Integral equations of the first kind

Secondly, the linearized inverse problem is, as well as known, ill-posed because it involves the solution of a Fredholm integral equation of the first kind. The solution must be regularized to yield a stable and physically plausible solution. In this apphcation, the classical smoothness constraint on the solution [8], does not allow to recover the discontinuities of the original object function. In our case, we have considered notches at the smface of the half-space conductive media. So, notche shapes involve abrupt contours. This strong local correlation between pixels in each layer of the half conductive media suggests to represent the contrast function (the object function) by a piecewise continuous function. According to previous works that we have aheady presented [14], we 2584... 

Effectively, this constitutes a Fredholm integral equation of the first kind for exp[—f3G r) where we know the left-hand side, exp(—/M.4(7,)) =... 

J. G. McWhirter and E. R. Pike, On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind, J. Phys. A Math. Gen. 11, 1729-1745 (1978). 

Let us again consider the convolution integral. Equation (86) is an example of a Fredholm integral equation of the first kind. In such equations the kernel can be expressed as a more-general function of both x and x ... 

Phillips, D.L., A technique for the numerical solution of certain integral equations of the first kind, I. Assoc. Comput. Mach., 9, 84-97, 1962. 

To calculate the free energy distributions (/(AG)) of ion adsorption, the Langmuir equation was used as the kernel of the Fredholm integral equation of the first kind... 

The factor of one half appears because of a property of the Dirac delta function which is used in the derivation of Eq. (105). See also Duplantier [35] for another interpretation). Thus, if the surface charge is specified on the boundary then Eq. (Ill) is a Fredholm integral equation of the second kind [90] for the unknown potential at boundary points s. On the other hand, if the boundary potential is known then either Eq. (Ill) is used as a Fredholm integral equation of the first kind for the surfaces charge, n Vt/z, or the gradient of Eq. (105) evaluated on the boundary gives rise to a Fredholm equation... 

Equation (9.32) is a linear Fredholm integral equation of the first kind. It is also known as an unfolding or deconvolution equation. One can preanalyze the data and try to solve this first-kind integral equation. Besides the complexity of this equation, there is a paucity of numerical methods for determining the unknown function / (h) [208,379] with special emphasis on methods based on the principle of maximum entropy [207,380]. The so-obtained density function may be approximated by several models, gamma, Weibull, Erlang, etc., or by phase-type distributions. 

Wing, G., A Primer on Integral Equations of the First Kind The Problem of Deconvolution and Unfolding, Society for Industrial and Applied Mathematics, Philadelphia, 1991. 

This is a Fredholm integral equation of the first kind. The regularized solution to this equation has been applied to the measurement both for the moments and the size distribution of a wide range of latices [46]. K has been given by van de Hulst [45] in terms of particle size/refractive index domain. Mie theory applies to the whole domain but in the boundary regions simpler equations have been derived. 

The goal, then, is to determine the distribution function m0(r, t) for each voxel so that the intrinsic magnetization may be determined using Eq. (12). In what follows, the explicit dependence on position is dropped, with the understanding that the same analysis applies to each voxel throughout the sample. Equation (11) can be rewritten as a Fredholm integral equation of the first kind ... 

From the mathematical point of view, equation (1) is a linear Fredholm integral equation of the first kind, which can be written in a more general form as follows ... 

At the level of approximation invoked by the simple geometric model, the mathematical problem becomes one of inverting Eq. (14), a linear Fredholm integral equation of the first kind, to obtain the PSD. The kernel r(P, e) represents the thermodynamic adsorption model, r(P) is the experimental function, and the pore size distribution /(//) is the unknown function. The usual method of determining /(H) is to solve Eq. (14) numerically via discretization into a system of linear equations. 

The integral is defined with respect to logarithm of the pore width. Such a definition is preferable as the pore size usually varies over a wide range. Equation (11.38) is the Fredholm integral equation of the first kind, with p(p, H) being the kernel. The numerical inversion for determining PSD function fi) is achieved by discretizing the Fredholm equation as follows ... 

