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Modelowanie Matematyczne w Fizyce i Technice

A REVIEW OF NUMERICAL METHODS FOR FRACTIONAL YARIATIONAL EQUATIONS

Tomasz Zf/aszezykMariusz Ciesielski², Jacek Leszczynski³

¹ Institute of Mathematics, Częstochowa University of Technology, Częstochowa, Poland

² Institute of Computer and Information Sciences, Częstochowa University of Technology, Częstochowa, Poland

³ Department ofHydrogen Energy, AGH University of Science and Technology, Kraków, Poland

[e-mail], [e-mail], [e-mail]

There are two different approaches to the formulation of differential eąuations containing derivatives of fractional order. In the first approach, the integer order derivatives in differential eąuations are simply replaced by the fractional derivatives. In the second approach, one modifies the variational principle by replacing the integer order derivative by a fractional one [5, 6], The variational functional

I for any differentiable Lagrangian L satisfying some smoothness properties with y(x)eC[a,b], a,be R is of the form

I - Jl(x,y(x),Df₊y(x),D^y(x))dx    (1)

Then, minimization of the functional (1) leads also to the fractional differential eąuations which are known in literaturę as fractional Euler-Lagrange differential eąuations

8L

OL Cr>«

— + D_h_

dy    dD“₊ y

^cD^. ——jj— = O

dD'l y

(2)

subject to the adeąuate boundary conditions.

This second approach, in recent years, seems to become increasingly important and different approaches have been developed by considering different types of Lagrangians, e.g., depending on the Riemann-Liouville Df₊, D,f_ or Caputo ^CD“₊, ^cD_b^/?_ derivatives.

The main feature of the fractional variational eąuations is that these eąuations contain simultaneously the left and right derivatives. This is also a fundamental problem in finding Solutions to eąuations of a variational type [5, 6], Conseąuently, numerical methods have been devoted to solve fractional variational problems [1,2, 3, 4].

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