Kirchhoff Boundary Research Articles

A note on Kirchhoff type boundary value problem involving Riemann-Liouville fractional derivative

In this paper, we study some nonlinear Kirchhoff boundary value problems of fractional differential equations involving Riemann Liouville operator. Under appropriate assumptions on the functions in the given problem, we establish the existence of solutions using variational methods combined with the mountain pass theorem. Moreover, an illustrative example is presented to prove the validity of the main result.

Well posedness of linear parabolic partial differential equations posed on a star-shaped network with local time Kirchhoff's boundary condition at the vertex

The main purpose of this work is to provide an existence and uniqueness result for the solution of a linear parabolic system posed on a star-shaped network, which presents a new type of Kirchhoff's boundary transmission condition at the junction. This new type of Kirchhoff's condition - that we decide to call here local-time Kirchhoff's condition - induces a dynamical behavior with respect to an external variable that may be interpreted as a local time parameter, designed to drive the system only at the singular point of the network. The seeds of this study point towards a theoretical inquiry of a particular generalization of Walsh's random spider motions, whose spinning measures would select the available directions according to the local time of the motion at the junction of the network.

Modeling and experiments on the vibro-acoustic analysis of ring stiffened cylindrical shells with internal bulkheads: A comparative study

The vibro-acoustic response of ring stiffened cylindrical shells with internal bulkheads under forced excitation is presented. The numerical analysis model is established using the Jacobi Ritz-Boundary element method. The first-order shear deformation theory, multi-segment technique and artificial spring technology are applied to establish the theoretical model, and the Jacobi orthogonal polynomials are introduced to represent the displacement functions. The Newmark-β integration method is used to obtain the vibration response of the structure, and the time-domain Kirchhoff boundary integral formulation is applied to describe the exterior acoustic field. The vibration and sound radiation of the ring stiffened cylindrical shell under impact and narrow band random loading were measured. The comparative study reveals that the results obtained from the proposed method agree well with the experimental results. The stiffeners and bulkheads have a significant effect on the vibration characteristics of the structure. Additionally, some physical insights into the resonant peaks and sound pressure directivity of the cylindrical shells are provided.

The dynamics of pulse solutions for reaction diffusion systems on a star shaped metric graph with the Kirchhoff's boundary condition

In this paper, we consider motions of localized patterns for reaction-diffusion systems of general types on a metric star graph which consists of several half-lines with a common end point called 'the junction point', where the Kirchhoff boundary condition is imposed. Assuming the existence and the stability of pulse and front like patterns for corresponding 1dimensional problems of reaction-diffusion systems, we rigorously derive ordinary differential equations describing the motions of them on a metric star graph. As the application, the attractive motion of a single pulse solution for the Gray-Scott model toward the junction point is shown. It is also shown that a single front solution of Allen-Cahn equation is repulsive against the junction point. The motion of multi pulse solutions and front solutions are also treated.

Asymptotic behavior for the long-range nonlinear Schrödinger equation on the star graph with the Kirchhoff boundary condition

We consider the cubic nonlinear Schr\"{o}dinger equation on the star graph with the Kirchhoff boundary condition. We prove modified scattering for the final state problem and the initial value problem. Moreover, we also consider the failure of scattering for the Schr\"{o}dinger equation with power-type long-range nonlinearities. These results are extension of the results for NLS on the one dimensional Euclidean space.

Impedance computation of cryogenic vacuum chambers using boundary element method

A two-dimensional boundary element method for calculating the impedance of infinitely long cryogenic vacuum chambers of general cross section is presented. The formulation is based on combining Kirchhoff's boundary integral representation of an electromagnetic field with the surface impedance boundary condition for the anomalous skin effect, which can occur in metals at cryogenic temperature. As a result, the electromagnetic field in the chamber is expressed as a superposition of the direct and indirect space charge fields and resistive-wall wakefield. This feature allows us to compute the corresponding three impedance contributions separately. This method does not assume a specific transverse charge density such as uniform and Gaussian distributions as well as the ultrarelativistic approximation. A technique for computing the impedances and the form factors in the method is also described. The presented method is applied to circular, rectangular, elliptical, and racetrack-type vacuum chambers. The geometric effect of the cryogenic vacuum chamber cross section on the resistive-wall impedance is shown. The effect of beam velocity is demonstrated for the racetrack-type cryogenic vacuum chamber.

