Asymptotic Boundary Value Problems Research Articles

Asymptotic path-independent integrals for the evaluation of crack-tip parameters in a neo-Hookean material

In this paper, we develop new asymptotic path-independent integrals for the evaluation of the crack tip parameters in a 2D neo-Hookean material. The new integrals are of both J-integral and interaction energy integral type and rely on the separation of the asymptotic boundary value problem into independent problems for each of the deformed coordinates. Both the plane stress and plane strain cases are considered. The integrals developed are used to compute the amplitude parameters of the asymptotic crack tip fields, which allows for direct extraction of these parameters from numerical results. A long strip with an edge crack under mixed loading modes is considered for both homogeneous and biomaterial cases. It is found that the asymptotic J-integrals produce good results for the first-order parameters while the interactions integrals produce good results for both the first and second-order parameters.

Topological structure of solution sets to asymptotic boundary value problems

Topological structure is investigated for second-order vector asymptotic boundary value problems. Because of indicated obstructions, the R δ -structure is firstly studied for problems on compact intervals and then, by means of the inverse limit method, on non-compact intervals. The information about the structure is furthermore employed, by virtue of a fixed-point index technique in Fréchet spaces developed by ourselves earlier, for obtaining an existence result for nonlinear asymptotic problems. Some illustrating examples are supplied.

Asymptotic boundary value problems for second-order differential systems

Topological methods are developed for the solvability of vector second-order boundary value problems on noncompact intervals. The solutions are located in given sets and enjoy prescribed properties. The main theorem is supplied by two illustrating examples.

On an Asymptotic Boundary Value Problem for Second Order Differential Equations

The existence of at least one solution to a nonlinear second order differential equation in on the semi-infinite with the first derivative vanishing at infinity is proved by using topological methods. The second boundary condition is x(0) = 0 or x′(0) = 0.

Asymptotic boundary value problems for evolution inclusions

When solving boundary value problems on infinite intervals, it is possible to use continuation principles. Some of these principles take advantage of equipping the considered function spaces with topologies of uniform convergence on compact subintervals. This makes the representing solution operators compact (or condensing), but, on the other hand, spaces equipped with such topologies become more complicated. This paper shows interesting applications that use the strength of continuation principles and also presents a possible extension of such continuation principles to partial differential inclusions.

Asymptotic boundary value problems in Banach spaces

A continuation principle is given for solving boundary value problems on arbitrary (possibly infinite) intervals to Carathéodory differential inclusions in Banach spaces. For this aim, the appropriate fixed point index is defined to condensing decomposable multivalued operators in Fréchet spaces. This index extends and unifies the one for compact maps in Andres et al. [Trans. Amer. Math. Soc. 351 (1999) 4861–4903] as well as the one for operators in Banach spaces in Bader [Ph.D. Thesis, University of Munich, 1995]. As an application, we prove the existence of an entirely bounded solution of a semilinear evolution inclusion.

Bounded solutions in a given set of differential systems

The criteria for an entirely bounded solution in a given set of a differential system with a quasi-linear part are developed sequentially via an asymptotic boundary value problem. This enables us to prove several bounded solutions separated by given functions. So, the multiplicity results can be also obtained in this way. For possible applications in epidemics and population dynamics, see e.g., the references in Gaines and Santanilla (Rocky Mountain J. Math. 12 (1982) 669–678).

Electron-scattering states at solid surfaces calculated with realistic potentials

Scattering states with low-energy-electron diffraction asymptotics are calculated for a general non-muffin-tin potential, as, e.g., for a pseudopotential with a suitable barrier and image potential part. The latter applies especially to the case of low lying conduction bands. The wave function is described with a reciprocal lattice representation parallel to the surface and a discretization of the real space perpendicular to the surface. The Schr\"odinger equation leads to a system of linear one-dimensional equations. The asymptotic boundary value problem is confined via the quantum transmitting boundary method to a finite interval. The solutions are obtained based on a multigrid technique that yields a fast and reliable algorithm. The influence of the boundary conditions, the accuracy, and the rate of convergence with several solvers are discussed. The resulting charge densities are investigated.

Asymptotic and Periodic Boundary Value Problems of Mixed FDEs and Wave Solutions of Lattice Differential Equations

We discuss the existence and approximation of solutions of asymptotic or periodic boundary value problems of mixed functional differential equations. Our approach is via monotone iteration and non-standard ordering in the profile set for asymptotic boundary value problems and viaS1-degree and equivariant bifurcation theory for periodic boundary value problems. Applications will be given to wave fronts and to slowly oscillatory spatially periodic traveling waves of lattice delay differential equations arising from population genetics, population dynamics, and neural networks.

Four-Point and Asymptotic Boundary Value Problems Via a Possible Modification of Poincaré's Mapping

Four-Point and Asymptotic Boundary Value Problems Via a Possible Modification of Poincaré's Mapping

Non-linear mass transfer in boundary layers—2. numerical investigation

The numerical solution of a class of problems for non-linear mass transfer in laminar diffusion boundary layers for a gas-liquid system is obtained utilizing a numerically stable continuation type shooting procedure for the numerical integration of asymptotic boundary value problems for systems of ordinary differential equations. The influence of gas solubility and non-linear effects in the gas on the hydrodynamics and the mass transfer have been studied. Numerical results are also reported.

Determination of asymptotic solutions of problems of nonlinear theories of thermoviscoelasticity

It is shown that the asymptotic solution of a problem of the nonlinear theory of thermoviscoelasticity, if it exists, can be found directly from the solution of the asymptotic boundary-value problem without completely solving the starting problem.