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We are concerned with the existence of homoclinic solutions for the nonlinear problems P u ′ ′ + ω u ′ - k u = f (t , u , u ′) , t ∈ R , lim | t | → + ∞ u (t) = 0 , where ω ∈ R , k > 0 are real constants, and f : R 3 → R is an L 1 - Carathéodory function. Under some suitable conditions, the existence of homoclinic solutions for problem (P) and the corresponding coupled systems are provided. The proofs of the main results are based on the method of upper and lower solutions. [ABSTRACT FROM AUTHOR]

Ordinary nonlinear differential equations with classical and fractional derivatives are used to simulate several real-world problems. Nonetheless, numerical approaches are used to acquire their solutions. While various have been proposed, they are susceptible to both disadvantages and advantages. In this paper, we propose a more accurate numerical system for solving nonlinear differential equations with classical and Caputo–Fabrizio derivatives by combining two concepts: the parametrized method and the predictor–corrector method. We gave theoretical analyses to demonstrate the method’s correctness, as well as several illustrated examples for both scenarios. [ABSTRACT FROM AUTHOR]

Tuberculosis (TB) has a long history as a serious disease induced by its causative agent Mycobacterium tuberculosis. This pathogen manipulates the host's immune response, thereby stimulating inflammatory processes, which leads to an even greater imbalance of specific enzymes/inhibitors that contribute to tissue destruction. This work addresses a model consisting of two ordinary differential equations obtained by reducing a previously developed large-scale model describing lung damage, taking into account key metabolic pathways controlled by bacteria. The resulting system is explored as a dynamical system simulating the interaction between bio-markers (matrix metalloproteinases) of tissue destruction and the pathogen. In addition to the analysis of the mathematical model's features, we qualitatively compared the model dynamics with real clinical data and discussed their mutual correspondence. [ABSTRACT FROM AUTHOR]

The heart is an organ with a limited capacity for regeneration and repair. In this paper, a new mathematical model is presented to study the left ventricular remodeling after myocardial infarction (MI) and followed stem cell therapeutic effort. The model represents the post-MI regeneration process of cardiomyocytes under stem cell therapy with oxygen restoration. The resulting system of nonlinear ordinary differential equations (ODE) is studied numerically in order to demonstrate the functionality and performance of the new model. The optimal time of stem cell injection for various oxygen restorations is determined. Moreover, the regeneration of cardiomyocytes is successfully correlated with improved left ventricle function observed in experiments. The proposed nonlinear ODE model is able to capture the complicated biological interactions in post-MI remodeling and can serve as a platform for in silico simulation and perturbation to optimize MI stem cell therapy. [ABSTRACT FROM AUTHOR]

Based on the Tao-Mason equation of state we have proposed a nonlinear ordinary differential equation that asymptotically converges to the compressibility factor of a pure substance or a mixture of chemical species. We have used the Dormand-Prince pair algorithm to solve the aforementioned differential equation in a purely numerical manner. Our method is devoid of the adverse convergence issues that are usually associated with the Newton-type solvers. We have provided two case studies concerning two industrially common compounds namely ethane and carbon dioxide, for the sake of exposition. For 96 points of different temperatures and pressures, our method succeeded at calculating the compressibility factor of carbon dioxide with an average absolute error of 6.53×10-5 and a maximum absolute error of 4.79×10-4. Unlike the previous root finding algorithms, we only need to perform “formal” polynomial deflations in our method, which circumvents the computation-intensive synthetic divisions, to obtain all compressibility factors offered by the Tao-Mason EOS.

We have developed a new procedure for converting nonlinear third-order ordinary differential equations into linear forms using point transformation. These linear equations are more general and easier to solve. By applying the point transformation again, we can obtain the general solutions for the original nonlinear equations. The key feature of this work is the use of examples to demonstrate the application of the derived linearization criteria to various existing problems, including third-order ordinary differential equations, secondorder ordinary differential equations under the Riccati transformation and third-order partial differential equations under travelling wave solutions. [ABSTRACT FROM AUTHOR]

