Initial-boundary-value problems

It should be noted that the normality conditions, arising from the work assumption applied to inelastic loading, ensure the existence and uniqueness of solutions to initial/boundary value problems for inelastic materials undergoing small deformations. Uniqueness of solutions is not always desirable, however. Inelastic deformations often lead to instabilities such as localized deformations. It is quite possible that the work assumption, which is essentially a stability postulate, is too strong in these cases. Normality is a necessary condition for the work assumption. Instabilities, while they may occur in real deformations, are therefore likely to be associated with loss of normality and violation of the work assumption. 

We solved the transient hydrogen diffusion initial/boundary-value problem coupled with the large strain elastoplastic boundary value problem for a pipe of an outer diameter 40.64 cm, wall thickness h = 9.52 mm, and with an axial crack of depth 0.2/i on the inner wall-surface. We obtained the solutions under hydrogen gas pressure of 15 MPa, material properties for an X70/80 type steel, and... 

K. S. Yee, Numerical solution of initial boundary value problems involving Maxwell s equations in isotropic media, IEEE Transactions on Antennas and Propagation 14, 302-307 (1966). 

Figure E.l represents a highly simplified view of an ideal structure for an application program. The boxes with the rounded borders represent those functions that are problem specific, while the square-comer boxes represent those functions that can be relegated to problem-independent software. This structure is well-suited to problems that are mathematically systems of nonlinear algebraic equations, ordinary differential equation initiator boundary-value problems, or parabolic partial differential equations. In these cases the problem-independent mathematical software is usually written in the form of a subroutine that in turn calls a user-supplied subroutine to define the system of equations. Of course, the analyst must write the subroutine that describes the particular system of equations. Moreover, for most numerical-solution algorithms, the system of equations must be written in a discrete form (e.g., a finite-volume representation). However, the equation-defining sub-...

This transformation allows for equal distribution in the y-space while concentrating the lines close to the x = L boundary. Parameter a sets the spacing of the lines. This technique is called MOL1D (Method Of Lines in 1 Dimension) and is suitable for solving parabolic and hyperbolic initial boundary value problems in one dimension. 

Using difference quotients, the partial differential equations become ordinary differential equations. The boundary conditions (e.g., inlet and outlet gas temperature of the fluidized bed) can easily be implemented using difference quotient in the entire differential equation system. So, initial/boundary value problems are transferred into initial value problems. Now, the ordinary differential equations of order 1... 

These mathematical representations are complex and it is necessary to use numerical techniques for the solution of the initial-boundary value problems associated with the descriptions of fluidized bed gasification. The numerical model is based on finite difference techniques. A detailed description of this model is presented in (11-14). With this model there is a degree of flexibility in the representation of geometric surfaces and hence the code can be used to model rather arbitrary reactor geometries appropriate to the systems of interest. [The model includes both two-dimensional planar and... 

In a recent work [42], Renardy characterizes a set of inflow boundary conditions which leads to a locally well-posed initial boundary value problem for the two-dimensional flow of an upper-convected Maxwell fluid transverse to a domain bounded by parallel planes. 

A typical result of numerical analysis is an estimate of the error U — Uh between the solution U of the continuous problem (. e., the solution of the initial boundary value problem) and the solution Uh of the discrete problem (also called approximate problem). In what follows the error estimates are obtained with the assumption that U is sufficiently regular. In many realistic situations the geometry of the flow has singularities (corners for example), the solution U is not regular, and these results do not apply. (As a matter of fact existence of a solution has not been shown yet in those singular situations.)... 

M. Renardy, Local existence of the Dirichlet initial boundary value problem for incompressible hypoelcistic materials, SIAM J. Math. Anal., 21 (1990) 1369-1385. 

Caracotsios, M., and W. E. Stewart, Sensitivity analysis of initial-boundary-value problems with mixed PDEs and algebraic equations. Applications to chemical and biochemical systems, Comput. Chem. Eng., 19, 1019-1030 (1995). 

Two methods are available for the numerical solution of initial-boundary-value problems, the finite difference method and the finite element method. Finite difference methods are easy to handle and require little mathematical effort. In contrast the finite element method, which is principally applied in solid and structure mechanics, has much higher mathematical demands, it is however very flexible. In particular, for complicated geometries it can be well suited to the problem, and for such cases should always be used in preference to the finite difference method. We will limit ourselves to an introductory illustration of the difference method, which can be recommended even to beginners as a good tool for solving heat conduction problems. The application of the finite element method to these problems has been described in detail by G.E. Myers [2.52]. Further information can be found in D. Marsal [2.53] and in the standard works [2.54] to [2.56]. 

The well-posedness of the two-fluid model has been a source of controversy reflected by the large number of papers on this issue that can be found in the literature. This issue is linked with analysis of the characteristics, stability and wavelength phenomena in multi -phase flow equation systems. The controversy originates primarily from the fact that with the present level of knowledge, there is no general way to determine whether the 3D multi-fluid model is well posed as an initial-boundary value problem. The mathematical theory of well posedness for systems of partial differential equations describing dispersed chemical reacting flows needs to be examined. 

The principle of well-posedness states that the description of the motion should be such that a solution to the initial boundary value problem exists and depends continuously on the initial and boundary conditions. 

K. S. Yee, Numerical solution of initial boundary value problems involving Maxwell s... 

DuhameTs principle helps to represent solutions to inhomogeneous initial boundary-value problems in terms of the solutions to homogeneous problems. The solution of the inhomogeneous equation. 