Hence, knowledge of the boundary velocity u(xs) and the form of the undisturbed flow evaluated at the body surface Uoo(xj) allows a direct calculation of the surface-force vector T n by means of a solution of the integral equation, (8-198). It is emphasized that we do not actually address the numerical problem of solving (8-198). We note, however, that it is an integral equation of the first kind, and it is known that there can be numerical difficulties with the solution of this class of integral equations. The reader who wishes to learn more about the details of numerical solution should consult one of the general reference books that were listed in the introduction to this section. 

This is a linear integral equation (of the first kind) that now can be solved directly, in principle, to determine the unknown surface temperature distribution 9S (x). The main advantage of this formulation, relative to solving the whole problem (11-6) with (11-108) as a boundary condition, is that (11-109) can be solved to determine 9S (x) directly, without any need to determine the temperature distribution elsewhere in the domain. This latter problem is only ID, in spite of the fact that the original problem was fully 2D. 

Differentiating the integral equation of the first kind with respect to the independent variable X yields (under suitable continuity and differentiability conditions on various functions in the integral equation)... 

In this subsection, the ill-posed nature of Fredholm integral equations of the first kind is demonstrated with an example equation. A technique known as regularization to overcome these issues is presented. The essential idea for regularization is to approximate the integral equation of... 

This section presents a simplified treatment of the technique of Tikhonov regularization. The idea of regularization is to convert a original ill-posed problem (which means that the error in the solution is magnified by errors in the input data) into a well-posed problem for which the error in the solution is under control. More specifically, consider the solution of the following Fredholm integral equation of the first kind. 

In mathematical terms the integral equation (92) is a Fredholm integral equation of the first kind, and the search for a solution of this equation falls into the class... 

As was shown previously [9] Eq. (1) is reduced to the Fredholm integral equation of the first kind, which yields function /(P) after solution via the Tikhonov regularization method. This inverse problem was solved on the basis of an algorithm from [9]. As a result, the function of the distribution over kinetic heterogeneity in /(lnP)-lnM coordinates with each maximum related to the functioning of AC of one type was obtained. 

Instead of the average values determined with Equation 10.46, the distribution functions of the energetic parameters can be calculated using Fredholm integral equation of the first kind with the kernel 0i similar to the right term in Equation 10.46 ... 

Equations 10.69 and 10.70 as Fredholm integral equations of the first kind were solved using a regularization procedure based on the CONTIN algorithm. The use of Equation 10.70 allows us to describe the sum relaxation, which can demonstrate certain deviation from the Arrhenius law because of, for example, the cooperative effects characteristic for such supramolecular systems as PVA or PVA/nanosilica. 

However, in the following the inversion problem shown in Eq. (16.11) was not solved by the above-mentioned Fourier space transformation and subsequent digital filtering. Alternatively, the convolution product, cf Eq. (16.13), can be regarded as a Fredholm integral equation of the first kind which can be described by the following equation [8, 17] ... 

Equation 4.33 is the Fredholm integral equation of the first kind. Except for a few special cases, no solution for G x) exists (Tricomi, 1985). Numerical solutions can be used. To solve for G x) from a given q P), one needs to have an individual pore isotherm, which must be related to the pore size. Moreover, the integral equation is ill-defined, that is, the solution for G x) is not unique, unless a functional form for G x) is assumed. It is clear then that there are as many solutions for the PSD as the number of assumed functional forms. 

In 1970s and 1980s several numerical methods were proposed in order to find the distribution energy functions of adsorption on the basis of tabulated data of experimental adsorption isotherm. From a mathematical point of view the integral adsorption equation is the Fredholm integral equation of the first kind. The particular nature of this equation poses severe difficulties to its solution and strict limits to the range of numerical methods that can be used in such a task. 

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