Based on the Tao-Mason equation of state, we have proposed a nonlinear ordinary differential equation that asymptotically converges to the compressibility factor of a pure substance or a mixture of chemical species. We have used the Dormand-Prince pair algorithm to solve the aforementioned differential equation in a purely numerical manner. Our method is devoid of the adverse convergence issues that are usually associated with Newton-type solvers. We have provided two case studies concerning two industrially common compounds namely ethane and carbon dioxide, for the sake of exposition. For 96 points of different temperatures and pressures, our method succeeded at calculating the compressibility factor of carbon dioxide with an average absolute error of 6.53Ã--10-5 and a maximum absolute error of 4.79Ã--10-4. Unlike the previous root-finding algorithms, we only need to perform "formal" polynomial deflations in our method, which circumvents the computation-intensive synthetic divisions, to obtain all compressibility factors offered by the Tao-Mason EOS. [ABSTRACT FROM AUTHOR]

This paper introduces a new numerical solution based on the least squares support vector machine (LS-SVR) for solving nonlinear ordinary differential equations of high dimensionality. We apply the quasilinearization method to linearize the magnetohydrodynamic (MHD) ow of nanouid around a stretching cylinder, thereby transforming it into a linear problem. We then utilize LS-SVR with fractional Hermite functions as basis functions to solve this problem over a semi-infinite interval. Our numerical results confirm the effectiveness of this approach. [ABSTRACT FROM AUTHOR]

We investigate the existence of self-similar solutions for the parabolic equation u t = Δ u + u m H u , with 0 ≤ m < 1 and H the Heaviside graph, coupled with the initial datum u x , 0 = − c x 2 1 1 − m , with c > 0. We analyze two cases: the problem in R n , n > 1 , with m = 0 and the problem in R when 0 ≤ m < 1. In the first case we extend the result of Gianni and Hulshof (1992) and show that there exist only two self-similar solutions changing sign, provided 0 < c < c c r , with c c r obtained solving a specific algebraic equation depending on n. In the second case we prove that there exist at least two self-similar solutions of problem u t = u x x + u m H u , u x , 0 = − c x 2 1 1 − m , changing sign and evolving region where u > 0. These solutions are of great interest. Indeed, on one hand they prove that the problem does not admit uniqueness and on the other they prove that a single point where u x , 0 = 0 , for an initial datum which is otherwise negative, can generate a region where u x , t is positive. [ABSTRACT FROM AUTHOR]

In this article, we present the Laguerre wavelet exact Parseval frame method (LWPM) for the two-dimensional flow of a rotating micropolar fluid in a porous channel with huge mass transfer. This flow is governed by highly nonlinear coupled partial differential equations (PDEs) are reduced to the nonlinear coupled ordinary differential equations (ODEs) using Berman's similarity transformation before being solved numerically by a Laguerre wavelet exact Parseval frame method. We also compared this work with the other methods in the literature available. Moreover, in the graphs of the velocity distribution and microrotation, we shown that the proposed scheme's solutions are more accurate and applicable than other existing methods in the literature. Numerical results explaining the effects of various physical parameters connected with the flow are discussed.

This paper provides the Laguerre wavelet method (LWM) for the 2-dimensional flow of a rotating micropolar fluid in a permeable channel with high mass transfer. The governing coupled nonlinear partial differential equations (PDEs) are reduced to the coupled nonlinear ordinary differential equations (ODEs) using Berman's similarity transformation before being solved numerically by a Laguerre wavelet method. We also compare this work with the Optimal Homotopy Asymptotic method (OHAM) and the Runge-Kutta method. Moreover, in the graphs of the F(η), F′(η) and G(η), we show that solutions obtained by the future method are more precise and applicable than other available methods in the literature. Numerical outcomes illuminating the properties of different physical parameters connected with the flow are discussed. [ABSTRACT FROM AUTHOR]

In this work, an approximate analytic solution which is called a semi-analytical solution to the pendulum equation is obtained. Moreover, the semi-analytical solution is compared to the approximate numerical solution obtained with the aid of Runge-Kutta method. [ABSTRACT FROM AUTHOR]

In this paper, the boundary layer flow of an Eyring-Powell non-Newtonian fluid over a linearly stretching sheet is solved using the combination of the quasilinearization method and the Fractional order of Rational Chebyshev function (FRC) collocation method on a semi-infinite domain. The quasilinearization method converts the equation into a sequence of linear equations then, using the FRC collocation method, these linear equations are solved. The governing nonlinear partial differential equations are reduced to the nonlinear ordinary differential equation by similarity transformations. The physical significance of the various parameters of the velocity profile is investigated through graphical figures. An accurate approximation solution is obtained and the convergence of numerical results is shown.

