Boundary Value Problems In Spaces Research Articles

A Hybrid Localized Meshless Method for the Solution of Transient Groundwater Flow in Two Dimensions

In this work, a hybrid localized meshless method is developed for solving transient groundwater flow in two dimensions by combining the Crank–Nicolson scheme and the generalized finite difference method (GFDM). As the first step, the temporal discretization of the transient groundwater flow equation is based on the Crank–Nicolson scheme. A boundary value problem in space with the Dirichlet or mixed boundary condition is then formed at each time node, which is simulated by introducing the GFDM. The proposed algorithm is truly meshless and easy to program. Four linear or nonlinear numerical examples, including ones with complicated geometry domains, are provided to verify the performance of the developed approach, and the results illustrate the good accuracy and convergency of the method.

Boundary value problems of elliptic and parabolic type with boundary data of negative regularity

We study elliptic and parabolic boundary value problems in spaces of mixed scales with mixed smoothness on the half-space. The aim is to solve boundary value problems with boundary data of negative regularity and to describe the singularities of solutions at the boundary. To this end, we derive mapping properties of Poisson operators in mixed scales with mixed smoothness. We also derive mathcal {R}-sectoriality results for homogeneous boundary data in the case that the smoothness in normal direction is not too large.

A Degree Theory for Lagrangian Boundary Value Problems

We study those nonlinear partial differential equations which appear as Euler-Lagrange equations of variational problems. On defining weak boundary values of solutions to such equations we initiate the theory of Lagrangian boundary value problems in spaces of appropriate smoothness. We also analyse if the concept of mapping degree of current importance applies to Lagrangian problems

Elliptic boundary value problems in spaces of continuous functions

In these notes we consider second order linear elliptic boundary value problems in the framework of different spaces on continuous functions. We appeal to a general formulation which contains some interesting particular cases as, for instance, a new class of functional spaces, called here Holog spaces and denoted by the symbol $\, C^{0,\,\lambda}_\alpha(\overline{\Omega}) \,,$ $\,0 \leq\,\lambda<\,1\,,$ and $\,\alpha \in\,\mathbb{R}\,.$ One has the following inclusions $$ C^{0,\,\lambda+\,\epsilon}(\overline{\Omega})\subset \,C^{0,\,\lambda}_\alpha(\overline{\Omega})\subset \,C^{0,\,\lambda}(\overline{\Omega}) \subset \,C^{0,\,\lambda,}_{-\alpha}(\overline{\Omega}) \subset\,C^{0,\,\lambda-\,\epsilon}(\overline{\Omega})\,, $$ for $\,\alpha>\,0\,$ ($\epsilon >\,0\,$ arbitrarily small). Roughly speaking, for each fixed $\,\lambda\,,$ the family $\, C^{0,\,\lambda}_\alpha(\overline{\Omega}) \,$ is a refinement of the single Holder classical space $\, C^{0,\,\lambda}(\overline{\Omega})=\,C^{0,\,\lambda}_0(\overline{\Omega})\,.$ On the other hand, for $\,\lambda=\,0\,$ and $\,\alpha>\,0\,,$ $\,C^{0,\,0}_\alpha(\overline{\Omega})=\,\, D^{0,\,\alpha}(\overline{\Omega})\,$ is a Log space. The more interesting feature is that, as for classical Holder (and Sobolev) spaces, full regularity occurs. namely, for each $\,\lambda>\,0\,$ and arbitrary real $\,\alpha\,,$ $\,\nabla^2\,u $ and $\,f\,$ enjoy the same $\, C^{0,\,\lambda}_\alpha(\overline{\Omega}) \,$ regularity. All the above setup is presented as part of a more general picture.

Numerical simulation of the freezing process in a moist soil

A numerical method is implemented for the solution of the unsteady three-dimensional heat transfer problem in moist soils with consideration of phase transitions in a region of subzero temperatures. The corresponding numerical model takes into account the physical nonlinearity of the soil. For the description of phase transitions in frozen soils, a model that consists of the four characteristic temperature zones where the basic thermodynamic soil parameters vary according to certain laws is used. The numerical simulation is carried out using a finite element method to discretize the boundary value problem in space and a finite difference scheme to discretize it in time. The obtained numerical solutions are compared with the analytical solutions to the Stefan problem.

The effect of geopotential perturbations of GOCE on its observations — A numerical study

In solution of gradiometric boundary value problem in space a regular grid of satellite gravity gradiometry data is required. This grid is considered on a sphere with radius of the mean Earth sphere and altitude of satellite. However, the gravitational gradients are measured by a gradiometer mounted on GOCE satellite and orbital perturbations of the satellite influence GOCE observations as well. In this study we present that these effects are about 2 E on GOCE data. Also numerical studies on the gravitational gradients in orbital frame show that the perturbations of co-latitude are more significant than that of inclination. The effect of perturbed inclination is less than −9 mE while the effect of perturbed co-latitude is within −173 mE in one day revolution of GOCE.

Positive solutions to fourth-order singular boundary value problems with integral boundary conditions in abstract spaces

Under various weaker conditions, we establish various results on the existence of positive solutions to fourth-order singular boundary value problems with integral boundary conditions in abstract spaces. The arguments are based upon a specially constructed cone and the fixed point theory in cone for a strict set contraction operator. Our results improve and extend that obtained in [X. Zhang, M. Feng, W. Ge, Existence results for nonlinear boundary value problems with integral boundary conditions in Banach spaces, Nonlinear Anal. 69 (2008) 3310–3321; Y. Liu, Multiple positive solutions to fourth-order singular boundary value problems in abstract spaces, Electron. J. Diff. Eqns. 120 (2004) 1–13], even in the scalar spaces.

