Problems with singular boundaries

In this section we define trace spaces at boundaries and consider Green s formulae. The statements formulated are applied to boundary value problems for solids with cracks provided that inequality type boundary conditions hold at the crack faces. 

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Submitting the main topic, we deal with models of solids with cracks. These models of mechanics and geophysics describe the stationary and quasi-stationary deformation of elastic and inelastic solid bodies having cracks and cuts. The corresponding mathematical models are reduced to boundary value problems for domains with singular boundaries. We shall use, if it is possible, a variational formulation of the problems to apply methods of convex analysis. It is of importance to note the significance of restrictions stated a priori at the crack surfaces. We assume that nonpenetration conditions of inequality type at the crack surfaces are fulfilled, which improves the accuracy of these models for contact problems. We also include the modelling of problems with friction between the crack surfaces. 

The differential equations (7.138) and (7.139) for the top-fired furnace, however, describe a boundary value problem with the boundary conditions (7.140) and (7.141) that can be solved via MATLAB s built-in BVP solver bvp4c that uses the collocation method, or via our modified BVP solver bvp4cf singhouseqr. m which can deal with singular Jacobian search matrices referred to in Chapter 5. On the other hand, the differential equation (7.142) is a simple first-order IVP. 

Thus, we see that the newly constructed finite difference schemes are indeed effective and that they allow us to approximate the solution and the normalized diffusion fluxes g-uniformly for both Dirichlet and Neumann boundary value problems with singular perturbations. 

Whether one gets problems with singularities depends on the boundary conditions of uq and u for the K-matrix ... 

To calculate the profiles and the differential capacitance of the interface numerically we have to choose a differential equation solver. However, the usual packages require that the problem is posed on a finite interval rather than on a semi-infinite interval as in our problem. In principle, we can transform the semi-infinite interval into a finite one, but the price to pay is a loss of translational invariance of the equations and the point mapped from that at infinity is singular, which may pose a problem on the solver. Most of the solvers are designed for initial-value problems while in our case we deal with a boundary-value problem. To circumvent these inconveniences we follow a procedure strongly influenced by the Lie group description. 

If A is not a square matrix and we command A b in MATLAB, then the SVD is invoked and finds the least squares solution to the minimization problem min., Ax — b. A slight variant that uses only the QR factorization mentioned in subsection (F) for a singular but square system matrix A Rn,n is used inside our modified boundary value solver bvp4cf singhouseqr. m in Chapter 5 in order to deal successfully with singular Jacobian matrices inside its embedded Newton iteration. 

Let us now consider the same flow domain (represented in Fig. l.a) with the boundary conditions of vanishing velocity on F] and F2 (the fluid is sticking at the wall on Fi and F2). This problem too has been largely studied for a Newtonian fluid. In this case, singular solutions of the homogeneous Stokes problem exist if a is a solution of the following equation (14) 5in(am) 2 sin((0) 2 aof) 0)... 

A more convenient, but entirely equivalent, problem for analysis is to consider the position and shape of the interface to be fixed, with the boundary translating at a velocity U. If we calculate the velocity and pressure fields for an incompressible, Newtonian fluid, assuming no-slip at the solid wall and the kinematic plus no-slip conditions at the interface, we find that the tangential stress component on the boundary exhibits a nonintegrable singularity as the distance to the contact point goes to zero, i.e.,... 

In the previous sections we have seen several examples of transport problems that are amenable to analysis by the method of regular perturbation theory. As we shall see later in this book, however, most transport problems require the use of singular-perturbation methods. The high-frequency limit of flow in a tube with a periodic pressure oscillation provided one example, which was illustrative of the most common type of singular-perturbation problem involving a boundary layer near the tube wall. Here we consider another example in which there is a boundary-layer structure that we can analyze by using the method of matched asymptotic expansions. 

The exact bounds for multiplicity can be determined by computationally solving the boundary-value problems with the use of the singularity theory. 

SINGULARLY PERTURBED PROBLEMS WITH BOUNDARY AND INTERIOR LAYERS THEORY AND APPLICATION ... 

This system and other problems for singularly perturbed ordinary differential equations will be investigated in Sections II-V. Solutions with boundary and/or interior layers will be considered. Our main goal will be the construction of an approximation to the solution valid outside the boundary (interior) layer as well as within the boundary (interior) layer, that is, so-called uniform approximation in the entire t domain. This approximation will have an asymptotic character. The definition of an asymptotic approximation with respect to a small parameter will be introduced in Section LB. 

