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The numerical solution of integral equations using Chebyshev polynomials

Published online by Cambridge University Press:  09 April 2009

David Elliott
Affiliation:
Mathematics Department, University of Adelaide, Adelaide, S.A.
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An investigation has been made into the numerical solution of non-singular linear integral equations by the direct expansion of the unknown function f(x) into a series of Chebyshev polynomials of the first kind. The use of polynomial expansions is not new, and was first described by Crout [1]. He writes f(x) as a Lagrangian-type polynomial over the range in x, and determines the unknown coefficients in this expansion by evaluating the functions and integral arising in the equation at chosen points xi. A similar method (known as collocation) is used here for cases where the kernel is not separable. From the properties of expansion of functions in Chebyshev series (see, for example, [2]), one expects greater accuracy in this case when compared with other polynomial expansions of the same order. This is well borne out in comparison with one of Crout's examples.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1960

References

[1]Crout, P. D., An Application of Polynomial Approximations to the Solution of Integral Equations arising in Physical Problems. Journal of Math, and Phys. 29 (1940) 3492.CrossRefGoogle Scholar
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