Overdetermined partial boundary value problems on finite networks

In this study, we define a class of non-self-adjoint boundary value problems on finite networks associated with Schrödinger operators. The novel feature of this study is that no data are prescribed on part of the boundary, whereas both the values of the function and of its normal derivative are given on another part of the boundary. We show that overdetermined partial boundary value problems are crucial for solving inverse boundary value problems on finite networks since they provide the theoretical foundations for the recovery algorithm. We analyze the uniqueness and the existence of solution for overdetermined partial boundary value problems based on the nonsingularity of partial Dirichlet-to-Neumann maps. These maps allow us to determine the value of the solution in the part of the boundary where no data was prescribed. We also execute full conductance recovery for spider networks.     

Efficient long-time computations of time-domain boundary integrals for 2D and dissipative wave equation

Linear hyperbolic partial differential equations in a homogeneous medium, e.g., the wave equation describing the propagation and scattering of acoustic waves, can be reformulated as time-domain boundary integral equations. We propose an efficient implementation of a numerical discretization of such equations when the strong Huygens’ principle does not hold.For the numerical discretization, we make use of convolution quadrature in time and standard Galerkin boundary element method in space. The quadrature in time results in a discrete convolution of weights W_j with the boundary density evaluated at equally spaced time points. If the strong Huygens’ principle holds, W_j converge to 0 exponentially quickly for large enough j. If the strong Huygens’ principle does not hold, e.g., in even space dimensions or when some damping is present, the weights are never zero, thereby presenting a difficulty for efficient numerical computation.In this paper we prove that the kernels of the convolution weights approximate in a certain sense the time domain fundamental solution and that the same holds if both are differentiated in space. The tails of the fundamental solution being very smooth, this implies that the tails of the weights are smooth and can efficiently be interpolated. Further, we hint on the possibility to apply the fast and oblivious convolution quadrature algorithm of Schädle et al. to further reduce memory requirements for long-time computation. We discuss the efficient implementation of the whole numerical scheme and present numerical experiments.     

Inverse Problems for Time-Dependent Singular Heat Conductivities: Multi-Dimensional Case

We consider an inverse boundary value problem for the heat equation ? _t u = div (γ? _x u) in (0, T) × Ω, u = f on (0, T) × ?Ω, u| _t=0 = u ₀, in a bounded domain Ω ? ? ⁿ , n ≥ 2, where the heat conductivity γ(t, x) is piecewise constant and the surface of discontinuity depends on time: γ(t, x) = k ² (x ∈ D(t)), γ(t, x) = 1 (x ∈ Ω?D(t)). Fix a direction e* ∈ 𝕊 ^n?1 arbitrarily. Assuming that ?D(t) is strictly convex for 0 ≤ t ≤ T, we show that k and sup {e*·x; x ∈ D(t)} (0 ≤ t ≤ T), in particular D(t) itself, are determined from the Dirichlet-to-Neumann map : f → ?_ν u(t, x)|_{(0, T)×?Ω}. The knowledge of the initial data u ₀ is not used in the proof. If we know min_0≤t≤T (sup _x∈D(t) x·e*), we have the same conclusion from the local Dirichlet-to-Neumann map. Numerical examples of stationary and moving circles inside the unit disk are shown. The results have applications to nondestructive testing. Consider a physical body consisting of homogeneous material with constant heat conductivity except for a moving inclusion with different conductivity. Then the location and shape of the inclusion can be monitored from temperature and heat flux measurements performed at the boundary of the body. Such a situation appears for example in blast furnaces used in ironmaking.     

Stability estimate for an inverse problem for the magnetic Schrödinger equation from the Dirichlet-to-Neumann map

We consider the problem of stability estimate of the inverse problem of determining the magnetic field entering the magnetic Schrödinger equation in a bounded smooth domain of Rn with input Dirichlet data, from measured Neumann boundary observations. This information is enclosed in the dynamical Dirichlet-to-Neumann map associated to the solutions of the magnetic Schrödinger equation. We prove in dimension n?2 that the knowledge of the Dirichlet-to-Neumann map for the magnetic Schrödinger equation measured on the boundary determines uniquely the magnetic field and we prove a Hölder-type stability in determining the magnetic field induced by the magnetic potential.     

