Fredholm integral equations

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... 

In this subsec tion is considered a method of solving numerically the Fredholm integral equation of the second land ... 

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 ... 

It is now possible to see that the matrix formulation has the potential for describing the more-general Fredholm integral equation. This equation corresponds in spectroscopy to the situation where the functional form of s(x) varies across the spectral range of interest. In these circumstances, s is expressed as a function of two independent variables. Although we proceed with the present treatment formulated in terms of convolutions, the reader should bear the generalization in mind. 

Past Methods Used To Solve The Fredholm Integral Equation... 

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. 

Atkinson, K. E. A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the Second Kind, SIAM, Philadelphia (1976). 

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. 

For r e r, this relationship leads to a Fredholm integral equation of the second type for the field Ey within the region F. The Green s function for the background sequence, Gb, which is present in equation (9.21) can be calculated from a simple recursive formula. Details about this procedure can be found in a monograph by Berdichevsky and Zhdanov (1984). 

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 ... 

Whatever the chosen calculation process, the isotherm analysis is based on a simple physical model describing, in the simplest way, the global experimental isotherm as a sum of partial isotherms corresponding to homogeneous adsorption patches. Hence, the amount of adsorbed molecules (probe) is given by the following Fredholm integral equation ... 

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 ... 

Eq. (5.19) is a Fredholm integral equation of the second kind. The Fredholm integral equation is easily solved if the kernel, Aa, is separable, that is,... 

If the phase shifts are given as a function of the energy a unique solution can only be obtained if, in addition, the bound state energies and the normalization constants of the bound state wave function are known (Gelfand and Levitan, 1951). The explicit construction of the potential leads to a complicated solution of a Fredholm integral equation. Other procedures (Agranovich and Marchenko, 1963 Hylleraas, 1963) may be more convenient for a practical application (see Benn and Scharf, 1967, O Brien and Bernstein,... 

Integral equations are equations that contain the integral of the unknown functions. There are two types of integral equations Volteira and Fredholm integral equations. Fredholm equations feature integrals with fixed limits, while Volterra equations have integrals in which the limits of integration are the independent variables. These equations can be obtained by direct formulation or by the reduction of differential equations. Volterra equations are reduced from initial-value problems, and Fredholm equations from boundary-value problems. 

Substitution for T(t b) from Equation (2.39) results in the Fredholm integral equation ... 

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