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2 Bulletin of the Polish Academy of Sciences. Technical Sciences

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High-accuracy numerical integration methods for fractional order derivatives and integrals computations

Brzeziński D. W., Ostalczyk P.

Bulletin of the Polish Academy of Sciences. Technical Sciences

2014

Vol. 62, nr 4

723--733

In this paper the authors present highly accurate and remarkably efficient computational methods for fractional order derivatives and integrals applying Riemann-Liouville and Caputo formulae: the Gauss-Jacobi Quadrature with adopted weight function, the Double Exponential Formula, applying two arbitrary precision and exact rounding mathematical libraries (GNU GMP and GNU MPFR). Example fractional order derivatives and integrals of some elementary functions are calculated. Resulting accuracy is compared with accuracy achieved by applying widely known methods of numerical integration. Finally, presented methods are applied to solve Abel’s Integral equation (in Appendix).

Numerical evaluation of fractional differ-integrals of some periodical functions via the IMT transformation

Brzeziński D., Ostalczyk P.

Bulletin of the Polish Academy of Sciences. Technical Sciences

2012

Vol. 60, nr 2

285-292

This paper presents accuracy evaluation of the numerical calculations of the fractional differ-integrals. We focus on applying the Riemann-Liouville formula, on singularity, which appears while using classical form of this formula. To calculate it we use the Newton-Cotes’ Quadrature and additionally two Gaussian rules. Using this different approach to the IMT Transformation, transforming the “core” integrand of Riemann-Liouville formula, we point the possible way of increasing the accuracy of the calculations. We use our own tools and compare obtained results with, where possible, exact values, where not – values obtained using an excellent method of integration incorporated in Mathematica.