybrid **differential** **equations** that includes the perturbations of original dif- ferential **equations** in different ways, have fundamental importance as they include several dynamical systems as special cases. The study of hybrid **differential** equa- tions is implicit in the works of Krasnose´ lskii, Dhage and Lakshmikantham and extensively studied by many researchers, we refer [9, 13, 14, 17, 28]. Dhage and Lakshmikanathm [9] studied existence and uniqueness results for the following first order hybrid **differential** equation

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The paper is organized as follows. We define our model in Section 2. In Section 3, we introduce non-comoving coordinates and present the corresponding Einstein **equations** and the matter **equations**. In the course of integration of these **equations** in Section 4, we derive a **nonlinear** ODE for the gravitational potential. We reformulate this ODE in Section 5 as a Lienard **differential** equation and transform it into a rational Abel **differential** equation of the first kind. In Section 6, we list some open mathematical problems. Finally, possible physical applications (galactic halos, dark energy stars) and related open problems are outlined in Section 7.

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impossible to integrate in (14) in a closed form. Thus, we use the Chebyshev interpola- tion of the integrand to construct the sequence of successive approximations also in this case. In the last example, we use again the polynomial version of presented method in order to approximate a solution of the **nonlinear** Dirichlet problem contain- ing p-Laplacian.

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On the solutions in generalized functions of some ordinary differential equations with polynomial coefficients, Dokl.. On the solutions in generalized functions of ordinary differential [r]

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Power series solution is a powerful and easiest tool to solve complicated **nonlinear** **coupled** **differential** **equations**. In this paper exact solutions for **nonlinear** **coupled** **differential** **equations** of Boussinesq system are obtained by power series method and applied Pade to find singularities and show the convergence. Fig 1 and 2 depict exact power series solution up to 25 terms and fig 3 and 4 are the graphs by numerical method which shows exact matching. In fig 5, 7 and 9 Singularities are identified by using Pade approximations for u graph. In fig 6, 8 and 10 we have shown that again same singularities are seen for H graph. In our future work we would like analyse these solutions about the singular points.

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