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We propose an extension of the cable equation by introducing a Caputo time fractional derivative. The fundamental solutions of the most common boundary problems are derived analitically via Laplace Transform, and result be written in terms…

While it is known that one can consider the Cauchy problem for evolution equations with Caputo derivatives, the situation for the initial value problems for the Riemann-Liouville derivatives is less understood. In this paper we propose new…

In this paper, we are primarily concerned with the study of entire and analytical solutions of abstract degenerate (multi-term) fractional differential equations with Caputo time-fractional derivatives. We also analyze systems of such…

In this paper, we consider a class of the Caputo fractional stochastic differential equations of fractional order $\alpha \in (\frac{1}{2},1]$. Our aim is to analyze of the continuous dependence of solutions on the fractional order…

We present a new numerical tool to solve partial differential equations involving Caputo derivatives of fractional variable order. Three Caputo-type fractional operators are considered, and for each one of them an approximation formula is…

We present a new fractional Taylor formula for singular functions whose Caputo fractional derivatives are of bounded variation. It bridges and ``interpolates" the usual Taylor formulas with two consecutive integer orders. This enables us to…

We prove Euler-Lagrange fractional equations and sufficient optimality conditions for problems of the calculus of variations with functionals containing both fractional derivatives and fractional integrals in the sense of Riemann-Liouville.

We present a method to solve fractional optimal control problems, where the dynamic depends on integer and Caputo fractional derivatives. Our approach consists to approximate the initial fractional order problem with a new one that involves…

Since the seminal work of Emmy Noether it is well know that all conservations laws in physics, \textrm{e.g.}, conservation of energy or conservation of momentum, are directly related to the invariance of the action under a family of…

This paper deals with initial value problems for fractional functional differential equations with bounded delay. The fractional derivative is defined in the Caputo sense. By using the Schauder fixed point theorem and the properties of the…

The Caputo time-derivative is usually defined pointwise for well-behaved functions, say, for continuously differentiable functions. Accordingly, in the theory of the partial fractional differential equations with the Caputo derivatives, the…

In this paper, we study the problems of minimizing a functional depending on the Caputo fractional derivative of order $0< \alpha \leq 1$ and the Riemann- Liouville fractional integral of order $\beta >0$ under certain constraints. A…

In this paper we study linear and nonlinear fractional differential equations involving the Caputo fractional derivative with Mittag-Leffler non-singular kernel of order $0<\alpha<1.$ We first obtain a new estimate of the fractional…

We extend in this paper the definition of Caputo derivatives of order in $(0,1)$ to a certain class of locally integrable functions using a convolution group. Our strategy is to define a fractional calculus for a certain class of…

In this work we look at the original fractional calculus of variations problem in a somewhat different way. As a simple consequence, we show that a fractional generalization of a classical problem has a solution without any restrictions on…

In this work, we consider boundary value problems involving Caputo and Riemann-Liouville fractional derivatives of order $\alpha\in(1,2)$ on the unit interval $(0,1)$. These fractional derivatives lead to non-symmetric boundary value…

The present work was primarily motivated by our findings in the literature of some flaws within the proof of the second-order Legendre necessary optimality condition for fractional calculus of variations problems. Therefore we were eager to…

In this paper the Green formula for the operator of fractional differentiation in Caputo sense is proved. By using this formula the integral representation of all regular in a rectangular domains solutions is obtained in the form of the…

We consider fractional isoperimetric problems of calculus of variations with double integrals via the recent modified Riemann-Liouville approach. A necessary optimality condition of Euler-Lagrange type, in the form of a multitime fractional…

We prove Euler-Lagrange type equations and transversality conditions for generalized infinite horizon problems of the calculus of variations on time scales. Here the Lagrangian depends on the independent variable, an unknown function and…