The method of integral equations

The analysis of the electromagnetic field of a vertical magnetic dipole located either on the axis of cylindrical interfaces (formations of an infinite thickness) or in a medium with horizontal interfaces only allows us to investigate the influence of the borehole and the invasion zone, as well as the effect caused by a finite thickness of the formation. For such models application of the separation of variables method is the most natural approax h enabling us to present the field in a explicit form by known functions. It is a much more complicated problem when the vertical magnetic dipole is located on the borehole axis and the formation has a finite thickness. In this case the method of separation of variable cannot be used, since both cylindrical and horizontal interfaces are present and it is more appropriate to apply such numerical methods as integral equations or finite elements. 

In fact, during the last 30 years the use of integral equations has allowed us to move significantly forward in the theory and interpretation of induction logging. This is the main reason why we will describe here only this numerical method. At the same time it is reasonable to point out that both methods have been used, provided that a model of the medium and a field have cylindrical symmetry with the common axis. Until now this restriction has not permitted us to investigate a field behavior in the case when the boundaries between a formation and a surrounding medium are not perpendicular to the borehole axis. 

Now let us suppose that a vertical magnetic dipole is located on the borehole axis and the medium possesses axial symmetry (Fig. 3.4). In accord with the Biot-Savart law the current of the magnetic dipole creates the primary magnetic field and its change with time generates the primary vortex electric field. Due to the axial symmetry this electric field does not intersect boundaries between media with different conductivities. Because of this no electric charges develop and as a result of the existence of the vortex electric field currents arise at every point, of the conductive medium with a density given by  

Inasmuch as electric charges are absent, the induced currents as well as the primary vortex electric field Eq, have only an azimuthal component in the cylindrical system of coordinates r, 0, (Fig. 3.4). It is obvious that interaction between current filaments does not change the direction of current flow in this case. Thus the total electric field is  

As was shown in Chapter 1 a circular current filament passing an elementary current tube at the point q creates the vortex electric field at point p (Fig. 3.4c) equal to  

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As was pointed out, the remarkable simplicity of Doll s theory is related to the fact that interaction of induced currents is neglected. In order to take into account this effect and improve the quality of an interpretation of logging data, a new approach, also an approximate one. was suggested (Kaufman, 1962). This method allows us relatively quickly to evaluate the field, subjected to an influence of the skin effect in a formation, when there are both cylindrical and horizontal interfaces. Much later, this rather complicated problem was solved by V. Dimitriev, L. Tabarovsky, V. Zakharov using the method of integral equations. 

It is appropriate to notice that there are several very well-developed numerical techniques allowing us to solve this problem, such as the method of integral equations, the method of finite differences, and others. 

The method of integral equations with respect to tangential components of the electric and magnetic fields. 

A second general technique for treating the angular distribution of the neutron flux is presented in Sec. 7.4. This is the method of integral equations. Solutions for the directed flux 0(r,Q) are derived on the basis of the one-velocity model for various media of infinite extent. The application of these solutions for the infinite medium to systems of finite size is demonstrated in the case of the homogeneous slab and sphere. 

A second general approach to the problem of determining the angular distribution of the neutron flux may be developed by utilizing the methods of integral equations. These methods are generally quite powerful, and formal solutions are readily derived even for nonhomogeneous systems with complex source distributions. Unfortunately, the mathematical techniques required for these applications are of a somewhat sophisticated... 

Abstract. In this study we examined the numerical methods of solving the direct problem of electrical sormding with direct current for a layered model with complex relief contact boundaries. The solution was obtained by the method of integral equations. The system of integral equations for the solution of the direct problem of electrical soimding with direct current for a layered relief medium was estabhshed. Numerical simulation of the field for two-layered medium with various shapes of relief contact boundaries was conducted. We obtained the density of distribution of secondary sources on contact bormdaries. 

Cases of ground surface relief of the Earth were not considered in investigations. Or the cases were not brought to systematical numeral modeling. Currently available methods of relief form corrections have approximate pattern. In this study, for the calculation of fields in layered relief medium, we use the method of integral equations, well-established when performing modeling in resistivity method [7],[8],[9]. 

In general, the solution of integral equations is not easy, and a few exact and approximate methods are given here. Often numerical methods must be employed, as discussed in Numerical Solution of Integral Equations. ... 

