Exact boundary controllability of nodal profile for quasilinear wave equations in a planar tree‐like network of strings

Based on the theory of semi‐global piecewise C² solutions to 1D quasilinear wave equations, the local exact boundary controllability of nodal profile for quasilinear wave equations in a planar tree‐like network of strings with general topology is obtained by a constructive method. The principles of providing nodal profiles and of choosing and transferring boundary controls are presented, respectively. Copyright © 2013 John Wiley & Sons, Ltd.     

Inverse nodal problem for p–laplacian dirac system

In this study, we solve an inverse nodal problem for p‐Laplacian Dirac system with boundary conditions depending on spectral parameter. Asymptotic formulas of eigenvalues, nodal points and nodal lengths are obtained by using modified Prüfer substitution. The key step is to apply modified Prüfer substitution to derive a detailed asymptotic estimate for eigenvalues. Furthermore, we have shown that the functions r(x) and q(x) in Dirac system can be established uniquely by using nodal parameters with the method used by Wang et al. Obtained results are more general than the classical Dirac system. Copyright © 2016 John Wiley & Sons, Ltd.     

An inverse problem for a differential pencil using nodal points as data

An inverse nodal problem is studied for a differential pencil with non-separated boundary conditions. We prove that a dense subset of nodal points uniquely determines the boundary data and potential functions. We also provide a constructive procedure for the solution of the inverse nodal problem.     

AnO(h 6) quintic spline collocation method for fourth order two-point boundary value problems

AnO(h ⁶) collocation method based on quintic splines is developed and analyzed for general fourth-order linear two-point boundary value problems. The method determines a quintic spline approximation to the solution by forcing it to satisfy a high order perturbation of the original boundary value problem at the nodal points of the spline. A variation of this method is formulated as a deferred correction method. The error analysis of the new method and its numerical behavior is presented.This research was supported by AFOSR grant 84-0385.     

Exact boundary controllability of nodal profile for 1-D quasilinear wave equations

Based on the theory of semi-global C ² solution for 1-D quasilinear wave equations, the local exact boundary controllability of nodal profile for 1-D quasilinear wave equations is obtained by a constructive method, and the corresponding global exact boundary controllability of nodal profile is also obtained under certain additional hypotheses.     

具有特征参数多项式边界条件的Sturm-Liouville 方程的逆结点问题

逆结点问题是通过特征函数的零点重构算子. 本文主要讨论具有特征参数多项式边界条件的 Sturm-Liouville 方程的逆结点问题. 20世纪50年代以后,人们发现在许多工程领域, Sturm-Liouville 问题的谱参数不仅出现在方程中, 而且也出现在边界条件中,因此带参数边界条件的逆结点问题对数学物理方面的研究有重要意义. 本文讨论区间 $[0,1]$ 上边界条件为参数多项式的 Sturm-Liouville 方程的逆结点问题, 并证明在 $[0,b]$ \big($ b\in \big(\frac{1}{2},1\big]$\big) 上结点的稠密子集可唯一确定 $[0,1]$ 上的势函数和边界条件中多项式的未知系数.     

Adaptive boundary element methods for strongly elliptic integral equations

Summary Integral operators are nonlocal operators. The operators defined in boundary integral equations to elliptic boundary value problems, however, are pseudo-differential operators on the boundary and, therefore, provide additional pseudolocal properties. These allow the successful application of adaptive procedures to some boundary element methods. In this paper we analyze these methods for general strongly elliptic integral equations and obtain a-posteriori error estimates for boundary element solutions. We also apply these methods to nodal collocation with odd degree splines. Some numerical examples show that these adaptive procedures are reliable and effective.This work was carried out while Dr. De-hao Yu was an Alexander-von-Humboldt-Stiftung research fellow at the University of Stuttgart in 1987, 1988     

Modified nodal cubic spline collocation for elliptic equations

We consider the modified nodal cubic spline collocation method for a general, variable coefficient, second order partial differential equation in the unit square with the solution subject to the homogeneous Dirichlet boundary conditions. The bicubic spline approximate solution satisfies both the Dirichlet boundary conditions and a perturbed partial differential equation at the nodes of a uniform partition of the square. We prove existence and uniqueness of the approximate solution and derive an optimal fourth order maximum norm error bound. The resulting linear system is solved efficiently by a preconditioned iterative method. Numerical results confirm the expected convergence rates. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2011     

Inverse Nodal Problem for Dirac System with Spectral Parameter in Boundary Conditions

An inverse nodal problem lies in constructing operators from the given zeros of their eigenfunctions. In this work, we deal with an inverse nodal problem of reconstructing the Dirac system with the spectral parameter in the boundary conditions. We prove that a set of nodal points of one of the components of the eigenfunctions uniquely determines all the parameters of the boundary conditions and the coefficients of the Dirac equations. We also provide a constructive procedure for solving this inverse nodal problem.     

Reconstruction of the Dirac-Type Integro-Differential Operator From Nodal Data

Abstract

The inverse nodal problem for Dirac type integro-differential operator with the spectral parameter in the boundary conditions is studied. We prove that dense subset of the nodal points determines the coefficients of differential part of operator and gives partial information for integral part of it.     

