# Equations, Initial Conditions, and Boundary Conditions

Equations (12,13) and (12,14) together then provide (n+ 1) partial differential equations in the unknowns c, T. They may be solved subject Co boundary conditions specified at the pellet surface at all times, and Initial conditions specified throughout the interior of the pellet at one particular time. [c.162]

Step 4 - it is initially assumed that the flow field in the entire domain is incompressible and using the initial and boundary conditions the corresponding flow equations are solved to obtain the velocity and pressure distributions. Values of the material parameters at different regions of the domain are found via Equation (3.70) using the pseudo-density method described in Chapter 3, Section 5.1. [c.145]

The crack is said to have a zero opening in this case. As it turned out there is no singularity of the solution provided the crack has a zero opening. What this means is the solution of (3.144), (3.147), (3.148) coincides with the solution of (3.140)-(3.142) found in the domain Q with the initial and boundary conditions (3.144), (3.145) (and without (3.143)). In the last case the equations (3.141), (3.142) hold in Q. This removable singularity property is of local character. Namely, if O(x ) is a neighbourhood of the point and [c.215]

In the SMB operation, the countercurrent motion of fluid and solid is simulated with a discrete jump of injection and collection points in the same direction of the fluid phase. The SMB system is then a set of identical fixed-bed columns, connected in series. The transient SMB model equations are summarized below, with initial and boundary conditions, and the necessary mass balances at the nodes between each column. [c.223]

The initial and boundary conditions are the same presented before and, for x = 0 (7 = 0), Equation (9.4) becomes [c.224]

The initial and boundary conditions are the same presented before and, for x = 0, Equation (20) becomes [c.227]

Solutions for this second-order differential equation are known for a number of initial and boundary conditions [4]. [c.163]

Smoluchowski theory [29, 30] and its modifications fonu the basis of most approaches used to interpret bimolecular rate constants obtained from chemical kinetics experiments in tenus of difhision effects [31]. The Smoluchowski model is based on Brownian motion theory underlying the phenomenological difhision equation in the absence of external forces. In the standard picture, one considers a dilute fluid solution of reactants A and B with [A] [B] and asks for the time evolution of [B] in the vicinity of A, i.e. of the density distribution p(r,t) = [B](rl)/[B] 2i ] r(t))l ] Q ([B] is assumed not to change appreciably during the reaction). The initial distribution and the outer and inner boundary conditions are chosen, respectively, as [c.843]

However, it is important to make sure that satisfies the desired boundary conditions initially and finally. Part of this is familiar already, since we have already demonstrated in equation (A3.11.3k equation (A3. IPS ) [c.961]

Optical conduits as described above are generally not practical. The most common waveguide is tire slab dielectric waveguide. In tliese devices, a high-transmission material is surrounded by a media of a lower refractive index. The light is guided into tire device by total internal reflection. The basic stmcture is shown in figure C2.15.11. As witli our initial description, light rays making an angle 0 with tlie z-axis experience multiple total internal reflections at tlie interfaces, provided tliat 0 is smaller than the complement of the critical angle. The slab boundaries define all of tlie properties of tlie guide. As before, rays making angles larger tlian tlie complement of tlie critical angle refract, losing a fraction of tlieir optical power at each reflection and eventually vanishing (tlie unguided waves of figure C2.15.11). The detailed analysis of tlie waveguide modes requires a full solution of Maxwell s equations botli inside and outside tlie high-index core witli tlie appropriate boundary conditions. Such an analysis, which is beyond tlie scope of tliis review, can be found in [1, 23]. We will summarize tlie results here. As witli our first analysis, a twice-reflected wave undergoes a phase shift tliat must be zero or a multiple of 2it to be self-consistent. The number of TE modes allowed is [c.2866]

The equilibrium problem for a plate is formulated as some variational inequality. In this case equations (3.92)-(3.94) hold, generally speaking, only in the distribution sense. Alongside (3.95), other boundary conditions hold on the boundary F the form of these conditions is clarified in Section 3.3.3. To derive them, we require the existence of a smooth solution to the variational inequality in question. On the other hand, if we assume that a solution to (3.92)-(3.94) is sufficiently smooth, then the variational inequality is a consequence of equations (3.92)-(3.94) and the initial and boundary conditions. All these questions are discussed in Section 3.3.3. In Section 3.3.2 we prove an existence theorem for a solution to the variational equation and in Section 3.3.4 we establish some enhanced regularity properties for the solution near F. [c.200]

