Fourier Truncation Method for Fractional Numerical Differentiation

ABSTRACT

We consider a ill-posed problem-fractional numerical differentiation with a new method. We propose Fourier truncation method to compute fractional numerical derivatives. A Holder-type stability estimate is obtained. A numerical implementation is described. Numerical examples show that the proposed method is effective and stable.

We consider a ill-posed problem-fractional numerical differentiation with a new method. We propose Fourier truncation method to compute fractional numerical derivatives. A Holder-type stability estimate is obtained. A numerical implementation is described. Numerical examples show that the proposed method is effective and stable.

KEYWORDS

Inverse Problems, Ill-Posed Problem, Fractional Numerical Differentiation, Fourier Regularization, Error Estimate

Inverse Problems, Ill-Posed Problem, Fractional Numerical Differentiation, Fourier Regularization, Error Estimate

Cite this paper

nullA. Qian and J. Mao, "Fourier Truncation Method for Fractional Numerical Differentiation,"*Applied Mathematics*, Vol. 2 No. 7, 2011, pp. 914-917. doi: 10.4236/am.2011.27124.

nullA. Qian and J. Mao, "Fourier Truncation Method for Fractional Numerical Differentiation,"

References

[1] J. Baumeister, “Stable Solution of Inverse Problems,” F. Vieweg and Sohn, Braunschweig, 1987.

[2] R. Gorenflo and S. Vessela, “Abel Integral Equations,” Springer-Verlag, Berlin, Heidelberg, New York, 1991.

[3] A. K. Louis, “Inverse und Schlecht Gesteellte Problem,” Teubner Verlag, Wiesbaden, 1989.

[4] D. A. Murio, “Automatic Numerical Differentiation by Discrete Molification,” Computers and Mathematics with Applications, Vol. 13, No. 4, 1987, pp. 381-386. doi:10.1016/0898-1221(87)90006-X

[5] D. A. Murio and L. Guo, “Discrete Stability Analysis of Molification Method for Numerical Differentiation,” Journal of Computational Applied Mathematics, Vol. 19, No. 6, 1990, pp. 15-25.

[6] I. Daubechies, “Ten Lectures on Wavelets (CBMS-NSF Regional Conference Series in Applied Mathematics),” SIAM: Society for Industrial and Applied Mathematics, Philadelphia, 1992.

[7] J. N. Lyness, “Has Numerical Differentiation a Future?” Utilitas Mathematics, Winnipeg, 1978.

[8] J. Oliver, “An Algorithm for Numerical Differentiation of a Funcion of One Real Variable,” Journal of Computational Applied Mathematics, Vol. 6, No. 2, 1980, pp. 145-160. doi:10.1016/0771-050X(80)90008-X

[9] T. Strom and J. N. Lyness, “On Numerical Differentiation,” BIT Numerical Mathematics, Vol. 15, No. 3, 1975, pp. 314-322. doi:10.1007/BF01933664

[10] D. N. Hao, “A Molification Method for Ill-Posed Problems,” Numerische Mathematik, Vol. 68. No. 4, 1994, pp. 469-506. doi:10.1007/s002110050073

[11] D. A. Murio, C. E. Mejia and S. Zhan, “Discrete Molification and Automatic Numerical Differentiation," Computers and Mathematics Application, Vol. 35, No. 5, 1998, pp. 1-16. doi:10.1016/S0898-1221(98)00001-7

[12] L. Elden, F. Berntsson and T. Reginska, “Wavelet and Fourier Methods For Solving the Sideways Heat Equation,” SIAM Journal on Scientific Computing, Vol. 21, No. 6, 2000, pp. 2187-2205.

[1] J. Baumeister, “Stable Solution of Inverse Problems,” F. Vieweg and Sohn, Braunschweig, 1987.

[2] R. Gorenflo and S. Vessela, “Abel Integral Equations,” Springer-Verlag, Berlin, Heidelberg, New York, 1991.

[3] A. K. Louis, “Inverse und Schlecht Gesteellte Problem,” Teubner Verlag, Wiesbaden, 1989.

[4] D. A. Murio, “Automatic Numerical Differentiation by Discrete Molification,” Computers and Mathematics with Applications, Vol. 13, No. 4, 1987, pp. 381-386. doi:10.1016/0898-1221(87)90006-X

[5] D. A. Murio and L. Guo, “Discrete Stability Analysis of Molification Method for Numerical Differentiation,” Journal of Computational Applied Mathematics, Vol. 19, No. 6, 1990, pp. 15-25.

[6] I. Daubechies, “Ten Lectures on Wavelets (CBMS-NSF Regional Conference Series in Applied Mathematics),” SIAM: Society for Industrial and Applied Mathematics, Philadelphia, 1992.

[7] J. N. Lyness, “Has Numerical Differentiation a Future?” Utilitas Mathematics, Winnipeg, 1978.

[8] J. Oliver, “An Algorithm for Numerical Differentiation of a Funcion of One Real Variable,” Journal of Computational Applied Mathematics, Vol. 6, No. 2, 1980, pp. 145-160. doi:10.1016/0771-050X(80)90008-X

[9] T. Strom and J. N. Lyness, “On Numerical Differentiation,” BIT Numerical Mathematics, Vol. 15, No. 3, 1975, pp. 314-322. doi:10.1007/BF01933664

[10] D. N. Hao, “A Molification Method for Ill-Posed Problems,” Numerische Mathematik, Vol. 68. No. 4, 1994, pp. 469-506. doi:10.1007/s002110050073

[11] D. A. Murio, C. E. Mejia and S. Zhan, “Discrete Molification and Automatic Numerical Differentiation," Computers and Mathematics Application, Vol. 35, No. 5, 1998, pp. 1-16. doi:10.1016/S0898-1221(98)00001-7

[12] L. Elden, F. Berntsson and T. Reginska, “Wavelet and Fourier Methods For Solving the Sideways Heat Equation,” SIAM Journal on Scientific Computing, Vol. 21, No. 6, 2000, pp. 2187-2205.