The temperature distribution in a semi-infinite rod with one end of the rod kept at the prescribed temperature is described by the solution of the following initial-boundary-value problem ... 

The system of partial differential equations of first order, Eqs. (44), usually has to be treated as an initial-boundary-value problem on an appropriate energy region 0 < U < U°° and for times t > 0, where the time represents the evolution direction of the kinetic problem. Initial values for each of the distributions fo(U, i) and/,( /, t), suitable for the problem under consideration, have to be fixed, for example at t = 0. Appropriate boundary conditions for the system are given by the requirements /o([7 > U°°, t) = 0 and /,(0, t) = 0. 

The numerical solution of the initial-boundary-value problem based on the equation system (44) can be performed (Winkler et al, 1995) by applying a finite-difference method to an equidistant grid in energy U and time t. The discrete form of the equation system (44) is obtained using, on the rectangular grid, second-order-correct centered difference analogues for both distributions f iU, i)/n and f U, t)/n and their partial derivatives of first order. 

As detailed below, the parabolic equation, Eq. (54), describes the evolution of the isotropic distribution and has to be solved as an initial-boundary-value problem on a nonrectangular solution region whose boundaries are partly determined by the spatial course of the electric field and thus by the specific kinetic problem considered. The parabolic problem has to be completed by appropriate initial and boundary conditions, which are briefly described below. 

Basic aspects of the spatial relaxation of the electrons in collision-dominated plasmas can be revealed when the evolution of the electrons whose velocity distribution has been disturbed at a certain space position is studied under the action of a space-independent electric field (Sigeneger and Winkler, 1997a Sigeneger and Winkler, 1997b). Sufficiently far from this position in the field acceleration direction of the electrons, a uniform state finally becomes established. Such relaxation problems can be analyzed on the basis of the parabolic equation for the isotropic distribution, Eq. (54), when the initial-boundary-value problem is adopted to the relaxation model. 

The numerical solution of Eq. (54) as an initial-boundary-value problem, specified to the spatial relaxation problem in uniform electric fields, can be obtained (Sigeneger and Winkler, 1996) by using a finite-difference approach according to the well-known Crank Nicholson scheme for parabolic equations. 

Abstract This contribution deals with the modeling of coupled thermal (T), hydraulic (H) and mechanical (M) processes in subsurface structures or barrier systems. We assume a system of three phases a deformable fractured porous medium fully or partially saturated with liquid and a gas which remains at atmospheric pressure. Consideration of the thermal flow problem leads to an extensively coupled problem consisting of an elliptic and parabolic-hyperbolic set of partial differential equations. The resulting initial boundary value problems are outlined. Their finite element representation and the required solving algorithms and control options for the coupled processes are implemented using object-oriented programming in the finite element code RockFlow/RockMech. 

Additionally, a heat output curve, boundary and initial conditions are prescribed in order to complete the description of the initial boundary value problem to be solved. 

The mathematical modelling of the T-H-M phenomena uses initial - boundary value problems for differential or variational equations involving the physical principles. We shall assume that these problems are discretized by the finite element or similar methods. What we want to point out is that the numerical solution can be computationally very expensive due to... 

The initial-boundary value problem of thermoelasticity can be discretized by the finite elements in space and the finite differences in time. Using the linear finite elements and the simplest time discretization, we come to the computation of... 

It is shown that the development of the equations governing THM processes in elastic media with double porosity can be approached in a systematic manner where all the constitutive equations governing deformability, fluid flow and heat transfer are combined with the relevant conservation laws. The double porosity nature of the medium requires the introduction of dependent variables applicable to the deformable solid, and the fluid phases in the two void spaces. The governing partial differential equations are linear in view of the linearized forms of the constitutive assumptions invoked in the formulations. The linearity of these governing equations makes them amenable to solution through conventional mathematical techniques applicable to the study of initial boundary value problems in mathematical physics (Selvadurai, 2000). Such solutions should serve as benchmarks for appropriate computational developments. 

The above dispersion relation exhibits the existence of curious waves, such that the group velocity is positive but the phase velocity is negative and the amplitude increases in the direction of the phase velocity. Thus, it is difficult to predict what waves are generated by imposing disturbance and how they propagate subsequently. Then, we return to the original differential equation and we consider the initial-boundary value problem, where a disturbance with a certain frequency is imposed at a location in the uniform flow and from an initial instant and to a specified direction. 

Davis, S. F., Flaherty, J. E., An Adaptive Finite Element Method for Initial-Boundary Value Problems for Parabolic Equation, SIAM J. Sci. Stat. Comput., 3, (1982), 6-27... 

Considering chemical application problems, a large number of them yields mathematical models that consist of initial-value problems (IVPs) for ordinary differential equations (ODEs) or of initial-boundary-value problems (IBVPs) for partial differential equations (PDEs). Special problems of this kind, which we have treated, are diffusion-reaction processes in chemical kinetics (various polymerizations), polyreactions in microgravity environment (photoinitiated polymerization with laser beams) and drying procedures of hygroscopic porous media. 

The initial-boundary value problem represented by Eq. 7.1 can be transmuted into a boundary integral equation by several different methods. Brebbia and Walker (1980) and Curran et al. (1980) approximated the time derivative in the equation in a finite difference form, thus changing the original parabolic partial differential equation to an elliptical partial differential equation, for which the standard boundary integral equation may be established. 

The structural models and the aforementioned additional loading of the plate and the beams are shown in Fig. 3. On the base of the above considerations the response of the plate and of the beams may be described by the following initial boundary value problems. 

The following hyperbolic, initial boundary-value problem serves as a starting point for the subsequent developments ... 

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