Sinc numerical methods

In numerical analysis and applied mathematics, sinc numerical methods are numerical techniques[1] for finding approximate solutions of partial differential equations and integral equations based on the translates of sinc function and Cardinal function C(f,h) which is an expansion of f defined by

where the step size h>0 and where the sinc function is defined by

Sinc approximation methods excel for problems whose solutions may have singularities, or infinite domains, or boundary layers.

The truncated Sinc expansion of f is defined by the following series:

.

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.

Indeed, Sinc are ubiquitous for approximating every operation of calculus

In the standard setup of the sinc numerical methods, the errors (in big O notation) are known to be with some c>0, where n is the number of nodes or bases used in the methods. However, Sugihara[2] has recently found that the errors in the Sinc numerical methods based on double exponential transformation are with some k>0, in a setup that is also meaningful both theoretically and practically and are found to be best possible in a certain mathematical sense.

Reading

  • Stenger, Frank (2011). Handbook of Sinc Numerical Methods. Boca Raton, Florida: CRC Press. ISBN 9781439821596.
  • Lund, John; Bowers, Kenneth (1992). Sinc Methods for Quadrature and Differential Equations. Philadelphia: Society for Industrial and Applied Mathematics (SIAM). ISBN 9780898712988.
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References

  1. Stenger, F. (2000). "Summary of sinc numerical methods". Journal of Computational and Applied Mathematics. 121: 379–420. doi:10.1016/S0377-0427(00)00348-4.
  2. Sugihara, M.; Matsuo, T. (2004). "Recent developments of the Sinc numerical methods". Journal of Computational and Applied Mathematics. 164-165: 673. doi:10.1016/j.cam.2003.09.016.


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