Search results for "Boundary value problem"
showing 10 items of 551 documents
Boundary value problem with integral condition for a Blasius type equation
2016
The steady motion in the boundary layer along a thin flat plate, which is immersed at zero incidence in a uniform stream with constant velocity, can be described in terms of the solution of the differential equation x'''= -xx'', which satisfies the boundary conditions x(0) = x'(0) = 0, x'(∞) = 1. The author investigates the generalized boundary value problem consisting of the nonlinear third-order differential equation x''' = -trx|x|q-1x'' subject to the integral boundary conditions x(0) = x'(0) = 0, x'(∞) = λ∫0ξx(s) ds, where 0 0 is a parameter. Results on the existence and uniqueness of solutions to boundary value problem are established. An illustrative example is provided.
Partial data inverse problems and simultaneous recovery of boundary and coefficients for semilinear elliptic equations
2021
We study various partial data inverse boundary value problems for the semilinear elliptic equation $\Delta u+ a(x,u)=0$ in a domain in $\mathbb R^n$ by using the higher order linearization technique introduced in [LLS 19, FO19]. We show that the Dirichlet-to-Neumann map of the above equation determines the Taylor series of $a(x,z)$ at $z=0$ under general assumptions on $a(x,z)$. The determination of the Taylor series can be done in parallel with the detection of an unknown cavity inside the domain or an unknown part of the boundary of the domain. The method relies on the solution of the linearized partial data Calder\'on problem [FKSU09], and implies the solution of partial data problems fo…
Shape optimization for Stokes problem with threshold slip boundary conditions
2017
This paper deals with shape optimization of systems governed by the Stokes flow with threshold slip boundary conditions. The stability of solutions to the state problem with respect to a class of domains is studied. For computational purposes the slip term and impermeability condition are handled by a regularization. To get a finite dimensional optimization problem, the optimized part of the boundary is described by B´ezier polynomials. Numerical examples illustrate the computational efficiency. peerReviewed
River-bed erosion due to changing boundary conditions: performance of a protective measure
2018
Abstract. Due to the introduction of man-made sediment barriers along a river, the amount of sediment load entering the downstream river reach is different to that leaving the reach, and erosion processes occur downstream of the barrier itself. Designers are often required to take into account the scouring process and to include adequate protective measures against the local scour. This paper addresses the performance of bio-engineering protective measures against the erosion process. In particular, a green carpet, realized with real flexible vegetation, has been used as the protective measure against erosion processes downstream of a rigid bed. Analyses are based on experimental work carri…
Multiplicity results for asymmetric boundary value problems with indefinite weights
2004
We prove existence and multiplicity of solutions, with prescribed nodal properties, to a boundary value problem of the formu″+f(t,u)=0,u(0)=u(T)=0. The nonlinearity is supposed to satisfy asymmetric, asymptotically linear assumptions involving indefinite weights. We first study some auxiliary half-linear, two-weighted problems for which an eigenvalue theory holds. Multiplicity is ensured by assumptions expressed in terms of weighted eigenvalues. The proof is developed in the framework of topological methods and is based on some relations between rotation numbers and weighted eigenvalues.
Solvability of a first order system in three-dimensional non-smooth domains
1985
summary:A system of first order partial differential equations is studied which is defined by the divergence and rotation operators in a bounded nonsmooth domain $\Omega\subset \bold R^3$. On the boundary $\delta\Omega$, the vanishing normal component is prescribed. A variational formulation is given and its solvability is investigated.
A new approach to calibrate the thermal conditions in space charge measurements on HVDC mini-cables
2019
The PEA method is currently widespread used for space charge measurements in mini-cables in order to qualify the behavior of semicon-dielectric-semicon compound under electric and thermal stress. The main goal of this research is to quantitatively evaluate the relationship between the thermal boundary conditions applied to cables or mini-cables and the maximum local electric field due to the accumulated space charge. Hitherto, several research groups have obtained thermal gradients over the dielectric's radius heating the conductor by Joule effect due to an induced current. In this paper, a numerical approach is offered to calibrate the heat exchange boundary conditions to apply to a sample…
Numerical construction of the density-potential mapping
2018
We demonstrate how a recently developed method Nielsen et al. [Nielsen et al., EPL 101, 33001 (2013)] allows for a comprehensive investigation of time-dependent density functionals in general, and of the exact time-dependent exchange-correlation potential in particular, by presenting the first exact results for two- and three-dimensional multi-electron systems. This method is an explicit realization of the Runge–Gross correspondence, which maps time-dependent densities to their respective potentials, and allows for the exact construction of desired density functionals. We present in detail the numerical requirements that makes this method efficient, stable and precise even for large and rap…
Biharmonic obstacle problem: guaranteed and computable error bounds for approximate solutions
2020
The paper is concerned with a free boundary problem generated by the biharmonic operator and an obstacle. The main goal is to deduce a fully guaranteed upper bound of the difference between the exact minimizer u and any function (approximation) from the corresponding energy class (which consists of the functions in $H^2$ satisfying the prescribed boundary conditions and the restrictions stipulated by the obstacle). For this purpose we use the duality method of the calculus of variations and general type error identities earlier derived for a wide class of convex variational problems. By this method, we define a combined primal--dual measure of error. It contains four terms of different natu…
On some partial data Calderón type problems with mixed boundary conditions
2021
In this article we consider the simultaneous recovery of bulk and boundary potentials in (degenerate) elliptic equations modelling (degenerate) conducting media with inaccessible boundaries. This connects local and nonlocal Calderón type problems. We prove two main results on these type of problems: On the one hand, we derive simultaneous bulk and boundary Runge approximation results. Building on these, we deduce uniqueness for localized bulk and boundary potentials. On the other hand, we construct a family of CGO solutions associated with the corresponding equations. These allow us to deduce uniqueness results for arbitrary bounded, not necessarily localized bulk and boundary potentials. T…