Numerical Solution Methods

Abstract

AbstractIn order to obtain solutions of the integral equations arising in electroanalytical chemistry, numerical methods are often necessary. Such methods are usually based on the general idea of replacing integrals by finite sums. The methods for one-dimensional integral equations can be divided into quadrature methods, product integration methods, and methods based on discrete semi-integration. Of particular interest are product integration methods, such as the step function method and the Huber method, used in conjunction with nonuniform time grids. An adaptive version of the Huber method has also been developed, which generates solutions automatically with a prescribed accuracy. The method based on the Grünwald definition of differintegration proves slightly more accurate than the step function method, but less accurate than the Huber method. In all these methods the computational time increases quadratically with the number of grid nodes used. An alternative, approximate (degenerate) kernel method is known, for which the computational time increases linearly with the number of grid nodes, but the method works for selected kernels only, and its error cannot be fully controlled. Numerical methods applicable to two- and higher-dimensional integral equations are less developed. The Gladwell and Coen method for singular integral equations was used by Cope and Tallman. Mirkin and Bard proposed a quadrature-type method. The boundary integral method typically uses boundary elements for numerical approximation.KeywordsIntegral EquationGrid NodeUniform GridMachine AccuracyTrue ErrorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.