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Analytical and numerical study for a fractional boundary value problem with a conformable fractional derivative of Caputo and its fractional integral

Bekkouche M. Moumen, Guebbai H.

Journal of Applied Mathematics and Computational Mechanics

2020

Vol. 19, nr 2

31-42

We study the existence and uniqueness of the solution of a fractional boundary value problem with conformable fractional derivation of the Caputo type, which increases the interest of this study. In order to study this problem we have introduced a new definition of fractional integral as an inverse of the conformable fractional derivative of Caputo, therefore, the proofs are based upon the reduction of the problem to a equivalent linear Volterra-Fredholm integral equations of the second kind, and we have built the minimum conditions to obtain the existence and uniqueness of this solution. The analytical study is followed by a complete numerical study.

Existence of a solution for the fractional forced pendulum

Torres C.

Journal of Applied Mathematics and Computational Mechanics

2014

Vol. 13, nr 1

125--142

In this work we study the fractional forced pendulum equation with combined fractional derivatives - tDαT 0Dαt u( t ) + g ( u ( t )) = f ( t ), t ∈ ( 0, T ) ( 0. 1 ) u ( 0 ) = u ( T ) = 0 where ½ < α < 1, g ∈ C ( R, R ), bounded f ∈ C [ 0, T ]. Using minimization techniques form variational calculus we show that ( 0. 1 ) has a nontrivial solution.