In the this paper, a new modified method is proposed for solving linear and nonlinear Lane-Emden type equations using first kind Chebyshev operational matrix of differentiation. The properties of first kind Chebyshev polynomial and their shifted polynomial are first presented. These properties together with the operation matrix of differentiation of first kind Chebyshev polynomial are utilized to obtain numerical solutions of a class of linear and nonlinear LaneEmden type singular initial value problems (IVPs). The absolute error of this method is graphically presented. The proposed framework is different from other numerical methods and can be used in differential equations of the same type. Several examples are illuminated to reveal the accuracy and validity of the proposed method.

In this research work, numerical time perturbation methods are applied on nonlinear ODE. Solutions are sought in the form of power series using time as the perturbation parameter. This time integration approach with continuation procedures allows to obtain analytical continuous approximated solutions. Asymptotic Numerical Method and new resummations techniques of divergent series namely Borel–Padé–Laplace and Inverse Factorial series are studied. A comparison with classic integration scheme is presented in order to evaluate the robustness and the effectiveness of these algorithms. Full details are given regarding first- and second-order derivative of resummation techniques. [ABSTRACT FROM AUTHOR]

In this paper, we consider continuous‐time Markov chains with a finite state space under nonlinear expectations. We define so‐called Q‐operators as an extension of Q‐matrices or rate matrices to a nonlinear setup, where the nonlinearity is due to model uncertainty. The main result gives a full characterization of convex Q‐operators in terms of a positive maximum principle, a dual representation by means of Q‐matrices, time‐homogeneous Markov chains under convex expectations, and a class of nonlinear ordinary differential equations. This extends a classical characterization of generators of Markov chains to the case of model uncertainty in the generator. We further derive an explicit primal and dual representation of convex semigroups arising from Markov chains under convex expectations via the Fenchel–Legendre transformation of the generator. We illustrate the results with several numerical examples, where we compute price bounds for European contingent claims under model uncertainty in terms of the rate matrix. [ABSTRACT FROM AUTHOR]

The paper presents an analytical solution for the centric viscoelastic impact of two smooth balls. The contact period has two phases, compression and restitution, delimited by the moment corresponding to maximum deformation. The motion of the system is described by a nonlinear Hunt–Crossley equation that, when compared to the linear model, presents the advantage of a hysteresis loop closing in origin. There is only a single available equation obtained from the theorem of momentum. In order to solve the problem, in the literature, there are accepted different supplementary hypotheses based on energy considerations. In the present paper, the differential equation is written under a convenient form; it is shown that it can be integrated and a first integral is found—this being the main asset of the work. Then, all impact parameters can be calculated. The effect of coefficient of restitution upon all collision characteristics is emphasized, presenting importance for the compliant materials, in the domain of small coefficients of restitution. The results (variations of approach, velocity, force vs. time and hysteresis loop) are compared to two models due to Lankarani and Flores. For quasi-elastic collisions, the results are practically the same for the three models. For smaller values of the coefficient of restitution, the results of the present paper are in good agreement only to the Flores model. The simplified algorithm for the calculus of viscoelastic impact parameters is also presented. This algorithm avoids the large calculus volume required by solving the transcendental equations and definite integrals present in the mathematical model. The method proposed, based on the viscoelastic model given by Hunt and Crossley, can be extended to the elasto–visco–plastic nonlinear impact model.

In this paper, two efficient computational algorithms based on Rational and Exponential Bessel (RB and EB) functions are compared to solve several well-known classes of nonlinear Lane-Emden type models. The problems, which are define in some models of non-Newtonian fluid mechanics and mathematical physics, are nonlinear ordinary differential equations of second-order over the semi-infinite interval and have a singularity at x = 0. The nonlinear Lane-Emden equations are converted to a sequence of linear differential equations by utilizing the quasilinearization method (QLM), and then these linear equations are solved by RB and EB collocation methods. Afterwards, the obtained results are compared with the solution of other methods to demonstrate the efficiency and applicability of the proposed methods. [ABSTRACT FROM AUTHOR]

