Numerical Solution of Integral Equations

In this subsection is considered a method of solving numerically the Fredholm integral equation of the second kind  

The method discussed arises because a definite integral can be closely approximated by any of several numerical integration formulas (each of which arises by approximating the function by some polynomial over an interval). Thus the definite integral in Eq. (3-81) can be replaced by an integration formula, and Eq. (3-81) may be written 

These Uj may be solved for by the methods under Numerical Solution of Linear Equations and Associated Problems and substituted into Eq. (3-82) to yield an approximate solution for Eq. (3-81). 

Because of the work involved in solving large systems of simultaneous linear equations it is desirable that only a small number of u s be computed. Thus the gaussian integration formulas are useful because of the economy they offer. 

Solutions for Volterra equations are done in a similar fashion, except that the solution can proceed point by point, or in small groups of points depending on the quadrature scheme. See Linz, P., Analytical and Numerical Methods for Volterra Equations, SIAM, Philadelphia (1985). There are methods that are analogous to the usual methods for 

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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. ... 

Spiegel MR (1965) Theory and problems of Laplace transforms. McGraw-Hill, New York Nicholson RS, Olmstead ML (1972) Numerical solutions of integral equations. In Matson JS, Mark HB, MacDonald HC (eds) Electrochemistry calculations, simulations and instrumentation, vol 2. Marcel Dekker, New York, p 119... 

A detailed treatment of the theoretical approach used in treating LSV and CV boundary value problems can be found in the monograph by MacDonald [23], More specific information on the numerical solution of integral equations common to electrochemical methods is available in the chapter by Nicholson [30]. The most commonly used method for the calculation of the theoretical electrochemical response, at the present time, is digital simulation which has been well reviewed by Feldberg [31, 32], Prater [33], Maloy [34], and Britz [35]. 

Miller, G.F. In Numerical Solutions of Integral Equations, Delves, L.M., Walsh, J., Eds. Oxford Univ. Press London, 1974. 

The difference between and the full renormalized potential is a well-behaved function that is evaluated numerically. The interest in the renormalization procedure is now mainly a theoretical one as formal results regarding screening and other thermodynamic parameters can be obtained this way. Results applicable to both pure one-component fluids or mixtures can be obtained. The numerical solution of integral equations, such as the SSOZ and CSL equations, for sites with charge interactions should no longer use the renormalization method but rather the method we are about to describe. 

The discussion in the previous sections was based on assumption of a macroscopic (i.e., infinitely large compared to the tip) uniformly reactive substrate. This assumption works well for most kinetic SECM experiments, but it is not appropriate when the substrate is small [e.g., another microdisk electrode (3)], contains small three-dimensional features, or consists of small patches of different reactivities. It was shown in Ref. 3 that the magnitude of either positive or negative feedback produced by a microscopic substrate is lower than the effect that would be produced under the same conditions by the large substrate. The current-distance curves in Fig. 7 computed for various values of h using both numerical solution of integral equations and the ADI simulations (3). Apparently, the size of the substrate is most important when h < 1. 

L. M. Delves and J. Welsh, Numerical Solution of Integral Equations, Clarendon Press, Oxford, 1974. 

A standard way to improve description of the particle shape in numerical solution of integral equations is adaptive discretization, using smaller dipoles near the particle surface. Application of this idea to the DDA is discussed in Ref. [40], but it is incompatible with the FFT acceleration (Sec. 2.4.2.2). Therefore, the only practically viable option is to keep the regular grid of cubical dipoles, but adjust the properties of the boundary dipoles. 

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