Solution of the boundary value problem

we imagine that there are fictitious masses at points of the surface S and they are distributed with surface density a in such way that the potential of their field is equal T. 

in accordance with the Newton s law of attraction this potential is 

One can say that we have expressed the disturbing potential in terms of an unknown density. Now we demonstrate that this transition is justified because it is possible to obtain the integral equation with respect to a. In Chapter 1, it was shown that the discontinuity of the normal components of the field at both sides of the surface masses is equal to —2nka. Correspondingly, we have 

Here the first and second terms describe the field on different sides of the surface and a is the angle between the direction of the v line and the normal to the surface S. It is obvious that the function T, given by Equation (2.302), is harmonic and regular at infinity. Next, we have to choose among an infinite number of such functions T one which also obeys the boundary condition on S. The last equation allows us to rewrite Equation (2.301) in the form 

we have derived the integral equation with respect to the function a. As in the case of Stokes problem it is possible to apply the spherical approximation, that is, the magnitude of the normal field at points of the surface S is 

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Difference Green s function. Further estimation of a solution of the boundary-value problem for a second-order difference equation will involve its representation in terms of Green s function. The boundary-value problem for the differential equation... 

VVe say that a scheme is stable with respect to coefficients (costable) if a solution of the boundary-value problem has slight variations under small perturbations of the scheme coefficients. In order to avoid misunderstanding, we focus our attention on the scheme with coefficients a, d, ip... 

Determine the exact solutions of the boundary-value problem... 

However, due to the maximum principle a solution of the boundary-value problem (29)-(30) is non-negative ... 

As we already know a determination of the function G q, p) satisfying all these conditions represents a solution of the boundary value problem and in accordance with the theorem of uniqueness these conditions uniquely define the function G q, p). In general, a solution of this problem is a complicated task, but there are exceptions, including the important case of the plane surface Sq, when it is very simple to find the Green s function. Let us introduce the point s, which is the mirror reflection of the point p with respect to the plane of the earth s surface, Fig. 1.10, and consider the function G (p, q,. s) equal to... 

Earlier we solved the boundary value problem for the spheroid of rotation and found the potential of the gravitational field outside the masses provided that the outer surface is an equipotential surface. Bearing in mind that, we study the distribution of the normal part of the field on the earth s surface, where the position of points is often characterized by spherical coordinates, it is natural also to represent the potential of this field in terms of Legendre s functions. This task can be accomplished in two ways. The first one is based on a solution of the boundary value problem and its expansion into a series of Legendre s functions. We will use the second approach and proceed from the known formula, (Chapter 1) which in fact originated from Legendre s functions... 

Earlier we found the equation for the outer surface of the ideal Earth, proceeding from a solution of the boundary value problem. It is useful also to make use of Equation (2.228) and to derive the same result. To simplify derivations we... 

Solution by Shooting Solution of the boundary value problem described by Eq. 6.59 is usually accomplished numerically by a shooting method. To implement a shooting method, the third-order equations is transformed to a system of three first-order equations as... 

It can be seen that the solution of the problem of the energy-optimal guiding of the system from a chaotic attractor to another coexisting attractor requires the solution of the boundary-value problem (33)-(34) for the Hamiltonian dynamics. The difficulty in solving these problems stems from the complexity of the system dynamics near a CA and is related, in particular, to the delicate problems of the uniqueness of the solution, its behaviour near a CA, and the boundary conditions at a CA. 

Figure 18. The most probable escape path (bottom solid curve) from S5 to the SI, found in the numerical simulations. The stable limit cycle is shown by rombs see Fig. 16 for other symbols. Parameters were h = 0.13, cty = 0.95,o>o 0.597,D = 0.0005. Top optimal force (solid line) corresponding to the optimal path after filtration [169]. The optimal path and optimal force from numerical solution of the boundary-value problem are shown by dots.

The composition boundary values entering into Eqs. (All) represent external values for Eqs. (A10). With some further assumptions concerning the diffusion and reaction terms, this allows an analytical solution of the boundary-value problem [Eqs. (A10) and (All)] in a closed matrix form (see Refs. 58 and 135). On the other hand, the boundary values need to be determined from the total system of equations describing the process. The bulk values in both phases are found from the balance relations, Eqs. (Al) and (A2). The interfacial liquid-phase concentrations xj are related to the relevant concentrations of the second fluid phase, y , by the thermodynamic equilibrium relationships and by the continuity condition for the molar fluxes at the interface (57,135). 