Boundary value problems in spaces of distributions on smooth and polygonal domains

We study boundary value problems of the form - Δ u = f on Ω and Bu = g on the boundary ∂ Ω , with either Dirichlet or Neumann boundary conditions, where Ω is a smooth bounded domain in R n and the data f , g are distributions. This problem has to be first properly reformulated and, for practical applications, it is of crucial importance to obtain the continuity of the solution u in terms of f and g. For f = 0 , taking advantage of the fact that u is harmonic on Ω , we provide four formulations of this boundary value problem (one using nontangential limits of harmonic functions, one using Green functions, one using the Dirichlet-to-Neumann map, and a variational one); we show that these four formulations are equivalent. We provide a similar analysis for f ≠ 0 and discuss the roles of f and g, which turn to be somewhat interchangeable in the low regularity case. The weak formulation is more convenient for numerical approximation, whereas the nontangential limits definition is closer to the intuition and easier to check in concrete situations. We extend the weak formulation to polygonal domains using weighted Sobolev spaces. We also point out some new phenomena for the “concentrated loads” at the vertices in the polygonal case.

Estimates of deviations from exact solutions for boundary-value problems with incompressibility condition

Methods of estimating the difference between exact and approximate solutions are considered for boundary-value problems in spaces of solenoidal functions. The estimates obtained apply to any functions in the energy space of the respective problem, and their computation requires solving only finite-dimensional problems. In the paper, two different methods are considered: one involves variational formulations and duality theory, and in the other, estimates are obtained from the integral identities that define generalized solutions of the problems in question. It is shown that estimates of deviations from an exact solution must include an additional penalty term with a factor determined by the constant in the Ladyzhenskaya–Babuska–Brezzi condition. §

On Well-Posedness of the Nonlocal Boundary Value Problems for Elliptic Equations

The nonlocal boundary value problem in an arbitrary Banach space E with the positive operator A is considered. The well-posedness of this boundary value problem in the spaces of smooth functions is established. The new exact Schauder's estimates of solutions of the boundary value problems for elliptic equations are obtained.

Difference equations in abstract spaces

AbstractExistence results are presented for second order discrete boundary value problems in abstract spaces. Our analysis uses only Sadovskii's fixed point theorem.

Singularities in Quasi–Banach Spaces with Applications to Semilinear Elliptic Equations II

AbstractThis paper continues the investigation about the singularity theory in dual rich quasi–Banach spaces given in T. Runst [Ru 2]. The abstract results are applied to the study of the solution structure of semilinear elliptic boundary value problems in spaces of Besov – Triebel – Lizorkin type.

Higher order elliptic boundary value problems in spaces with homogeneous norms

AbstractWe consider general boundary value problems for homogeneous elliptic partial differential operators with constant coefficients. Under natural conditions on the operators, these problems give rise to isomorphisms between the appropriate spaces with homogeneous norms. We also consider operators which are not properly elliptic and boundary systems which do not satisfy the complementing condition and determine when they give rise to left or right invertible operators. A priori inequalities and regularity results for the corresponding boundary value problems in Sobolev spaces are then readily obtained.

Second order elliptic boundary value problems in spaces with homogeneous norms

AbstractWe consider the interior and Dirichiet problems and problems with first order boundary conditions, for a second order homogeneous elliptic partial differential operator with constant coefficients. Under natural conditions on the operators, these problems give rise to isomorphisms between the appropriate spaces with homogeneous norms. From there we obtain a priori estimates and regularity results for boundary value problems in Sobolev spaces.

A new technique to solve models for the reaction of solid particles and for parallel catalyst deactivation

Gas—solid non-catalytic reactions and/or parallel catalyst deactivation can often be described locally by a rate expression proportional to both solid reactant concentration (or catalyst activity) and gas reactant concentration. The reaction (deactivation) of a whole solid pellet is then represented by a coupled system of non-linear partial differential equations with prescribed boundary and initial conditions. Computations of solid conversion or the catalyst effectiveness factor by finite difference methods are often very lengthy and time consuming. In this paper the above system of partial differential equations is reduced by an integral transformation to a two-point non-linear boundary value problem in space for an ordinary differential equation. It is shown further that for a slab geometry an equivalent initial value problem in time can be formulated which allows rapid computations and solutions not only for the solid conversion and catalyst effectiveness factor but also for point gas and solid concentrations. This technique improves the computational procedure for these problems considerably. Extensions to other general rate forms and pellet geometries are presented.

Optimization of a non-linear distributed parameter system using periodic boundary control†

Conditions for loeal optimality are worked out using simple calculus of variations. To find the optimum control, a two-point boundary value problem in space and time has to be solved, which involves the solution of the adjoint differential equation together with the prooess equation. The method is applied to the optimization of a periodic process, consisting of a tubular reactor where a second-order homogeneous reaction takes place and a periodicity oondition of the state is satisfied everywhere along the reactor. Tho plug flow and the diffusion model are assumed. In the first case an exact solution is carried out. The improvements in yield compared with steady-state conditions are obtained and shown in graphs.