Stability estimate in determination of a coefficient in transmission wave equation by boundary observation

In this paper, we study the global stability in determination of a coefficient in the transmission wave equation from data of the solution in a subboundary over a time interval. Providing regular initial data, we prove a hölder stability estimate in the inverse problem with a single measurement. Moreover, the exponent in the stability estimate depends on the regularity of initial data.     

Determination of point sources in vibrating plate by boundary measurements

We consider an inverse problem of determining point sources in vibrating plate by boundary measurements. We show that the boundary observation of the domain determines uniquely the sources for an arbitrarily small time of observation and we establish a conditional stability.     

Stabilization of the wave equation with variable coefficients and a delay in dissipative boundary feedback

We consider the stabilization of the wave equation with variable coefficients and a delay in the dissipative boundary feedback. By virtue of the Riemannian geometry methods, the energy-perturbed approach and the multiplier skills, we establish the uniform stability of the energy of the closed-loop system.     

Determination of the source parameter in a heat equation with a non-local boundary condition

In this article we consider the inverse problem of identifying a time dependent unknown coefficient in a parabolic problem subject to initial and non-local boundary conditions along with an overspecified condition defined at a specific point in the spatial domain. Due to the non-local boundary condition, the system of linear equations resulting from the backward Euler approximation have a coefficient matrix that is a quasi-tridiagonal matrix. We consider an efficient method for solving the linear system and the predictor–corrector method for calculating the solution and updating the estimate of the unknown coefficient. Two model problems are solved to demonstrate the performance of the methods.     

Stability estimate for the hyperbolic inverse boundary value problem by local Dirichlet-to-Neumann map

In this paper we consider the stability of the inverse problem of determining a function q(x) in a wave equation in a bounded smooth domain in Rn from boundary observations. This information is enclosed in the hyperbolic (dynamic) Dirichlet-to-Neumann map associated to the solutions to the wave equation. We prove in the case of n?2 that q(x) is uniquely determined by the range restricted to a subboundary of the Dirichlet-to-Neumann map whose stability is a type of double logarithm.     

Cloaking via impedance boundary condition for the 2-D Helmholtz equation

We consider control problems for the 2-D Helmholtz equation in an unbounded domain with partially coated boundary. Dirichlet boundary condition is given on one part of the boundary and the impedance boundary condition is imposed on another its part. The role of control in control problem under study is played by boundary impedance. Quadratic tracking–type functionals for the field play the role of cost functionals. Solvability of control problems is proved. The uniqueness and stability of optimal solutions with respect to certain perturbations of both cost functional and incident field are established.     

The pointwise estimates of solutions for semilinear dissipative wave equation in multi-dimensions

In this paper, we study the global-in-time existence and the pointwise estimates of solutions to the Cauchy problem for the dissipative wave equation in multi-dimensions. Using the fixed point theorem, we obtain the global existence of the solution. In addition, the pointwise estimates of the solution are obtained by the method of the Green function. Furthermore, we obtain the Lp, 1?p?∞, convergence rate of the solution.     

