Bu çalışmada hem sınırsız hem de sınırlı bir operatör içeren bir diferansiyel operatörün özdeğerlerinin sayısı için asimtotik formüller elde edilecektir.

In this work we investigate the resolvent operator and completeness of eigenfunctions of a Sturm–Liouville problem with discontinuities at two points. The problem contains an eigenparameter in one of the boundary conditions. For operator-theoretic formulation of the considered problem we define an equivalent inner product in the Hilbert space \( L_{2} [ - 1,1] \oplus C \) and suitable self-adjoint linear operator in it.

In this work we obtain the regularized trace formula for an even-order differential operator with unbounded operator coefficient.

In this work, with the aim of determining Green’s solution or generalized Green’s solution, we propose a novel constructive approach by which a linear or specific nonlinear problem involving general linear nonlocal condition for a first-order functional ordinary integro-differential equation with general nonsmooth coefficients satisfying some general properties such as p-integrability and boundedness is reduced to an integral equation. A system of two integro-algebraic equations, called “the adjoint system,” is constructed for this problem. Green’s functional for the problem with trivial kernel and generalized Green’s functional for the problem with nontrivial kernel are the unique solutions to the specific cases of this adjoint system. Green’s functional and generalized Green’s functional have two components. Their first components correspond to Green’s function and generalized Green’s function for the problem, respectively. Some illustrative applications are provided with known and unknown results.

In the present paper, we investigate some interesting properties including several special polynomials arising from Caputo-fractional derivative. From our investigation, we derive a lot of interesting identities of several special polynomials.

We consider the following boundary-value problem of nonlinear fractional differential equation with p-Laplacian operator D0+β(ϕp(D0+αu(t)))+a(t)f(u)=0, 01, ϕp-1=ϕq, 1/p+1/q=1,0⩽γ<1, 0⩽h⩽1, λ,μ>0 are parameters, a:(0,1)→[0,+∞), and f:[0,+∞)→[0,+∞) are continuous. By the properties of Green function and Schauder fixed point theorem, several existence and nonexistence results for positive solutions, in terms of the parameters λ and μ are obtained. The uniqueness of positive solution on the parameters λ and μ is also studied. In the final section of this paper, we derive not only new but also interesting identities related special polynomials by which Caputo fractional derivative.

It is generally accepted that slip boundary conditions that are higher-order in Knudsen number can extend the validity of slip models beyond the conventionally-accepted upper limit of Kn = 0.1. The present paper investigates oscillatory rarefied flows over nonplanar surfaces and demonstrates that the effects of the Knudsen layer cannot explain why slip models perform poorly for nonplanar flow cases in comparison to the equivalent planar case; nor the presence of Knudsen layers can explain why slip models show larger discrepancies over convex surfaces. The paper highlights the dramatic effects of curvature; and the profound influence of the S-layer in microscale.

In this article, we consider the boundary-value problem of nonlinear fractional differential equation with p-Laplacian operator. By the properties of Green function and Schauder fixed point theorem, several existence and nonexistence results for positive solutions, in terms of two parameters are obtained. The uniqueness of positive solution on these parameters is also studied.

The breakdown of the no-slip boundary condition at a fluid–solid interface has been recognized in micro/nanofluidics for many years. However, the relationship between the curvature of the surface and the degree of boundary slip has not been understood sufficiently well. The present study reveals that the degree of slip depends effectively on the surface curvature, which is having an opposing effect over rotating concave and convex surfaces. The results show that as surface curvature increases, the boundary slip becomes negligible over a concave surface, while it becomes increasingly important over a convex surface. In addition, boundary slip formulae are proposed that can accurately predict the boundary slips over convex and concave surfaces. These formulae are found to be in very good agreement with DSMC data for a range of accommodation coefficients and boundary curvatures. The present study then explains the mechanism of the intriguing phenomenon of velocity inversion which has, until the present study, often been mistakenly attributed solely to the effects of boundary curvature.

In this work, by Green’s functional concept, in order to obtain Green’ssolution we concentrate on a new constructive technique by which alinear completely nonhomogeneous nonlocal problem for a second-orderloaded differential equation with generally variable coefficients satisfying some general properties such as p-integrability and boundedness istransformed into one and only one integral equation. A system of threeintegro-algebraic equations called the special adjoint system is obtainedfor this problem. A solution of this special adjoint system is Green’sfunctional which enables us to determine Green’s function and Green’ssolution for the problem. Two illustrative applications are provided.

