Nyström method

The Nyström method of discretizing an integral equation uses a quadrature rule; i.e. applying the quadrature rule

$\int_a^b h (x) \;\mathrm d x \approx \sum_{k=1}^n w_k h (x_k)$

to, for example, the inhomogeneous Fredholm equation of the second kind

$f (x) = \lambda u (x) - \int_a^b K (x, x') u (x') \;\mathrm d x'$,

results in

$f (x) \approx \lambda u (x) - \sum_{k=1}^n w_k K (x, x_k) u (x_k)$.

References

• E. J. Nyström, Über die praktische Auflösung von Integralgleichungen mit Anwendungen auf Randwertaufgaben, Acta Mathematica 54 pp. 185-204 (1930).
• L. M. Delves & J. Walsh (eds) 1974 Numerical Solution of Integral Equations, Clarendon, Oxford.
• H.-J. Reinhardt 1985 Analysis of Approximation Methods for Differential and Integral Equations, Springer, New York.

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