Certain nodal surfaces of the eigenfunctions of the Helmholtz equation in a cubic region

Certain general properties of the nodal surfaces of the linear combination of eigenfuctions of the first homogeneous boundary value problem of the Helmholtz equation in a cubic region, corresponding to the simplest triple eigenvalues, are considered.Institute of Polymer Mechanics, Academy of Sciences of the Lavian, SSR, Riga. Translated from Mekhanika Polimerov, No. 1, pp. 109–115, January–February, 1970.     

Exact boundary controllability of partial nodal profile for network of strings

Based on the theory of exact boundary controllability of nodal profile for hyperbolic systems, the authors propose the concept of exact boundary controllability of partial nodal profile to expand the scope of applications. With the new concept, we can shorten the controllability time, save the number of controls, and increase the number of charged nodes with given nodal profiles. Furthermore, we introduce the concept of in-situ controlled node to deal with a new situation that one node can be charged and controlled simultaneously. The minimum number of boundary controls on the entire tree-like network is determined by using the concept of ‘degree of freedom of charged nodes’ introduced. And the concept of ‘control path’ is introduced to appropriately divide the network, so that we can determine the infimum of controllability time. General frameworks of constructive proof are given on a single interval, a star-like network, a chain-like network and a planar tree-like network for linear wave equation(s) with Dirichlet, Neumann, Robin and dissipative boundary conditions to establish a complete theory on the exact boundary controllability of partial nodal profile.     

Nodal solutions of multi-point boundary value problems

We study the nonlinear boundary value problem consisting of the equation y^″+w(t)f(y)=0 on [a,b] and a multi-point boundary condition. By relating it to the eigenvalues of a linear Sturm-Liouville problem with a two-point separated boundary condition, we obtain results on the existence and nonexistence of nodal solutions of this problem. We also discuss the changes in the existence question for different types of nodal solutions as the problem changes.     

Nodal adjoint method for multigroup diffusion in heterogeneous slab reactors

A variational adjoint nodal method is proposed to yield an approximation theory for the eigenvalues of the multigroup neutron diffusion boundary value problem of a heterogeneous quasicritical one-dimensional slab reactor. Semianalytical eigensolutions are constructed for the associated whole-reactor group nodal fluxes. The method appears to be mathematically more consistent, computationally more straightforward, and practically more convenient than alternative nodal or finite element schemes. © 1992 John Wiley & Sons, Inc.     

Nodal volumes for eigenfunctions of analytic regular elliptic problems

We establish sharp upper bounds on the (n−1)-dimensional Hausdorff measure of the zero (nodal) sets and on the maximal order of vanishing corresponding to eigenfunctions of a regular elliptic problem on a bounded domain Ω ⊆ ℝ ⁿ with real-analytic boundary. The elliptic operator may be of an arbitrary even order, and its coefficients are assumed to be real-analytic. This extends a result of Donnelly and Fefferman ([DF1], [DF3]) concerning upper bounds for nodal volumes of eigenfunctions corresponding to the Laplacian on compact Riemannian manifolds with boundary.     

Global Bifurcation on Time Scales

We consider the structure of the solution set of a nonlinear Sturm-Liouville boundary value problem defined on a general time scale. Using global bifurcation theory we show that unbounded continua of nontrivial solutions bifurcate from the trivial solution at the eigenvalues of the linearization, and we show that certain nodal properties of the solutions are preserved along these continua. These results extend the well-known results of Rabinowitz for the case of Sturm-Liouville ordinary differential equations.     

Reconstruction of the Dirac Operator From Nodal Data

Inverse nodal problems consist in constructing operators from the given nodes (zeros) of their eigenfunctions. In this work, we deal with the inverse nodal problem of reconstructing the Dirac operator on a finite interval. We prove that a dense subset of nodal points uniquely determine the parameters of the boundary conditions, the mass of a particle and the potential function of the Dirac system. We also provide a constructive procedure for the solution of the inverse nodal problem.     

Modified nodal cubic spline collocation for three-dimensional variable coefficient second order partial differential equations

We formulate a fourth order modified nodal cubic spline collocation scheme for variable coefficient second order partial differential equations in the unit cube subject to nonzero Dirichlet boundary conditions. The approximate solution satisfies a perturbed partial differential equation at the interior nodes of a uniform $N\times N\times N$ partition of the cube and the partial differential equation at the boundary nodes. In the special case of Poisson’s equation, the resulting linear system is solved by a matrix decomposition algorithm with fast Fourier transforms at a cost $O(N^3\log N)$ . For the general variable coefficient diffusion-dominated case, the system is solved using the preconditioned biconjugate gradient stabilized method.     

Asymptotic stability of the exact boundary controllability of nodal profile for 1D quasi-linear wave equations

In this paper, we consider the asymptotic stability of the exact boundary controllability of nodal profile for 1D quasi-linear wave equations. First, for 1D quasi-linear hyperbolic systems with zero eigenvalues, we establish the existence and uniqueness of semiglobal classical solution to the one-sided mixed initial-boundary value problem on a semibounded initial axis and discuss the asymptotic behavior of the corresponding solutions under different hypotheses on the initial data. Based on these results, we obtain the asymptotic stability of the exact boundary controllability of nodal profile for 1D quasi-linear wave equations on a semibounded time interval.     

Existence and qualitative properties of concentrating solutions for the sinh-Poisson equation

We prove the existence of nodal solutions for – u = sinhu with Dirichlet boundary conditions in bounded, 2D, smoothand non-smooth domains. Indeed, for positive and small enough,we show that there exist at least two pairs of solutions, whichchange sign exactly once, whose nodal lines intersect the boundary.