It is noteworthy that the original equilibrium problem for a plate with a crack can be stated twofold. On the one hand, it may be formulated as variational inequality (3.98). In this case all the above-derived boundary conditions are formal consequences of such a statement under the supposition of sufficient smoothness of a solution. On the other hand, the problem may be formulated as equations (3.92)-(3.94) given initial and boundary conditions (3.95)-(3.97) and (3.118)-(3.122). Furthermore, if we assume that a solution is sufficiently smooth then from (3.92)-(3.97) and (3.118)-(3.122) we can derive variational inequality (3.98). [c.208]

The considered problem is formulated as a variational inequality. In general, the equations (3.140)-(3.142) hold in the sense of distributions. In addition to (3.143), complementary boundary conditions will be fulfilled on F, X (0,T). The exact form of these conditions is given at the end of the section. The assumption as to sufficient solution regularity requires the variational inequality to be a corollary of (3.140)-(3.142), the initial and all boundary conditions. The relationship between these two problem formulations is discussed in Section 3.4.4. We prove an existence of the solution in Section 3.4.2. In Section 3.4.3 the main result of the section concerned with the cracks of minimal opening is established. [c.212]

In this case all the written boundary conditions (i.e. (3.166)-(3.169)) are the corollary of this formulation. Of course, in deriving (3.166)-(3.169) we should assume an additional regularity of the solution. On the other hand, the problem admits the formulation in the form of equations (3.140)-(3.142) with the initial and boundary conditions (3.143)-(3.145), (3.166)-(3.169). If a solution of the last boundary problem is smooth enough the formulation (3.144), (3.147), (3.148) follows from (3.140)-(3.145), (3.166)-(3.169). In fact, let (If, tZ)) be a smooth function belonging to K. We multiply equations (3.141), (3.142) taken for a fixed t G (0,T) by tZ) — w t), w — w t), respectively. We next integrate the relations over By using the Green formulae like (3.151), (3.152) and by [c.219]

The term operational method implies a procedure of solving differential and difference equations by which the boundary or initial conditions are automatically satisfied in the course of the solution. The technique offers a veiy powerful tool in the applications of mathematics, but it is hmited to linear problems. [c.462]

Example Consider the diffusion equation, with boundary and initial conditions. [c.479]

It is plain from the description above that the boundary between crack initiation and crack growth is not clear cut. Indeed many would regard the distinction as semantic, or state that most of the fatigue life is spent in crack propagation, however small those cracks might be. Nevertheless, the proportion of cyclic life occupied by the various stages can vary greatly with metallurgical structure, magnitude of the applied cyclic and mean stress, geometry and environment. Only stage II of the growth process (Fig. 8.61) can be properly characterised in terms of the linear elastic parameter, the cyclic range of the crack-tip stress-intensity factor, Even this is subject to the conditions that the crack-tip plasticity be contained within an elastic continuum and that the crack is large compared to microstructural dimensions. When these conditions are satisfied and the environment is benign, the familiar Paris equation can characterise the crack growth rate per cycle, da/dN, over a wide range of AK (Fig. 8.59). [c.1291]

In a potential-step experiment, the potential of the working electrode is instantaneously stepped from a value where no reaction occurs to a value where the electrode reaction under investigation takes place and the current versus time (cluonoamperometry) or the charge versus time (cluonocoulometry) response is recorded. The transient obtained depends upon the potential applied and whether it is stepped into a diffrision control, in an electron transfer control or in a mixed control region. Under diffusion control the transient may be described by the Cottrell equation obtained by solving Pick s second law witii the appropriate initial and boundary conditions [1, 2, 3, 4, 5 and 6] [c.1929]

The solutions of such partial differential equations require infomiation on the spatial boundary conditions and initial conditions. Suppose we have an infinite system in which the concentration flucPiations vanish at the infinite boundary. If, at t = 0 we have a flucPiation at origin 5C(f,0) = AC (f), then the diflfiision equation [c.721]

This treatment of reaction at the limit of bulk diffusion control is essentially the same as that presented by HugoC 69j. It is attractive computationally, since only a single two-point boundary value problem must be solved, namely that posed by equations (11.15) and conditions (11.16). This must be re-solved each time the size of the pellet is changed, since the pellet radius a appears in the boundary conditions. However, the initial value problem for equations (11.12) need be solved only once as a preliminary to solving (11.15) and (11.16) for any number of different pellet sizes. [c.117]

