For these two equations, we have a = –1, δ = 2 and a = 0, δ = 2, respectively. Unfortu- nately, when considering the one-dimensional wave and Schrödinger equations with pe- riodic **boundary** **conditions**, the multiplicity of the eigenvalues becomes an obstacle. In the 1990s, the people even doubted that KAM could not handle multiple normal fre- quency case. To overcome this diﬃculty, using the Nash–Moser iteration and Lyapunov– Schmidt decomposition, Craig and Wayne [10, 11] gave a good estimation of the inverse of an inﬁnite-dimensional matrix with small eigenvalues, and thus the quasi-**periodic** solu- tions were successfully constructed to the equations with **periodic** **boundary** **conditions**. Further developing Craig and Wayne’s method, Bourgain [5–8] proved the existence of quasi-**periodic** solutions to higher-dimensional wave and Schrödinger equations. We need to emphasize that the methods of Craig, Wayne, and Bourgain are very eﬀective methods to prove the existence of quasi-**periodic** solutions but do not get linear stability, so their results are weaker than those obtained by KAM method. On the other hand, You [35] es- tablished a KAM theorem for lower-dimensional tori in the ﬁnite-dimensional case and applied it to multiple normal frequency case. In 2000, Chierchia and You [9] proved an inﬁnite-dimensional theorem that can deal with multiple normal frequency case. For more detail, we refer the readers to [22, 32, 33, 36, 37] and the references therein.

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The stochastic Burgers equation with **periodic** **boundary** **conditions** has already been studied numerically by numerous authors, including its long-time statistics (usually from a physical more than mathematical point of view). Let us mention [11, 12], where some interesting physical motivations are explained. The numerical scheme which we propose here does not seem to have been used yet, in particular for large time simulations.

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Up to this point, nothing has changed since the original problem except the formulation. To proceed, we now assume separation of scales, i.e. that the subscale feature has a length scale much smaller than that of the macroscale. Furthermore, we also make the assump- tion that v and p S are **periodic** over, and continuous inside, each F , thus replacing the condition on continuity over the boundaries F . As an intermediate step, we note that by removing continuity over F , reaction forces arise, which eventually will contribute to the subsequent macrohomogeneity condition. In [3] it is shown that **periodic** **boundary** **conditions** satisfy the aforementioned condition. In order to impose periodicity (either in weak or strong form), we start out by following along the lines of [3] and split the sub- scale **boundary** into two parts; = + ∪ − where the + / − sign is the sign of the normal to that part of the **boundary** c . Furthermore, we introduce the jump operator

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We have presented an up-to-date review of the results on **boundary** value problems of nonlinear fractional-order diﬀerential equations, inclusions and coupled systems supple- mented with a variety of anti-**periodic** (and anti-**periodic** type) **boundary** **conditions**. In Section , we have given some basic deﬁnitions of fractional calculus and model equa- tions involving fractional-order derivatives. In Section , we have collected a variety of results on classical anti-**periodic** **boundary** value problems of nonlinear fractional diﬀer- ential equations, inclusions and impulsive equations. The concept of parametric type anti- **periodic** **boundary** **conditions** is also outlined. The relationship between the Green’s func- tions of lower- and higher-order anti-**periodic** fractional **boundary** value problems is also described. Some new results related to further generalization of classical anti-**periodic** problems are discussed in detail and illustrated with examples. Section contains some recent results on **boundary** value problems of Liouville-Caputo (Caputo) type sequential fractional diﬀerential equations supplemented with anti-**periodic** type (non-separated) two-point and nonlocal multipoint **boundary** **conditions**. In Section , some existence results for a new kind of **boundary** value problem of coupled Caputo type fractional dif- ferential equations equipped with non-separated coupled **boundary** **conditions** are given. Some results involving fractional order anti-**periodic** **boundary** **conditions** are elaborated in Section . We recall that anti-**periodic** **boundary** **conditions** appear in numerous situ- ations such as interpolation problems, anti-**periodic** wavelets, mathematical problems of ordinary, partial and impulsive diﬀerential equations, problems in physics, etc. Keeping in view the importance of anti-**periodic** type **boundary** value problems occurring in several disciplines, the present survey provides a detailed description of the work on the topic completed over a period of the last decade and may serve as a platform for the researchers who are interested in exploring more and more insights in this topic.