In this work, we investigate a sequence of approximations converging to the existing unique solution of a multi-point boundary value problem(BVP) given by a linear fourth-order ordinary differential equation with variable coeffcients involving nonlocal integral conditions by using reproducing kernel method(RKM). Obtaining the reproducing kernel of the reproducing kernel space by using the original conditions given directly by RKM may be troublesome and may introduce computational costs. Therefore, in these cases, initially considering more admissible conditions which will allow the reproducing kernel to be computed more easily than the original ones and then taking into account the original conditions lead us to satisfactory results. This analysis is illustrated by a numerical example. The results demonstrate that the method is still quite accurate and effective for the cases with both derivative and integral conditions even if the accuracy is less compared to the cases with just derivative conditions.

In this work, we investigate a linear completely nonhomogeneous nonlocal multipoint problem for an m-order ordinary differential equation with generally variable nonsmooth coefficients satisfying some general properties such as p-integrability and boundedness. A system of m + 1 integro-algebraic equations called the special adjoint system is constructed for this problem. Green's functional is a solution of this special adjoint system. Its first component corresponds to Green's function for the problem. The other components correspond to the unit effects of the conditions. A solution to the problem is an integral representation which is based on using this new Green's functional. Some illustrative implementations and comparisons are provided with some known results in order to demonstrate the advantages of the proposed approach.

Although many gas-phase microfluidic devices contain curved surfaces, relatively little research has been conducted on the degree of slip over nonplanar surfaces. The present study demonstrates the influence of the surface shape (i.e., convex/concave) on the velocity slip and formation of the Knudsen layer. In addition, the study reveals that there is a simple relationship between the shear stress exerted on the surface and the velocity defect in the Knudsen layer.

In this work we investigate the resolvent operator and completeness of eigenfunctions of a Sturm-Liouville problem with discontinuities at two points. The problem contains an eigenparameter in the one of boundary conditions. For operator-theoretic formulation of the considered problem we define an equivalent inner product in the Hilbert space and suitable self-adjoint lineer operator in it.

In this work, we present a new constructive technique which is based on Green's functional concept. According to this technique, a linear completely nonhomogeneous nonlocal problem for a second-order ordinary differential equation is reduced to one and only one integral equation in order to identify the Green's solution. The coefficients of the equation are assumed to be generally variable nonsmooth functions satisfying some general properties such as p-integrability and boundedness. A system of three integro-algebraic equations called the special adjoint system is obtained for this problem. A solution of this special adjoint system is Green's functional which enables us to determine the Green's function and the Green's solution for the problem. Some illustrative applications and comparisons are provided with some known results.

Although many microfluidic devices contain curved surfaces, relatively little research has been conducted on nonplanar gas-phase microflows. In this study, the existence of the Knudsen layer over a concave surface has been investigated using a generic nonplanar second-order velocity-slip boundary condition and the direct simulation Monte Carlo (DSMC) method.

Collaboration in science is one of the key components of world-class research. The European Commission supports collaboration between institutions and funds young researchers appointed by these partner institutions. In these networks, the mobility of the researchers is enforced in order to enhance the collaboration. In this study, based on a real Marie Curie Initial Training Network, an algorithm to construct a collaboration network is investigated. The algorithm suggests that a strongly efficient expansion leads to a star-like network. The results might help the design of efficient collaboration networks for future Initial Training Network proposals.

This work investigates the approximate solution for fourth‐order multi‐point boundary value problem represented by linear integro‐differential equation involving nonlocal integral boundary conditions by using the reproducing kernel method (RKM). The investigated solution is represented in the form of a series with easily computable components in the reproducing kernel space. When the used algorithm for approximation is applied directly for the given original conditions, it can be very troublesome to compute the reproducing kernel of space. Therefore firstly, it is considered more appropriate conditions to be computed the kernel easily than original ones. Nextly, the original conditions are taken into account. Analysis is illustrated by a numerical example. The results demonstrate that the method is quite accurate and effective.