Telegraph systems on networks and port-Hamiltonians. Ⅱ. Network realizability

<p style='text-indent:20px;'>Hyperbolic systems on networks often can be written as systems of first order equations on an interval, coupled by transmission conditions at the endpoints, also called port-Hamiltonians. However, general results for the latter have been difficult to interpret in the network language. The aim of this paper is to derive conditions under which a port-Hamiltonian with general linear Kirchhoff's boundary conditions can be written as a system of <inline-formula><tex-math id="M1">\begin{document}$ 2\times 2 $\end{document}</tex-math></inline-formula> hyperbolic equations on a metric graph <inline-formula><tex-math id="M2">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>. This is achieved by interpreting the matrix of the boundary conditions as a potential map of vertex connections of <inline-formula><tex-math id="M3">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula> and then showing that, under the derived assumptions, that matrix can be used to determine the adjacency matrix of <inline-formula><tex-math id="M4">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>.</p>

Implementing non-scalar diffraction in Fourier optics via the Braunbek method.

Fourier optics is a powerful and efficient tool for solving many diffraction problems, but relies on the assumption of scalar diffraction theory and ignores the three-dimensional structure and material properties of the diffracting element. Recent experiments of sub-scale starshade external occulters revealed that the inclusion of these physical properties is necessary to explain the observed diffraction at 10-10 of the incident light intensity. Here, we present a methodology for implementing non-scalar diffraction while maintaining the efficiency and ease of standard Fourier optics techniques. Our methodology is based on that of Braunbek, in which the Kirchhoff boundary values are replaced with the exact field in a narrow seam surrounding the edge of the diffracting element. In this paper, we derive the diffraction equations used to implement non-scalar diffraction and outline the computational implementation used to solve those equations. We also provide experimental results that demonstrate our model can replicate the observational signatures of non-scalar diffraction in sub-scale starshades, in effect validating our model to better than 10-10 in relative intensity. We believe this method to be an efficient tool for including additional physics to the models of coronagraphs and other optical systems in which a full electromagnetic solution is intractable.

Nonlinear dynamic and acoustic analysis of orthogonally stiffened composite laminated cylindrical shells containing piecewise isolators

This paper presents a numerical method for analyzing the nonlinear structural and acoustic responses of a coupled stiffened composite laminated cylindrical shell and piecewise isolation system immersed in an infinite acoustic fluid. The cylindrical shell is reinforced by a series of circumferential rings and longitudinal stringers, which may be few or many in number, non-uniform or uniform in size, and arbitrarily distributed in space. The isolation system in the shell contains motion-limiting stops, and the restoring forces of the isolators are assumed as bi-linear functions of the isolator deformations. A modified variational approach for the shell combined with a discrete element method for the rings and the stringers is employed to establish the nonlinear dynamics model of the structural system. The exterior acoustic fluid is computed by a time-domain Kirchhoff boundary integral formulation. The structural and acoustic models are coupled together by considering the compatibility conditions on their interface. The validity of the present method for predicting the structural and acoustic responses of the coupled shell and isolator system is confirmed. The contribution of the grouped circumferential wave modes of the shell to the vibration and radiated sound of the coupled structural system is examined. The effects of the excitation frequency of the external load, as well as the stiffness ratio and the gap of the suspension springs in each isolator on the structural and acoustic responses of the coupled system are discussed.

Nonlinear vibro-acoustic behaviors of coupled sandwich cylindrical shell and spring-mass-damper systems

Abstract This paper is concerned with the structural-acoustic analysis of a coupled sandwich cylindrical shell and nonlinear spring-mass-damper system immersed in an infinite acoustic medium. The restoring force of each spring in the spring-mass-damper system is assumed as a general polynomial function of the spring deformation, from which the nonlinear quadratic, cubic and quartic springs can be directly recovered. A modified variational approach is employed to formulate the nonlinear dynamics model of the coupled structural system, and the exterior acoustic fluid is computed by a time-domain Kirchhoff boundary integral formulation. The structural and acoustic models are coupled together through the compatibility conditions on their interface. The proposed model is validated by comparing the solutions obtained from the finite element/boundary element method and those available in the literature. Very good agreement between the predicted results and the reference solutions is achieved. Individual contributions of circumferential wave modes of the sandwich cylindrical shell to the vibration and acoustic responses of the coupled nonlinear structural system are examined. The effects of the amplitude and excitation frequency of the external loads, and different types of nonlinear springs (including the quadratic, cubic and quartic nonlinear springs) on the vibration and sound radiation behaviors of the structural system are investigated.

