Brzeziński, Dariusz W.Ostalczyk, Piotr2016-02-032016-02-032014Bulletin of the Polish Academy of Sciences Technical Sciences, Vol. 62, Issue 4, 2014, Pages 723–733http://hdl.handle.net/11652/1072http://www.degruyter.com/view/j/bpasts.2014.62.issue-4/bpasts-2014-0078/bpasts-2014-0078.xml?format=INTIn 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).enaccuracy of numerical calculationsfractional order derivatives and integralsdouble exponential formulagauss-jacobi quadrature with adopted weight functionarbitrary precisionnumerical integrationabel’s integral equationArtykuł