Fredholm integral equation, methods

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

Past Methods Used To Solve The Fredholm Integral 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). 

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. 

In the second approach, Ambrosone et al. (28, 29) have developed a numerical procedure based on a solu tion of the Fredholm integral equation to resolve the distribution function P(K) without prior assumptions of its analytical type. The method involves the selection of a generating function for the numerical solution which may not be trivial in some cases. The method was successfully tested by computer simulation of E(S) for a hypothetical emulsion with bimodal distribution (31). Figure 5 shows the reconstruction of the true droplet size distribution used to test the method by calculating the size distribution from the correspond ing synthesized ii(5) relationship. 

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. 

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. 

Several numerical algorithms have been developed in order to solve the Fredholm integral equation and many of them have been applied for determining the adsorption energy distribution function from the experimental adsorption data. The following list includes the most popular and useful numerical methods ... 

A typical regularization method for the Fredholm integral equation of the first kind is to add a term Q(f> y) (with q a fixed number) to the left-hand side of Eq. (66), thus obtaining a Fredholm integral equation of the second kind, whose solution is stable. The difiieulty of this method is related to the fact that the formula expressing such a solution (the Neumann series ) does converge only for q > A, where A is the norm of the operator A [45]. 

If one measures Apj at different tube lengths, T j, the relative amounts of particles, AQ3, of the corresponding settling rate classes, Wg, may in principle be calculated from the linear set of equations, described by Eqn. 17. The different velocity increases, Aej, for particles of a certain settling rate at position, T j, can be calculated from a force balance. Eqn. 17 represents a so-called first order Fredholm integral equation, well known from several optical methods, for example, diffraction pattern analysis. The problems encountered in the solution of this equation /12/ occur here as well. We suffered from this at the beginning of 1970, when we tried to set up an instrument according to P. Bernutat s idea.At that time we did not succeed for a number of reasons. To obtain a mathematical... 

There are a few methods to solve Fredholm integral equations, which are not that prone to measurement faults [3, 12]. One of them is the inversion with linear smoothing according to Phillips-Twomey, which is used, for example, to analyze the measurement data of laser diffraction systems. 

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. 

This section discusses methods of unfolding, assuming that an energy spectrum is measured with a multichannel analyzer or any other device that divides the measured spectrum into energy bins. As stated at the beginning of Sec. 11.5, unfolding means to solve the Fredholm-type integral equation... 

The Tikhonov Regularization Method. We shall dwell briefly on the mathematical aspect of solving the first-kind Fredholm equation by the Tikhonov regularization method, referring basically to the works that deal with the use of this method in the EXAFS spectroscopy [41-44]. The integral equation (92) can be presented in the operator form... 

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