In this research, a hybrid numerical method combining cosine and sine (CAS) wavelets and Green's function approach is created to acquire the arrangements of fractional Lane–Emden Problem. The suggested methodology for detecting the solution of nonlinear equations dependent on variations of germinal algorithms is applied on nonlinear fractional Lane–Emden Problem under some smooth conditions and results in an iterative scheme of nonlinear equations Because of its efficiency, this technique is applied on a large variety of equations from the boundary value problems to the optimization. This paper is extending the suggested methodology technique for fractional Lane–Emden Problem. Moreover, the feature of the present novel method is utilized to convert the problem under observance into a system of algebraic equations that can be illuminated by suitable algorithms. A rapprochement of results has likewise been obtained using the present strategy and those reported using other techniques seem to indicate the precision and computational efficiency to establish the suitability of the Green-CAS wavelet method. [ABSTRACT FROM AUTHOR]

In this article, we introduce a fractional order of rational Bessel functions collocation method (FRBC) for solving the Thomas-Fermi equation. The problem is defined in the semi-infinite domain and has a singularity at $x = 0$ and its boundary condition occurs at infinity. We solve the problem on the semi-infinite domain without any domain truncation or transformation of the domain of the problem to a finite domain. This approach at first, obtains a sequence of linear differential equations by using the quasilinearization method (QLM), then at each iteration the equation is solves by FRBC method. To illustrate the reliability of this work, we compare the numerical results of the present method with some well-known results, to show that the new method is accurate, efficient and applicable.

Highlights • The heat transfer flow of a Walters’ liquid B in a porous medium with Navier slip is investigated. • The physical problem is modelled into nonlinear PDE’s to ODE’s via similarity transformation. • The analytical solutions derived for velocity and temperature fields by Kummer’s function. • Temperature profiles are analyzed for two type of boundary heating processes namely PST and PHF. • The temperature distribution inside the stretching sheet is found to be controlled by Pr, N R and N I. Abstract This paper investigates the steady-state momentum and radiation heat transfer flow of a viscoelastic fluid in a porous media in the presence of a linear Navier slip boundary condition. The velocity of the fluid over the linear stretching sheet is varied linearly with the axial distance while a Walters’ liquid-B model is assumed for the viscosity. A similarity transformation reduces the Navier–Stokes equations to a set of partial differential equations that are converted into ordinary differential equations and solved analytically for the velocity. Moreover, heat is balanced between a temperature dependent heat source and radiation and leads to a differential equation with variable coefficients. The temperature equation is transformed to a confluent hypergeometric differential equation using the Rosseland approximation for the radiation and solved analytically. Results are discussed for two boundary conditions of the sheet, the prescribed surface temperature and the wall heat flux. Parameters like the Reynolds number, the viscoelastic parameter and the boundary slip parameter are found to determine the flow field. In addition, the Prandtl number, the radiation number, the wall temperature and the heat source/sink parameters are found to control the temperature distribution inside the stretching sheet. [ABSTRACT FROM AUTHOR]

This article deals with a class of three-point nonlinear boundary-value problems (BVPs) with Neumann type boundary conditions which arises in bridge design. The source term (nonlinear term) depends on the derivative of the solution, it is also Lipschitz continuous. We use monotone iterative technique in the presence of upper and lower solutions for both well order and reverse order case. Under some sufficient conditions we prove existence results. We also construct two examples to validate our results. These result can be used to generate a user friendly package in Mathematica or MATLAB so that solutions of nonlinear boundary-value problems can be computed.

In this paper, the nonlinear singular Thomas-Fermi differential equation on a semi-infinite domain for neutral atoms is solved by using the generalized fractional order of the Chebyshev orthogonal functions (GFCFs) of the first kind. First, this collocation method reduces the solution of this problem to the solution of a system of nonlinear algebraic equations. Second, using solve a system of nonlinear equations, the initial value for the unknown parameter L is calculated, and finally, the value of L to increase the accuracy of the initial slope is improved and the value of y'(0) = -1.588071022611375312718684509 is calculated. The comparison with some numerical solutions shows that the present solution is highly accurate. [ABSTRACT FROM AUTHOR]

