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Integral Equations and Inverse Theory part 2

special quadrature rules, but they are also sometimes blessings in disguise, since they can spoil a kernel’s smoothing and make problems well-conditioned. In §§18.4–18.7 we face up to the issues of inverse problems. | 18.1 Fredholm Equations ofthe Second Kind 791 special quadrature rules but they are also sometimes blessings in disguise since they can spoil a kernel s smoothing and make problems well-conditioned. In 18.4-18.7 we face up to the issues of inverse problems. 18.4 is an introduction to this large subject. We should note here that wavelet transforms already discussed in 13.10 are applicable not only to data compression and signal processing but can also be used to transform some classes of integral equations into sparse linear problems that allow fast solution. You may wish to review 13.10 as part of reading this chapter. Some subjects such as integro-differential equations we must simply declare to be beyond our scope. For a review of methods for integro-differential equations see Brunner 4 . It should go without saying that this one short chapter can only barely touch on a few of the most basic methods involved in this complicated subject. CITED REFERENCES AND FURTHER READING Delves L.M. and Mohamed J.L. 1985 Computational Methods for Integral Equations Cambridge U.K. Cambridge University Press . 1 Linz P. 1985 Analytical and Numerical Methods for Volterra Equations Philadelphia S.I.A.M. . 2 Atkinson K.E. 1976 A Survey of Numerical Methods for the Solution of Fredholm Integral Equations ofthe Second Kind Philadelphia S.I.A.M. . 3 Brunner H. 1988 in Numerical Analysis 1987 Pitman Research Notes in Mathematics vol. 170 D.F. Griffiths and G.A. Watson eds. Harlow Essex U.K. Longman Scientific and Technical pp. 18-38. 4 Smithies F. 1958 Integral Equations Cambridge U.K. Cambridge University Press . Kanwal R.P. 1971 Linear Integral Equations New York Academic Press . Green C.D. 1969 Integral Equation Methods New York Barnes Noble . 18.1 Fredholm Equations of the Second Kind We desire a numerical solution for f t in the equation f t XÎ J a K t s f s ds g t 18.1.1 The method we describe a very basic one is called the Nystrom method. It requires the choice of some .