Fredholm equations

INTEGRAL EQUATIONS, F.G. Tricomi. Authoritative, well-written treatment of extremely useful mathematical tool with wide applications. Volterra Equations, Fredholm Equations, much more. Advanced undergraduate to graduate level. Exercises. Bibliography. 238pp. 5k x 8k. 64828-1 Pa. 6.95... 

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. 

There are two classes of integral equations, Fredholm and Volterra. In the following subsections, the methods for solving each class of integral equations are presented. 

The immediate result of a spectroscopic ensemble technique is a signal spec-tmm, i.e. the variation of the measured signal g over the spectral parameter (time, space or frequency). Each size fraction v possesses a characteristic spectrum k/s ), which in general covers the whole spectral range. Assuming that each size fraction contributes independently and linearly to the measured signal spectmm, the determination of the size distribution requires the inversion of a linear integral equation (Fredholm type) ... 

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 ... 

Another school has also developed and attempted to understand the functional dependence of adsorption on heterogeneous surfaces on the vapor pressure and temperature. Various empirical or semiempirical equations were proposed [24-26] and used later to represent experimental data and to evaluate EADF by inverting Eq. (1), which belongs to the class of linear Fredholm integrals of the first kind [27]. 

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). 

In this section, we will obtain the non-dimensional effective or upscaled equations using a two-scale expansion with respect to the transversal Peclet number Note that the transversal P let number is equal to the ratio between the characteristic transversal timescale and longitudinal timescale. Then we use Fredholm s alternative to obtain the effective equations. However, they do not follow immediately. Direct application of Fredholm s alternative gives hyperbolic equations which are not satisfactory for our model. To obtain a better approximation, we use the strategy from Rubinstein and Mauri (1986) and embed the hyperbolic equation to the next order equations. This approach leads to the effective equations containing Taylor s dispersion type terms. Since we are in the presence of chemical reactions, dispersion is not caused only by the important Peclet number, but also by the effects of the chemical reactions, entering through Damkohler number. 

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 ... 

Methods that have been developed for the solution of the Fredholm equation sometimes rely on continuous functional representations of s(x — x ) and i(x). These methods are limited in usefulness for the experimentalist, who wishes to apply deconvolution techniques by computer to digitized spectra. 

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. 

It should be appreciated here that the singularities at r =a and r =b makes the kernel of this equation practically separable of rank two. Fredholm theory shows that a solution to the homogeneous equation (23) requires a singular reciprocal kernel. Thus it holds that... 

Any proposed method for determining MWDs from the Fredholm equation must converge to the correct solution and must do so rapidly. Convergence to the correct solution is difficult because several local solutions can exist apart from the global solution and numerical methods often converge to a local solution. This problem is most serious with broad unimodal or mulitmodal MWDs. 

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

Although the methods previously discussed have shown some promise in solving the Fredholm equation, two recent methods were chosen for intensive study and comparison with our GEX function fit method. These methods were the constrained regularization method, and the polynomial subdistribution method. 

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. 

A different approach was proposed by Skaggs and Kabala [58]. They attempted to reconstruct the history of the plume using TR. A linear Fredholm equation of the first kind is defined by... 

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

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