The present study is an attempt to find a solution for Volterra's Population Model by utilizing Spectral methods based on Rational Christov functions. Volterra's model is a nonlinear integro-differential equation. First, the Volterra's Population Model is converted to a nonlinear ordinary differential equation (ODE), then researchers solve this equation (ODE).The accuracy of method is tested in terms of RES error and compare the obtained results with some well-known results.The numerical results obtained show that the proposed method produces a convergent solution. [ABSTRACT FROM AUTHOR]

In this paper, the nonlinear singular Thomas–Fermi differential equation for neutral atoms is solved using the fractional order of rational Chebyshev orthogonal functions (FRCs) of the first kind, F T n α ( t , L ) , on a semi-infinite domain, where L is an arbitrary numerical parameter. First, using the quasilinearization method, the equation be converted into a sequence of linear ordinary differential equations (LDEs), and then these LDEs are solved using the FRCs collocation method. Using 300 collocation points, we have obtained a very good approximation solution and the value of the initial slope y ′ ( 0 ) = − 1.5880710226113753127186845094239501095 , highly accurate to 37 decimal places. [ABSTRACT FROM AUTHOR]

In this article, we introduce a fractional order of rational Bessel functions collocation method (FRBC) for solving the Thomas-Fermi equation. The problem is defined in the semi-infinite domain and has a singularity at x = 0 and its boundary condition occurs at infinity. We solve the problem on the semi-infinite domain without any domain truncation or transformation of the domain of the problem to a finite domain. This approach at first, obtains a sequence of linear differential equations by using the quasilinearization method (QLM), then at each iteration the equation is solves by FRBC method. To illustrate the reliability of this work, we compare the numerical results of the present method with some well-known results, to show that the new method is accurate, efficient and applicable. [ABSTRACT FROM AUTHOR]

In this paper we review several approaches to mathematical modeling of simple battery cells and develop these ideas further with emphasis on charge recovery and the response behavior of batteries to given external load. We focus on models which use few parameters and basic battery data, rather than detailed reaction and material characteristics of a specific battery cell chemistry, starting with the coupled ODE linear dynamics of the kinetic battery model. We show that a related system of PDE with Robin type boundary conditions arises in the limiting regime of a spatial kinetic battery model, and provide a new probabilistic representation of the solution in terms of Brownian motion with drift reflected at the boundaries on both sides of a finite interval. To compare linear and nonlinear dynamics in kinetic and stochastic battery models we study Markov chains with states representing available and remaining capacities of the battery. A natural scaling limit leads to a class of nonlinear ODE, which can be solved explicitly and compared with the capacities obtained for the linear models. To indicate the potential use of the modeling we discuss briefly comparison of discharge profiles and effects on battery performance. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence of multiple positive solutions for the fourth-order boundary value problemu^(4)(t) = f(t,u(t)), 0 < t < 1,u(0) = u(1) = u''(0) = u''(1) = 0,where f : [0, 1] × [0,∞) → [0,∞) is continuous. Existence theorems are established via the theory of fixed point index in cones.

The Painlev'e equations are second order ordinary differential equations which can be grouped into six families, namely Painlev'e equation I, II,?, VI. This paper presents the series solution of second Painlevé equation via optimal homotopy asymptotic method (OHAM). This approach is highly efficient and it controls the convergence of the approximate solution. Comparison of the obtained solution via OHAM is provided with those obtained by Homotopy Perturbation Method (HPM), Adomian Decomposition Method (ADM), Sinc-collocation and Runge-Kutta 4 methods. It is revealed that there is an excellent agreement between OHAM and other published data which confirm the effectiveness of the OHAM. [ABSTRACT FROM AUTHOR]

This paper presents an analytical solution of nonlinear differential equation of micro-structures subjected to electrostatic fields. The constitutive equation of such a model is a second order differential equation (ODE). The problem is solved when the assumption of linear deflection is considered. However, deflection of micro cantilevers in practical applications is nonlinear. Moreover, the constitutive ODE is stiff and various numerical algorithms used to solve it yield non-consistent numerical solutions. A deduction order method - Lie group symmetry is employed to reduce the order of the ODE. Although the resulting first order ODE has no symmetry that would guarantee an explicit close form solution, it enables an analytical formulation for the no-damping assumption only. The restoring force term in the first order ODE reveals the pull-in voltage as expressed in classical MEMS textbooks. It is shown that the numerical solution for the second order ODE and the reduced first order ODE are same. Finding any symmetry other than translation, scaling or rotation will enable the reduction of the first order ODE and thus, the formulation of an analytical solution to this highly non-linear problem. [ABSTRACT FROM AUTHOR]