In accordance with boundary conditions (3.95), (3.121), and (3.122), the sum of the boundary integrals here is nonpositive, whence (3.108) ensues. From (3.108) we infer (3.100). Equation (3.92), together with initial and boundary conditions (3.96)-(3.97) and (3.118), leads to (3.99). As we noted many times, (3.99) and (3.100) imply (3.98), which proves the assertion that (3.92)-(3.97) and (3.118)-(3.122) yield (3.98). [c.208]

Onsager Model. The theory developed by Onsager (72) has been the standard model to use for analy2ing the electric field dependence of the charge-generation efficiency. The model solves the diffusion equation of the relative motion of an electron—hole pair, bounded by thek Coulomb interaction, under an electric field. The origin of the electron—hole pak and the pathway by which it is generated are not considered in this model. The model solves for the probabiUty that the pak separates toward infinity with a given initial separation distance, Tq. An important boundary condition and assumption for this model is that if the pak separation distance reaches 2ero, the pak recombines immediately. With this assumption, the charge-generation efficiency, ( )(rQ, E), in the presence of an electric field, E, is given by the following (72,73) [c.414]

Experience in the solution of these governing equations is extensive, primarily because they are used in weather forecasting. This experience has led to useful simplifications and to the realization that the system is very sensitive to initial conditions. The sensitivity to initial conditions leads to the temporal amplification of any errors in the initial or boundary conditions, or of computational errors. The growth of these errors seriously limits the period of time that can be simulated for use in air pollution studies unless a separate mechanism is used to dampen error growth. This same limitation makes longer range weather forecasts increasingly uncertain. A method that has been developed to reduce the growth of errors arising from initial and boundary conditions when reconstmcting historical conditions is to use observations to adjust the solution back towards the actual. This is referred to as "Newtonian nudging" or four-dimensional data assimilation (FDDA) (73). In essence, the predicted solution from the dynamic model is averaged, using various weightings, and the meteorological fields developed by objective analysis. [c.384]

The solution to this equation requites an initial condition (n 2itt = 0) and a boundary condition (n at a specific value of T). Assuming that crystals are formed at zero size gives the boundary condition [c.355]

These equations form a fourth-order system of differential equations which cannot be solved analytically in almost all interesting nonseparable cases. Further, according to these equations, the particle slides from the hump of the upside-down potential — V(Q) (see fig. 24), and, unless the initial conditions are specially chosen, it exercises an infinite aperiodic motion. In other words, the instanton trajectory with the required periodic boundary conditions, [c.60]

The MD simulations have been carried out in the microcanonical ensemble and the system was made up of 2560, 2800 and 2880 Cu particles arranged on a fee lattice with (100), (110), and (111) faces respectively. Periodic boundary conditions were imposed in the three space directions. The free surfaces were produced by fixing the dimensions of the simulation box at a value twice as large as the thickness of the crystal along the z-direction an infinite slab was thus produced delimited by two free surfaces. The equations of motion were integrated by means of the Verlet algorithm and a time step 6t=5xl0 s. With this integration step the total energy was conserved within 6E/E=10 . Initially, 5000 time steps were used to reach thermal equilibrium, while another 10000 steps were subsequently taken to calculate time-averages. [c.152]

The first equation in equation 9.16 is a three-dimensional nonlinear partial differential equation for the velocity field v. In general, due to its nonlinearity, it is extremely difficult to solve. While it is true that for small velocities, the Navier-Stokes equations can be linearized and, at least when the geometry of boundaries and/or obstacles is relatively simple, even be solved analytically, as velocities become larger, instabilities appear and make it increasingly difficult to obtain any kind of analytical results. Indeed, for some problems, it is nontrivial to obtain even numerical approximations. Since turbulent flow typically involves a complicated nesting of eddies and vortices (see below), literally ranging in scale from the macroscopic to the molecular, part of the difficulty stems from the need to simultaneously take into account these widely differing length scales, and therefore to use either extremely small or variable-sized meshes for numerical finite-difference calculations. On the purely theoretical side, it is also worth observing that while, for three-dimensions, it is known that smooth solutions depending continuously on the initial conditions exist for a short time, it remains an open problem as to whether solutions of the incompressible Navier-Stokes equations actually exist for all time [tem77]. [c.468]

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**Equations, Initial Conditions, and Boundary Conditions**:

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