When a slow steady-state autoxidation of a suitable hydrocarbon is disturbed by adding either a small amount of inhibitor or initiatory a new stationary state is established in a short time. The change in velocity during the non-steady state can be followed with sensitive manometric apparatus. With the aid of integrated equations describing the nonsteady state the individual rate constants of the autoxidation reaction can be derived from the results. Scope and limitations of this method are discussed. Results obtained for cumene, cyclohexene, and Tetralin agree with literature data. 

This is a first-order linear differential equation and can be solved using the method of integrating factors that we show below. Multiplying both sides by e1 2, we have... 

The highest integration of all variables and a certain combination of both sets of variables are possible by application of the method of simultaneous equations which is also called the structural equations method or path analysis. In this method the correlation of independent variables is explicitly accounted for. It is, furthermore, allowed that some independent variables are also considered as dependent variables within the same system of equations. As a result of both advantages the system equations are coupled at least by common error variables. 

The solution of integral equation (9.139) is similar to equation (9.125) and can also be obtained using the method of successive iterations, which is governed by the equations... 

Kupradze, V. D., 1934, Method of integral equations in the diffraction theory (in Russian) Mathematical Series, 41, 4. 

One of the many applications of the theory of complex variables is the application of the residue theorem to evaluate definite real integrals. Another is to use conformal mapping to solve boundary-value problems involving harmonic functions. The residue theorem is also very useful in evaluating integrals resulting from solutions of differential equations by the method of integral transforms. 

Integral transforms can be used to solve ordinary differential equations by converting them to algebraic equations. In what follows, the convolution properties of the different transforms have been listed, followed by the methods of integral transform used to solve (a) one-dimensional diffusion equations in the infinite and semi-infinite domains and (b) Laplace equations in the cylindrical geometries. 

There are a number of other methods which may be used to obtain approximate wave functions and energy levels. Five of these, a generalized perturbation method, the Wentzel-Kramers-Brillouin method, the method of numerical integration, the method of difference equations, and an approximate second-order perturbation treatment, are discussed in the following sections. Another method which has been of some importance is based on the polynomial method used in Section 11a to solve the harmonic oscillator equation. Only under special circumstances does the substitution of a series for 4 lead to a two-term recursion formula for the coefficients, but a technique has been developed which permits the computation of approximate energy levels for low-lying states even when a three-term recursion formula is obtained. We shall discuss this method briefly in Section 42c. 

Surfaces with Nonuniform Radiosity. If the radiosity across a given surface does not meet the assumption of uniformity, then the surface may be subdivided into subsurfaces, each of which approximates the condition of uniformity. In the limit, this reduces to relations in the form of integral equations. In this case, the net radiation method can be extended. Note that Eqs. 7.72 and 7.73 still apply to every position on surface k, but Eq. 7.74 must be modified to remove the assumption of uniform radiosity. The third equation for the net radiation method is the relation for incident radiation onto a particular location on surface k from all other surfaces, each of which can have a variable radiosity. The resulting relations are... 

In the method of integration we start with a rate equation which we think may be applicable. For example, if the reaction is believed to be a first-order reaction we start with... 

Comparison with calculations based on a solution of the system of integral equations has permitted us to establish boundaries of application of this approximate method, i.e. the range of parameters when induced currents in the borehole and invasion zone arise due to only the primary electric field and the skin effect in the formation and in the surrounding medium manifests in the same manner as in the horizontally layered medium. With error which does not exceed 10% this range of parameters is defined as ... 

A direct analytical solution technique exists for certain cases, using the method of integral transforms. It provides a systematic approach to a very difficult problem and is beginning to appear more and more frequently in the technical literature, as can be seen in the review of the models in Section V. A full description of the integral transform technique for solution of differential equations is presented in the references cited, as well as in numerous mathematical texts, including the one by Snedden 87), Cleary and Adrian 14) present a very thorough version of the solution process, as does Yeh (765). 

Computer simulations are not the only methods which can be used to calculate the dielectric constant of pure liquids. Other approaches are given by the use of integral equations, in particular, the hypemetted chain (HNC) molecular integral equation and the molecular Omstein-Zemike (OZ) theory (see Section 8.7.1 for details on such methodologies). 

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