Numerical solutions of the diffusion equation

Butler and Pillingf) calculated an exact numerical solution of the diffusion equation. They showed that the interpolation formula proposed by Gosele et al.e) reproduces the numerical solution with high precision. 

Finally, as described in Box 4.1 of Chapter 4, an exact numerical solution of the diffusion equation (based on Fick s second law with an added sink term that falls off as r-6) was calculated by Butler and Pilling (1979). These authors showed that, even for high values of Ro ( 60 A), large errors are made when using the Forster equation for diffusion coefficients > 10 s cm2 s 1. Equation (9.34) proposed by Gosele et al. provides an excellent approximation. 

Diffusion is intricately linked with all aspects of the radical-pair mechanism. The CIDNP kinetics for the reaction of a sensitizer with a large spherical molecule that has only a small reactive spot on its surface were studied theoretically. This situation is t)qjical for protein CIDNP, where only three amino acids are readily polarizable, and where such a polarizable amino acid must be exposed to the bulk solution to be able to react with a photoexcited dye. Goez and Heun carried out Monte Carlo simulations of diffusion for radical ion pairs both in homogeneous phase and in micelles. The advantage of this approach compared to numerical solutions of the diffusion equation is that it can easily accommodate arbitrary boundary conditions, such as non-spherical symmetry, as opposed to the commonly used "model of the microreactor" ° where a diffusive excursion starts at the micelle centre and one radical is kept fixed there. 

This technique originated in the numerical solution of the diffusion equation and other areas of fluid mechanics where numerical derivatives are required. In many areas of physics the algebraic expansion method is seen historically as related to the expansion in eigenfunctions method and is known as the spectral method because of the common use of Fourier transform techniques. 

Figures 13(a) and 14(a) depict, on a logarithmic scale, the theoretical CTs of Lii.sNi02and graphite electrodes, respectively, determined from the numerical solution of the diffusion equation for the conditions of Eqs. (7)-(9), by taking the values described in Section V.2. The theoretical CTs of Figures 13(a) and 14(a) compare fairly well with the corresponding experimental CTs of Figures 2(a) and 3(a), respectively, in value and shape.

II. Numerical Solutions of the Diffusion Equation with Prescribed Values on the Boundary... 

Numerical Experiments with the Classical Finite Difference Scheme Principles for Constructing Special Finite Difference Schemes Special Finite Difference Schemes for Problems (2.12), 2.13) and (2.14), (2.15) Numerical Experiments with the Special Difference Scheme Numerical Solutions of the Diffusion Equation with Prescribed Diffusion Fluxes on the Boundary... 

II. NUMERICAL SOLUTIONS OF THE DIFFUSION EQUATION WITH PRESCRIBED VALUES ON THE BOUNDARY... 

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