It is known that, in the case of singularly perturbed elliptic equations for which (as the parameter s equals zero) the equation does not contain any derivatives with respect to the space variable, the principal term in the singular part of the solution is described by an ordinary differential equation similar to Eq. (1.16a) (see, e.g., [3-6]). Thus, it can be expected that, when solving singularly perturbed elliptic and parabolic equations using classical difference schemes, one faces computational problems similar to the computational problems for the boundary value problem (1.16). 

Using the example of a boundary value problem for a singularly perturbed ordinary differential equation with Neumann boundary conditions, we discuss some principles of constructing special finite difference schemes. These principles will be used in Section III.D to construct special finite difference schemes for singularly perturbed equations of the parabolic type. 

Many processes of heat and mass transfer, for example, fast-running processes, lead to the investigation of singularly perturbed boundary value problems with a perturbation parameter e. For example, those problems arise in the analysis of heat and mass transfer for mechanical working of materials, in particular, metals. The use of classical methods for the numerical solution of such problems (see, e.g., [1,11,12]) leads us... 

The construction and investigation of special difference schemes for a particular singularly perturbed boundary value problem with a discontinuous initial condition were examined in [16-19]. 

The analysis of heat exchange processes, in the case of the plastic shear of a material, leads us to singularly perturbed boundary value problems with a concentrated source. Problems such as these were considered in Section IV, where it was shown that classical difference schemes give rise to errors, which exceed the exact solution by many orders of magnitude if the perturbation parameter is sufficiently small. Besides, a special finite difference scheme, which allows us to approximate both the solution and... 

Problem (5.35a) is a singularly perturbed boundary value problem with complicated conditions of exchange on the boundaries of the subdomains. We want to find the solution of the boundary value problem (5.35a) and also the quantity... 

The special finite difference schemes constructed here allow one to approximate solutions of boundary value problems and also normalized di sion fluxes. They can be used to solve effectively applied problems with boundary and interior layers, in particular, equations with discontinuous coefficients and concentrated factors (heat capacity, sources, and so on). Methods for the construction of the special schemes developed here can be used to construct and investigate special schemes for more general singularly perturbed boundary value problems (see, e.g., [4, 17, 18, 24, 35-39]). 

P.A. Farrell, P. W. Hemker, and G. 1. Shishkin, Discrete approximation for a singularly perturbed boundary value problem with parabolic layers, Stichting Mathematisch Centmm, Amsterdam, Report NM-R9502, February 1995. 

G. I. Shishkin, Finite difference approximations for singularly perturbed boundary value problems with diffusion layers. Dublin, Ireland, INCA Preprint No. 2, 1994. 

Singularly Perturbed Problems with Boundary and Interior Layers Theory and Application... 

In [157] the authors present an initial-value methodology for the numerical approximation of quasilinear singularly perturbed two point boundary value problems in ordinary differential equations. These problems have a boundary layer at one end (left or right) point. The techniaque which used by the authors is to reduce the original problem to an asymptotically equivalent first order initial-value problem. This is done with the... 

Equations (25)-(27) along with the boundary conditions (29)-(32) must be solved subject to initial conditions at = 0 for the velocity field (which should be solenoidal for consistency with (27)) and the surface surfactant concentration F(z, 0). As can be seen the problem is difficult due to the nonlinear coupling present. An additional difficulty which is of interest here, is the possibility of jet pinching which manifests itself as a finite-time singularity of the system we will describe how results from the analysis of such events using asymptotic methods can be used in practical applications. Finally, note that if F = 0, we have the case of clean interfaces with constant surface tension. 

In the present paper the fracture is treated as a real crack with infinite small distance between sides. For the solution of the elasticity problem with the cavity and the fracture, the modification of the Dual BEM with discontinuous elements is built [4]. It is the most optimal method with regard to the computational costs and the convenience of the integral equations approximation. Near the crack front special elements are used. They account the singularity of the elasticity problem solution. To improve the accuracy of the Stress Intensity Factors calculation, the special boundary elements near the crack front are accounted in the interpolation formulae. 

Local activism continued. The health hazards were neither Dynamit Nobel s in-house problems nor singular cases to be dealt with by occupational health and safety administration alone. Activists shifted the boundaries of the workplace beyond the factory to the city council, by pointing out that the workers of Dynamit Nobel were part of Troisdorf s citizenry ... 

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