Moving boundary value problems for the wave equation

We study certain boundary value problems for the one-dimensional wave equation posed in a time-dependent domain. The approach we propose is based on a general transform method for solving boundary value problems for integrable nonlinear PDE in two variables, that has been applied extensively to the study of linear parabolic and elliptic equations. Here we analyse the wave equation as a simple illustrative example to discuss the particular features of this method in the context of linear hyperbolic PDEs, which have not been studied before in this framework.     

decay problem for the dissipative wave equation in odd dimensions

Consider the Cauchy problem in odd dimensions for the dissipative wave equation: (□+∂_t)u=0 in with (u,∂_tu)|_t=0=(u₀,u₁). Because the L² estimates and the L^∞ estimates of the solution u(t) are well known, in this paper we pay attention to the L^p estimates with 1p<2 (in particular, p=1) of the solution u(t) for t0. In order to derive L^p estimates we first give the representation formulas of the solution u(t)=∂_tS(t)u₀+S(t)(u₀+u₁) and then we directly estimate the exact solution S(t)g and its derivative ∂_tS(t)g of the dissipative wave equation with the initial data (u₀,u₁)=(0,g). In particular, when p=1 and n1, we get the L¹ estimate: u(t)_L¹Ce^−t/4(u_0W^n,1+u_1W^n−1,1)+C(u_0L¹+u_1L¹) for t0.     

Strichartz estimates for the wave equation on manifolds with boundary

We prove certain mixed-norm Strichartz estimates on manifolds with boundary. Using them we are able to prove new results for the critical and subcritical wave equation in 4-dimensions with Dirichlet or Neumann boundary conditions. We obtain global existence in the subcritical case, as well as global existence for the critical equation with small data. We also can use our Strichartz estimates to prove scattering results for the critical wave equation with Dirichlet boundary conditions in 3-dimensions.     

Determination of a material constant in the impedance boundary condition for electromagnetic fields

We study a recovery problem for an unknown boundary data at the boundary part in static electromagnetism. Our computational area is a bounded domain Ω⊂Rⁿ with a Lipschitz continuous boundary. The problem for determining the coefficient λ is considered. This coefficient represents one of the ferromagnetic material characteristics occupying this domain. The existence and uniqueness of a weak solution are proved and a numerical method for its recovery is supported by numerical experiments.     

A boundary value problem for Bitsadze equation in the unit disc

A boundary value problem for the Bitsadze equation

$\frac{{\partial ^2 }}{{\partial \bar z^2 }}u(x,y) \equiv \frac{1}{4}\left( {\frac{\partial }{{\partial x}} + i\frac{\partial }{{\partial y}}} \right)^2 u(x,y) = 0$

in the interior of the unit disc is considered. It is proved that the problem is Noetherian and its index is calculated, and solvability conditions for the non-homogeneous problem are proposed. Some solutions of the homogeneous problem are explicitely found.

     

Lp Decay problem for the dissipative wave equation in even dimensions

We study L^p decay estimates of the solution to the Cauchy problem for the dissipative wave equation in even dimensions: (□+?_t)u=0 in ?^N × (0,∞) for even N=2n?2 with initial data (u,?_tu)∣_t=0 =(u₀,u₁). The representation formulas of the solution u(t)=?_tS(t)u₀ + S(t)(u₀+u₁) provide the sharp estimates on L^p norms with p?1. Copyright © 2004 John Wiley & Sons, Ltd.     

Blow-up solutions for a nonlinear wave equation with boundary damping and interior source

In this paper we consider the wave equation with nonlinear damping and source terms. We are interested in the interaction between the boundary damping −|y_t(L,t)|^m−1y_t(L,t) and the interior source |y(t)|^p−1y(t). We find a sufficient condition for obtaining the blow-up solution of the problem. Furthermore, we also obtain that the solution may blow up even if m≥p.     

A posteriori error estimates for finite element approximations to the wave equation with discontinuous coefficients

We derive residual‐based a posteriori error estimates of finite element method for linear wave equation with discontinuous coefficients in a two‐dimensional convex polygonal domain. A posteriori error estimates for both the space‐discrete case and for implicit fully discrete scheme are discussed in L^∞(L²) norm. The main ingredients used in deriving a posteriori estimates are new Clément type interpolation estimates in conjunction with appropriate adaption of the elliptic reconstruction technique of continuous and discrete solutions. We use only an energy argument to establish a posteriori error estimates with optimal order convergence in the L^∞(L²) norm.