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In this paper, we study the existence of classical solutions for a second-order impulsive diﬀerential equation with non-separated **periodic** **boundary** **conditions**. By using the variational method and critical point theory, we give some new criteria to guarantee that the impulsive problem has at least one solution under some diﬀerent **conditions**. Our results extend and improve some recent results.

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Abstract. We study the stabilization of global solutions of the linear Kawa- hara equation (K) with **periodic** **boundary** **conditions** under the effect of a localized damping mechanism. The Kawahara equation is a model for small amplitude long waves. Using separation of variables, the Ingham inequality, multiplier techniques and compactness arguments we prove the exponential decay of the solutions of the (K) model.

Theory of fractional diﬀerential equations has important application in many areas. It has become a new research ﬁeld in diﬀerential equations [1–3]. There are a lot of good research results on **boundary** value problems of fractional diﬀerential equations [4–24]. Recently fractional Langevin equations have been studied by some scholars (see, for ex- ample, [25–27]).

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Free convection heat transfer from inclined wavy surface has received attention because of its vast applications. Some of these applications include ground water flows, oil recovery processes, thermal insulation engineering food processing etc. Extensive literature on the topic is availed for porous media, Slimi et. al. 1998[1] studied two – dimensional and transient fluid flow and heat transfer by natural convection in a vertical cylinder opened at both ends filled with a saturated porous medium and heated with a uniform lateral heat flux. The study was carried out using the forchheimer – extended Darcy flow model. Taofik et.al. 1999[2] studied unsteady natural convection which occurs in a vertical cylindrical enclosure opened at both ends, filled with a fluid saturated porous medium with a **periodic** lateral heat flux density. The study was carried out by the use of the Darcy law and it takes in to account heat conduction in the wall. The set of equations was solved numerically by the standard finite volume method. Saaed 2000[3] proposed a simple numerical expression for average Nusselt numbers over isothermal horizontal cylinder for all Rayleigh by using Darcy flow model. And also Khalid Abd.Al-hussein 2001[4] obtained a simple relation for Nusselt number which is a strong function of modified Rayleigh number, time, radius ratio, and aspect ratio. AL-Najar 2004[5] used the finite difference method to investigate the steady free convection from a two separated horizontal cylinders embedded in saturated porous media bounded by rectangular cavity. The cylinders kept isothermally hot while the bounded cavity is isothermally. It found that the large heat gained to the cavity been at the upper horizontal wall above cylinders. Saleh 2008[6] studied numerically unsteady natural convection heat transfer through a fluid–saturated porous media in inclined pipe enclosure. The temperature at cylindrical sidewall Tw was

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INTRODUCTION Fourth order boundary value problems arise in the study of the equilibrium of an.. elastic beam under an external load, e.g., see.[r]

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P., Integral type asymptotic conditions for the solvability of a periodic fourth order boundary value problem, in Differential Equations, ed.. Aftabizadeh, Ohio University to appear..[r]

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and (2) if β = −1, α = −1 and k = 1, then they are **periodic** **boundary** **conditions**; if β = 1, α = 1 and k = 2 then they are antiperiodic **boundary** **conditions**. These bound- ary **conditions** are regular but not strongly regular. Note that the **boundary** **conditions** are strongly regular if and only if all large eigenvalues are far from each other [9]. This easify to investigate the perturbation theory and Riesz basis property. If the **boundary** **conditions** are not strongly regular then the eigenvalues are pairwise very close to each other. This situation complicates the investigation of the perturbation theory. Therefore the regular cases which are not strongly regular are still investigated. Only the special cases, the **periodic** and antiperiodic problems, were investigated in detail. In [8] we ob- tained the asymptotic formulas for the large eigenvalues by the asymptotic methods, but

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3.17 cm when **periodic** **boundary** condition is applied. But in the second twelve hours, PCM will be frozen. It is obvious from Figures 4 and 6 that the interface length is the same in times having the same distance from t = 12 hours. It can be found that in **periodic** **boundary** **conditions**, melting and freezing process in PCM happen at the same interface exactly. Also, temperature profiles are completely symmetry in freezing and melting process. This confirms that the results are obtained under **periodic** **boundary** condition. The inverse of melting process, the freezing rate in the second six hours is greater than first six hours.

