Presentation on theme: "International GeoGebra Conference for Southeast Europe January 15-16, Novi Sad, Serbia Fractional calculus and Geogebra Đurđica Takači Departman za matematiku."— Presentation transcript:
International GeoGebra Conference for Southeast Europe January 15-16, Novi Sad, Serbia Fractional calculus and Geogebra Đurđica Takači Departman za matematiku i informatiku Prirodno-matematički fakultet Univerzitet u Novom Sadu
Gamma Function Gamma function extends factorials to non-integer values GamaFunkcija Gamma function extends factorials to non-integer values GamaFunkcija
Convolution Convolution of two functions Convolution.ggb
On the power (order) of the derivative -- differential operator The origins of the fractional calculus go back to the end of the 17th century, when L'Hospital asked (in a letter) Leibniz about the sense of the notation the derivative of order 1/2. Leibniz's answer was "An apparent paradox, from which one day useful consequences will be drawn."
Fractional calculus is a natural generalization of calculus Many applications of fractional calculus in: viscoelasticity and damping, diffusion and wave propagation, electromagnetism, chaos and fractals, heat transfer, biology, electronics, signal processing, robotics, system identification, traffic systems, genetic algorithms, percolation, modeling and identification, telecommunications, chemistry, irreversibility, physics, control systems, economy, and finance,...
Fractional calculus Rimann-Liouville fractional integral operator of order Fractional-Integral
Fractional derivatives in Caputo sense Fractional Derivative Fractional Derivative
Fractional derivatives in Caputo sense Fractional Derivative
Delta Functions The Dirac delta function, or δ function (introduced by Paul Dirac), is a generalized function: Delta sequences Academ. Stevan Pilipovic
References: 1. Caputo, M., Linear models of dissipation whose Q is almost frequency independent- II, Geophys. J. Royal Astronom. Soc., 13, No 5 (1967), Mainardi, F., Yu. Luchko, Pagnini, G., The fundamental solution of the space-time fractional diffusion equation Fractional Calculus and its Application, 4, , Mainardi, F., Pagnini, G., The Wright functions as the solutions of time-fractional diffusion equation Applied Math. and Comp. Vol.141, Iss.1, 20 August 2003, Podlubny, I., Fractional Differential Equations, Acad. Press, San Diego (1999). 5. Ross, B., A brief history and exposition of fundamental theory of fractional calculus, In: "Fractional Calculus and Its Applications" (Proc. 1st Internat. Conf. held in New Haven, 1974; Lecture Notes in Math. 457, Springer-Verlag, N. York (1975), pp Samko, S.G., Kilbas, A.A., Marichev, O.I., Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Sci. Publ., Switzerland (1993).
7. Takači, Dj., Takači, A., On the approximate solution of mathematical model of a viscoelastic bar, Nonlinear Analysis, 67 (2007), Takači, Dj., Takači, A., On the mathematical model of a viscoelastic bar, Math. Meth. Appl. Sci. 2007; 30: Yulita M. R., Noorani,M.S.M., Hashim, I. Variational iteration method for fractional heat- and wave-like equations, Nonlinear Analysis, 10 (2009), Odibat Z., M. Analytic study on linear systems of fractional differential equations, Computers and Mathematics with Applications
Delta Functions The Dirac delta function, or δ function (introduced by Paul Dirac), is a generalized function: Academician Stevan Pilipovic Delta sequences