Nonlinear structural and acoustic responses of three-dimensional elastic cylindrical shells with internal mass-spring systems

Prediction of the vibration and sound radiation of an elastic cylindrical shell with nonlinear internal structures is of significant interest to marine and aerospace industries. This paper investigates the structural and acoustic behaviors of a finite cylindrical shell attached with a nonlinear spring-mass-damper system and embedded in an infinite acoustic medium. The restoring forces of the springs are expressed as general polynomial functions of the spring deformation, from which the nonlinear quadratic, cubic and quartic springs can be directly recovered. The exact three-dimensional theory of elasticity is employed for the structural modeling of the cylindrical shell. The discretized equations of motion of the coupled nonlinear structural system are established using an efficient modified variational method, and the fluid surrounding the shell is computed by a time-domain Kirchhoff boundary integral formulation. The two sets of structural and acoustic equations are coupled together through the compatibility conditions on the fluid-structure interface. The present results are validated by comparing the solutions obtained from the finite element method and those available in the literature. The vibration and sound pressure responses of the coupled structural system with nonlinear quadratic, cubic and quartic springs are investigated. The contributions of the circumferential wave modes of the shell to the nonlinear vibro-acoustic responses and sound directivity patterns of the nonlinear coupled structural system are discussed.

A Note on Sign-Changing Solutions to the NLS on the Double-Bridge Graph

We study standing waves of the NLS equation posed on the double-bridge graph: two semi-infinite half-lines attached at a circle. At the two vertices, Kirchhoff boundary conditions are imposed. We pursue a recent study concerning solutions nonzero on the half-lines and periodic on the circle, by proving some existing results of sign-changing solutions non-periodic on the circle.

Standing waves for the NLS on the double-bridge graph and a rational–irrational dichotomy

We study standing waves of NLS equation posed on the double-bridge graph: two semi-infinite half-lines attached at a circle. At the two vertices Kirchhoff boundary conditions are imposed. The configuration of the graph is characterized by two lengths, L1 and L2. We study the solutions with possibly nontrivial components on the half-lines and a cnoidal component on the circle. The problem is equivalent to a nonlinear boundary value problem in which the boundary condition depends on the spectral parameter ω. After classifying the solutions with rational L1/L2, we turn to L1/L2 irrational showing that there exist standing waves only in correspondence to a countable set of negative frequencies ωn. Moreover we show that the frequency sequence admits cluster points and any negative real number can be a limit point of frequencies choosing a suitable irrational geometry L1/L2. These results depend on basic properties of diophantine approximation of real numbers.

Nonlinear Partial Differential Equations on Graphs

One-dimensional metric graphs in two and three-dimensional spaces play an important role in emerging areas of modern science such as nano-technology, quantum physics, and biological networks. The workshop focused on the analysis of nonlinear partial differential equations on metric graphs, especially on the bifurcation and stability of nonlinear waves on complex graphs, on the justification of Kirchhoff boundary conditions, on spectral properties and the validity of amplitude equations for periodic graphs, and the existence of ground states for the NLS equation with and without potential.

Spectral stability of shifted states on star graphs

We consider the nonlinear Schrödinger (NLS) equation with the subcritical power nonlinearity on a star graph consisting of N edges and a single vertex under generalized Kirchhoff boundary conditions. The stationary NLS equation may admit a family of solitary waves parameterized by a translational parameter, which we call the shifted states. The two main examples include (i) the star graph with even N under the classical Kirchhoff boundary conditions and (ii) the star graph with one incoming edge and N − 1 outgoing edges under a single constraint on coefficients of the generalized Kirchhoff boundary conditions. We obtain the general counting results on the Morse index of the shifted states and apply them to the two examples. In the case of (i), we prove that the shifted states with even are saddle points of the action functional which are spectrally unstable under the NLS flow. In the case of (ii), we prove that the shifted states with the monotone profiles in the N − 1 edges are spectrally stable, whereas the shifted states with non-monotone profiles in the N − 1 edges are spectrally unstable, the two families intersect at the half-soliton states which are spectrally stable but nonlinearly unstable under the NLS flow. Since the NLS equation on a star graph with shifted states can be reduced to the homogeneous NLS equation on an infinite line, the spectral instability of shifted states is due to the perturbations breaking this reduction. We give a simple argument suggesting that the spectrally stable shifted states in the case of (ii) are nonlinearly unstable under the NLS flow due to the perturbations breaking the reduction to the homogeneous NLS equation.

