Geodesic
On a Differential Equation with Left and Right Fractional Derivatives
Fractional calculus and applied analysis, Tome 10 (2007) no. 2, pp. 139-150.

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We treat the fractional order differential equation that contains the left and right Riemann-Liouville fractional derivatives. Such equations arise as the Euler-Lagrange equation in variational principles with fractional derivatives. We reduce the problem to a Fredholm integral equation and construct a solution in the space of continuous functions. Two competing approaches in formulating differential equations of fractional order in Mechanics and Physics are compared in a specific example. It is concluded that only the physical interpretation of the problem can give a clue which approach should be taken.
Keywords: Left and Right Riemann-Liouville Fractional Derivatives, Fractional Differential Equation, Euler-Lagrange Equation, Variational Principle, 26A33, 70H03, 70S05, 49S05, 70H25 
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Atanackovic, Teodor; Stankovic, Bogoljub. On a Differential Equation with Left and Right Fractional Derivatives. Fractional calculus and applied analysis, Tome 10 (2007) no. 2, pp. 139-150. http://geodesic.mathdoc.fr/item/FCAA_2007_10_2_a3/