Sinc numerical methods

'Sinc Numerical Methods' are Numerical techniques for finding approximate solutions of partial differential equations(PDE's) as well as integral equations in Numerical Analysis and Applied mathematics. This methods are based on the [sinc] function defined as ${\text{sinc}}(x)={\frac {\sin(\pi x)}{\pi x}}$ which has many interesting properties.

For a detailed discussion see the recently published book "Handbook of Sinc numerical Methods" which has written by Frank Stenger who has written many excellent papers at this topic. He had published the first book in 1992.

Sinc numerical methods cover: *function approximation,

• approximation of derivatives,
• approximate definite and indefinite integration,
• approximate solution of initial and boundary value ordinary differential equation (ODE) problems,
• approximation and inversion of Fourier and Laplace transforms,
• approximation of Hilbert transforms,
• approximation of definite and indefinite convolution,
• approximate solution of partial differential equations,
• approximate solution of integral equations,
• construction of conformal maps.

In the standard setup of the Sinc numerical methods, the errors are known to be O(exp(−�√n)) with some �k>0 , where n is the number of nodes or bases used in the methods. However, Sugihara has recently found that the errors in the Sinc numerical methods are O(exp(−k' n/log n)) with some ��k'>0, in another setup that is also meaningful both theoretically and practically. It has also been found that the error bounds of O(exp(−k�n/log n)) are best possible in a certain mathematical sense.

Recent developments of the Sinc numerical methods Masaaki Sugihara, Takayasu Matsuo Journal of Computational and Applied Mathematics 164–165 (2004)