Thin film flow dynamics on fiber nets

We analyze existence and qualitative behavior of non-negative weak solutions for fourth order degenerate parabolic equations on graph domains with Kirchhoff's boundary conditions at the inner nodes and Neumann boundary conditions at the boundary nodes. The problem is originated from industrial constructions of spray coated meshes which are used in water collection and in oil-water separation processes. For a certain range of parameter values we prove convergence toward a constant steady state that corresponds to the uniform distribution of coating on a fiber net.

Bifurcations of Standing Localized Waves on Periodic Graphs

The nonlinear Schrodinger (NLS) equation is considered on a periodic metric graph subject to the Kirchhoff boundary conditions. Bifurcations of standing localized waves for frequencies lying below the bottom of the linear spectrum of the associated stationary Schrodinger equation are considered by using analysis of two-dimensional discrete maps near hyperbolic fixed points. We prove existence of two distinct families of small-amplitude standing localized waves, which are symmetric about the two symmetry points of the periodic graphs. We also prove properties of the two families, in particular, positivity and exponential decay. The asymptotic reduction of the two-dimensional discrete map to the stationary NLS equation on an infinite line is discussed in the context of the homogenization of the NLS equation on the periodic metric graph.

Ground State on the Dumbbell Graph

We consider standing waves in the focusing nonlinear Schrodinger (NLS) equation on a dumbbell graph (two rings attached to a central line segment subject to the Kirchhoff boundary conditions at the junctions). In the limit of small L norm, the ground state (the orbitally stable standing wave of the smallest energy at a fixed L norm) is represented by a constant solution. However, when the L norm is increased, this constant solution undertakes two bifurcations, where the first is the pitchfork (symmetry breaking) bifurcation and the second one is the symmetry preserving bifurcation. As a result of the first symmetry breaking bifurcation, the standing wave becomes more localized in one of the two rings. As a result of the second symmetry preserving bifurcation, the standing wave becomes localized in the central line segment. In the limit of large norm solutions, both standing waves are represented by a truncated solitary wave localized in either the ring or the central line segment. Although the asymmetric wave supported in the ring is a ground state near the symmetry breaking bifurcation of the constant solution, it is the symmetric wave supported in the central line segment which becomes the ground state in the limit of large L 2 norm. The analytical results are confirmed by numerical approximations of the ground state on the dumbbell graph.

Bifurcations and stability of standing waves in the nonlinear Schrödinger equation on the tadpole graph

We develop a detailed analysis of edge bifurcations of standing waves in the nonlinear Schrödinger (NLS) equation on a tadpole graph (a ring attached to a semi-infinite line subject to the Kirchhoff boundary conditions at the junction). It is shown in the recent work [] by using explicit Jacobi elliptic functions that the cubic NLS equation on a tadpole graph admits a rich structure of standing waves. Among these, there are different branches of localized waves bifurcating from the edge of the essential spectrum of an associated Schrödinger operator.We show by using a modified Lyapunov-Schmidt reduction method that the bifurcation of localized standing waves occurs for every positive power nonlinearity. We distinguish a primary branch of never vanishing standing waves bifurcating from the trivial solution and an infinite sequence of higher branches with oscillating behavior in the ring. The higher branches bifurcate from the branches of degenerate standing waves with vanishing tail outside the ring.Moreover, we analyze stability of bifurcating standing waves. Namely, we show that the primary branch is composed by orbitally stable standing waves for subcritical power nonlinearities, while all nontrivial higher branches are linearly unstable near the bifurcation point. The stability character of the degenerate branches remains inconclusive at the analytical level, whereas heuristic arguments based on analysis of embedded eigenvalues of negative Krein signatures support the conjecture of their linear instability at least near the bifurcation point. Numerical results for the cubic NLS equation show that this conjecture is valid and that the degenerate branches become spectrally stable far away from the bifurcation point.

Fast time domain modeling of surface scattering from reflectors and diffusers.

Time-domain prediction models have been developed for auditorium reflectors and room acoustic diffusers. The models are time-domain equivalents of the single-frequency formulations that exploit the Kirchhoff boundary conditions. Consequently, they are approximate, wave-based solutions to the Kirchhoff integral equation using surface meshes. The new time-domain formulations are validated by comparison to their frequency-domain equivalents for three different surfaces: a plane surface, a curved reflector, and a Schroeder diffuser. In terms of computation time and accuracy, the new models lie between the finite difference time domain and geometric room models.