Analysis and Applications of Composed Forms of Caputo Fractional Derivatives

• Autorzy: Błaszczyk T. , Kotela E. , Hall M. R. , Leszczyński J.
• Języki publikacji: EN

Abstrakty

EN

In this paper we consider two ordinary fractional differential equations with composition of the left and the right Caputo derivatives. Analytical solution of this type of equations is known for particular cases, having a complex form, and therefore is difficult in practical calculations. Here, we present two numerical schemes being dependent on a fractional order of equation. The results of numerical calculations are compared with analytical solutions and then we illustrate convergence of our schemes. Finally, we show an application of the considered equation.

Słowa kluczowe

EN

difference equation; Caputo derivatives

PL

równanie różniczkowe; pochodne Caputo

• Wydawca: Oficyna Wydawnicza Politechniki Białostockiej
• Czasopismo: Acta Mechanica et Automatica
• Rocznik: 2011
• Tom: Vol. 5, no. 2
• Strony: 11--14
• Opis fizyczny: Bibliogr.12 poz., Wykr.

Twórcy

autor

Błaszczyk T.

autor

Kotela E.

autor

Hall M. R.

autor

Leszczyński J.

• Czestochowa University of Technology Czestochowa,

Bibliografia

• 1. Agrawal O.P. (2002), Formulation of Euler-Lagrange equations for fractional variational problems, J. Math. Anal. Appl.,Vol. 272, 368-379.
• 2. Blaszczyk T. (2009), Application of the Rayleigh-Ritz method for solving fractional oscillator equation, Scientific Research of the Institute of Mathematics and Computer Science, Vol. 8, No. 2, 29-36.
• 3. Blaszczyk T. (2010), Zastosowanie rownania frakcyjnego oscylatora do modelowania efektu pamieci w materii granulowanej, rozprawa doktorska, Politechnika Czestochowska (in Polish).
• 4. Blaszczyk T., Ciesielski M. (2010), Fractional EulerLagrange equations - numerical solutions and applications of reflection operator, Scientific Research of the Institute of Mathematics and Computer Science, Vol. 9, No. 2, 17-24.
• 5. Blaszczyk T., Kotela E., Leszczynski J. (2011), Application of the fractional oscillator equation to simulations of granular flow In a silo, Computer Methods in Materials Science, Vol. 11, No. 1, 64-67.
• 6. Kilbas A.A., Srivastava H.M., Trujillo J.J. (2006), Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam.
• 7. Klimek M. (2002), Lagrangean and Hamiltonian fractional sequential mechanics, Czech. J. Phys., Vol. 52, 1247-1253.
• 8. Klimek M. (2007), Solutions of Euler-Lagrange equations in fractional mechanics, AIP Conference Proceedings 956. XXVI Workshop on Geometrical Methods in Physics. Bialowieza, 73-78.
• 9. Klimek M. (2008), G-Meijer functions series as solutions for certain fractional variational problem on a finite time interval, Journal Europeen des Systemes Automatises (JESA), Vol. 42, 653-664.
• 10. Leszczynski J., Blaszczyk T. (2010), Modeling the transition between stable and unstable operation while emptying a silo, Granular Matter, DOI: 10.1007/s10035-010-0240-5.
• 11. Riewe F. (1996), Nonconservative Lagrangian and Hamiltonian mechanics, Phys. Rev. E, Vol. 53, 1890-1899.
• 12. US Department of Transportation, Federal Transport Administration, (2009). LTPP Seasonal Monitoring Programme (SMP): Pavement Performance Database (PPDB).[electronic database], Standard Data Release 23.0, DVD Version, USA.