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4 B. Ahmad and V. Otero-Espinar, “Existence of solutions for fractional diﬀerential inclusions with anti- **periodic** **boundary** **conditions**,” **Boundary** Value Problems, vol. 2009, Article ID 625347, 11 pages, 2009. 5 B. Ahmad and J. J. Nieto, “Existence and approximation of solutions for a class of nonlinear impulsive functional diﬀerential equations with anti-**periodic** **boundary** **conditions**,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 10, pp. 3291–3298, 2008.

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Chebyshev polynomials are used here for the spectral representation in the GWRM. These have several desir- able qualities. They converge rapidly to the approxi- mated function, they are real and can easily be converted to ordinary polynomials and vice versa, their minimax property guarantees that they are the most economical polynomial representation, they can be used for non- **periodic** **boundary** **conditions** (being problematic for Fourier representations) and they are particularly apt for representing **boundary** layers where their extrema are locally dense [13,14]. The GWRM is fully spectral; all calculations are carried out in spectral space. The pow- erful minimax property of the Chebyshev polynomials [14] implies that the most accurate n-m order approxima- tion to an nth order approximation of a function is a sim- ple truncation of the terms of order > n-m. Thus nonlin- ear products are easily and efficiently computed in spec- tral space. Since the GWRM efficiently uses rapidly convergent Chebyshev polynomial representation for all time, space and parametrical dimensions, pseudospectral implementation [13] has so far not been pursued. The GWRM eliminates time stepping and associated, limiting grid causality **conditions** such as the CFL condition.

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This paper is organized as follows. In Sect. 2, we establish an existence theorem of solu- tions for PBVP (1) under nonlinear growth restriction of f . The key is an analytic technique from the theory of coincidence degree. In Sect. 3, we obtain the existence of positive so- lutions of (3) by Theorem 3. Two illustrative examples of nonlinear fractional problems with **periodic** **boundary** **conditions** are shown in Sect. 4.

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momenta are injected on the lattice using non-**periodic** **boundary** **conditions** [4, 5] and the matrix elements of the tensor current are determined for many kinematical **conditions**, in which parent and child mesons are either moving or at rest. The data exhibit a remarkable breaking of Lorentz symmetry due to hypercubic e ff ects for both D → π and D → K form factors. The presence of these e ff ects has already been observed in Ref. [1] for the vector and scalar form factors, and in that paper we presented a method to subtract the hypercubic artefacts and recover the Lorentz-invariant form factors in the continuum limit. Apart from Ref. [1], hypercubic e ff ects have never been observed in the context of the D → π(K) transitions. Previous lattice calculations, however, used only a limited number of kinematical **conditions** (typically the D-meson at rest). We argue that this may obscure the presence of hypercubic effects in the lattice data. These effects appear to be affected by the difference between the parent and the child meson masses. This is clearly a very important issue, which warrants further investigations. If this is the case, these effects will play an important role in the determination of the form factors governing semileptonic B-meson decays into lighter mesons and is therefore crucial to have them under control.

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Solutions of Hamiltonian systems are very important in applications. In recent years, the existence and multiplicity of solutions for Hamiltonian systems via critical point theory have been studied by many authors (see [2, 5–10, 12–22]). In particular, by means of criti- cal point theory, the least action principle, and the minimax method, the existence and multiplicity of **periodic** solutions for second-order Hamiltonian systems with **periodic** **boundary** **conditions** were extensively studied in the cases where the gradient of the non- linearity is bounded sublinearly and linearly, and many interesting results are given in [5, 9, 10, 13–19, 22]. In this paper, we discuss the existence and multiplicity of solutions for the following second-order Hamiltonian systems satisfying generalized **periodic** bound- ary value **conditions**:

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The monotonicity assumptions imply the comparison principle and then the uniqueness of **periodic** solution 6 and references therein and the continuous dependence with respect to the data 12 and references therein. Nonmonotone assumptions, especially on the zero-order term fu, originate multiplicity of solutions 25, 38, 39 and references therein. Sometimes the method of super and subsolution can be applied by passing through an auxiliary monotone framework and applying some iterating arguments 34, 40, 41, and references therein. This applies also to the case in which fu can be singular 42.

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