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Integral Equations and Inverse Theory part 4

This procedure can be repeated as with Romberg integration. The general consensus is that the best of the higher order methods is the block-by-block method (see [1]). Another important topic is the use of variable stepsize methods | 18.3 Integral Equations with SingularKernels 797 This procedure can be repeated as with Romberg integration. The general consensus is that the best of the higher order methods is the block-by-block method see 1 . Another important topic is the use of variable stepsize methods which are much more efficient if there are sharp features in K or f. Variable stepsize methods are quite a bit more complicated than their counterparts for differential equations we refer you to the literature 1 2 for a discussion. You should also be on the lookout for singularities in the integrand. If you find them then look to 18.3 for additional ideas. CITED REFERENCES AND FURTHER READING Linz P. 1985 Analytical and Numerical Methods for Volterra Equations Philadelphia S.I.A.M. . 1 Delves L.M. and Mohamed J.L. 1985 Computational Methods forIntegral Equations Cambridge U.K. Cambridge University Press . 2 18.3 Integral Equations with Singular Kernels Many integral equations have singularities in either the kernel or the solution or both. A simple quadrature method will show poor convergence with N if such singularities are ignored. There is sometimes art in how singularities are best handled. We start with a few straightforward suggestions 1. Integrable singularities can often be removed by a change of variable. F or example the singular behavior K t s s1 2 or s-1 2 near s 0 can be removed by the transformation z s1 2. Note that we are assuming that the singular behavior is confined to K whereas the quadrature actually involves the product K t s f s and it is this product that must be fixed. Ideally you must deduce the singular nature of the product before you try a numerical solution and take the appropriate action. Commonly however a singular kernel does not produce a singular solution f t . The highly singular kernel K t s S t s is simply the identity operator for example. 2. If K t s can be factored as w s K t s where w s is singular and K t s is smooth then a Gaussian quadrature based .