The Painlev'e equations are second order ordinary differential equations which can be grouped into six families, namely Painlev'e equation I, II,…, VI. This paper presents the series solution of second Painlevé equation via optimal homotopy asymptotic method (OHAM). This approach is highly efficient and it controls the convergence of the approximate solution. Comparison of the obtained solution via OHAM is provided with those obtained by Homotopy Perturbation Method (HPM), Adomian Decomposition Method (ADM), Sinc-collocation and Runge-Kutta 4 methods. It is revealed that there is an excellent agreement between OHAM and other published data which confirm the effectiveness of the OHAM. doi:10.14456/WJST.2015.43

We consider a system of nonlinear Maxwell-type equations in the plane, which arises in the modeling of mesoscopic scale electromagnetic phenomena. After reduction to a quasi-linear equation and symmetrization, we obtain an explicit local description of the solutions in the form of power series. Subsequently, we apply general methods to show global existence of the solutions and to analyze their oscillations and asymptotic behavior. [ABSTRACT FROM AUTHOR]

Volterra's model for population growth in a closed system includes an integral term to indicate accumulated toxicity in addition to the usual terms of the logistic equation. In this research, a new numerical algorithm is introduced for solving this model. The proposed numerical approach is based on the modified Bessel function of the first kind and the collocation method. In this method, we aim to solve the problems on the semi-infinite domain without any domain truncation, variable transformation in basis functions and shifting the problem to a finite domain. Accordingly, we employ two different collocation approaches, one by computing through Volterra's population model in the integro-differential form and the other by computing by converting this model to an ordinary differential form. These methods reduce the solution of a problem to the solution of a nonlinear system of algebraic equations. To illustrate the reliability of these methods, we compare the numerical results of the present methods with some well-known results in other to show that the new methods are efficient and applicable. [ABSTRACT FROM AUTHOR]

RS feedback models have been successful in explaining the observed phenomenon of clustering in autonomous oscillation in yeast, but current models do not include the biological reality of dynamical delay and do not have the related property of quorum sensing. Here an RS type ODE model for cell cycle feedback, including an explicit term modeling a chemical feedback mediating agent, is analyzed. New dynamics include population dependent effects: subcritical pitchfork bifurcations, and quorum sensing occur. The model suggests new experimental directions in autonomous oscillation in yeast. [ABSTRACT FROM AUTHOR]

In this paper we propose a method for solving some well-known classes of Lane-Emden type equations which are nonlinear ordinary differential equations on the semi-infinite domain. The proposed approach is based on an Unsupervised Combined Artificial Neural Networks (UCANN) method. Firstly, The trial solutions of the differential equations are written in the form of feed-forward neural networks containing adjustable parameters (the weights and biases); results are then optimized with the combined neural network. The proposed method is tested on series of Lane-Emden differential equations and the results are reported. Afterward, these results are compared with the solution of other methods demonstrating the efficiency and applicability of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we propose a Tau method for solving the singular Lane-Emden equation-a nonlinear ordinary differential equation on a semi-infinite interval. We applied collocation, Galerkin, and Tau methods for solving this problem, and according to the results, the solution of Tau method is the most accurate. The operational derivative and product matrices of the modified generalized Laguerre functions are presented. These matrices, in conjunction with the Tau method, are then utilized to reduce the solution of the Lane-Emden equation to that of a system of algebraic equations. We also present a comparison of this work with some well-known results and show that the present solution is highly accurate. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper presents a mathematical model for photo-excited carrier decay in a semiconductor. Due to the carrier trapping states and recombination centers in the bandgap, the carrier decay process is defined by the system of nonlinear differential equations. The system of nonlinear ordinary differential equations is approximated by linearized backward Euler scheme. Some a priori estimates of the discrete solution are obtained and the convergence of the linearized backward Euler method is proved. The identifiability analysis of the parameters of deep centers is performed and the fitting of the model to experimental data is done by using the genetic optimization algorithm. Results of numerical experiments are presented. [ABSTRACT FROM AUTHOR]