Time domain electromagnetic (EM) migration is based on downward extrapolation of the residual field in reverse time. In this section I will show that electromagnetic migration, as the solution of the boundary value problem for the adjoint Maxwell s equation, can be clearly associated with solution of the inverse problem in the time domain. In particular, I will demonstrate that the gradient of the residual field energy flow functional with respect to the perturbation of the model conductivity is equal to the vector cross-correlation function between the predicted field for the given... 

The definition of the electromagnetic migration field in time domain was introduced in the monograph by Zhdanov (1988). According to this definition, the migration field is the solution of the boundary value problem for the adjoint Maxwell s equations. For example, we can introduce the migration anomalous field E ", H " as the field, determined in reverse time t = —t, whose tangential components are equal to the anomalous field in reverse time at the surface of the earth S... 

The solution of the boundary value problem (11.69) and (11.70) for the concentration equation can be obtained with the aid of Green s tensor formula (F.IO) (see Appendix F). We assume that the volume V is bounded by the surface S, which... 

The eigenfunction F associated with the eigenvalue Hi has exactly (i — 1) zero points between the boundaries, or in other words in the interval (a,b). These properties will now be used in the following solutions of the boundary value problems for a plate, a cylinder and a sphere. 

The idea of using functions of the form frjr for approximate solutions of the boundary value problems has been used in a large number of papers. For example, this idea was used in [70] for the one-dimensional Schrodinger equation to prove the Hull-Julius relation (4.21) and to study the confined IlJ molecule. The direct use of the Kirkwood-Buckingham relation for variational calculations of the hydrogen atom in a half space was... 

On account of the boundary conditions /f(0) = 0 and ip l) = 0 (string of length I with ends fixed) the cosine vibration drops out at once as being inconsistent with the first condition. But even the sine vibration is not a solution of the boundary value problem, unless l /X is an exact integral multiple of tt, so that ifj vanishes when x = 1. It is only for definite values of A (the proper values) that we obtain possible forms of vibration these are given in fact by... 

In general, for arbitrary kinetics, a numerical solution of the balance equation taking into account the boundary conditions is necessary. From the obtained concentration profiles one is able to obtain the effectiveness factor. This is completely feasible with the tools of the modern computing technology. Analytical and semi-analytical expressions for the effectiveness factor, rji, are, however, always favoured if they are available, since the numerical solution of the boundary value problem is not a trivial task. The solutions for different types of Langmuir-Hinshelwood kinetics were presented in the literature, for instance by R. Aris and P. Schneider. 

Eq. (5.248) and its modification for a deformed surface [154], together with the corresponding equations for AT [152] and Eq. (5.243) are the only analytical results obtained as solution of the boundary value problem for the diffusion equations of micelles and monomers. An approximate relation for Ay can be also obtained without integration of the diffusion equations with the help of the penetration theory [155], In this case the derivative on the right hand side of Eq. (5.237) is replaced by the ratio of finite differences... 

Application of numerical methods have been rather seldom in studies of adsorption kinetics from micellar solutions. The main difficulties are probably connected with the large number of independent parameters. The first work belongs to Miller [146]. Fainerman and Rakita also published numerical results of the solution of the boundary value problem (5.236), (5.237), (5.245) [85]. Recently Danov et al. proposed an original method for solving the boundary value problem for the diffusion of micelles and monomers [92]. The system of equations was reduced to a system of ordinary differential equations by using a model concentration profile in the bulk phase. The obtained results agree better with dynamic surface tensions of micellar solutions than equation (5.248). 

Note that even if the result Eej = E (h) can be obtained from experiments, a theoretical approach will in general not be consistent if material parameters depend on geometrical data, such as the film thickness h. In the present example the situation is different according to the extension of the theory by the parameter K, a consistent formulation is obtained. The material parameters are constant values but due to the enhancement the model is able to predict boundary layers as part of the solution. The introduction of the effective stiffness according to Eq. (26) is only a re-interpretation of the results, i.e., it is based on the solution of the boundary value problem. 

So, the solution of the boundary value problem (11.76), (11.77) gives us FJ, and rH as functions of sizes and velocities of particles, distances between them, and the viscosity of the external liquid. If the problem of hydrodynamic interaction... 

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