A fractional wave equation replaces the second time derivative by a Caputo derivative of order between one and two. In this paper, we show that the fractional wave equation governs a stochastic model for wave propagation, with deterministic time replaced by the inverse of a stable subordinator whose index is one half the order of the fractional time derivative.

We introduce the fractional integral corresponding to the new concept of fractional derivative recently introduced by Caputo and Fabrizio and we study some related fractional differential equations.

We correct a recent result concerning the fractional derivative at extreme points. We then establish new results for the Caputo and Riemann-Liouville fractional derivatives at extreme points.

In this paper, we present a numerical computational approach for solving Caputo type fractional differential equations. This method is based on approximation of Caputo derivative in terms of integer order derivatives and waveform relaxation method. The utility of the method is shown by applying it to several examples. A comparative study indicates that our approach is more efficient and accurat...

In this paper, we investigate numerical approximations of the scalar conservation law with the Caputo derivative, which introduces the memory effect. We construct the first order and the second order explicit upwind schemes for such equations, which are shown to be conditionally 1 contracting and TVD. However, the Caputo derivative leads to the modified CFL-type stability condition, ( t) = O( x...

in this paper we introduce a type of fractional-order polynomials basedon the classical chebyshev polynomials of the second kind (fcss). also we construct the operationalmatrix of fractional derivative of order $ gamma $ in the caputo for fcss and show that this matrix with the tau method are utilized to reduce the solution of some fractional-order differential equations.

The formulations of Riemann-Liouville and Caputo derivatives in the complex plane are presented. Two versions corresponding to the whole or half plane. It is shown that they can be obtained from the Grünwald-Letnikov derivative.

In this paper, the fractional Sturm-Liouville problems, in which the second order derivative is replaced by a fractional derivative, are derived by the Homotopy perturbation method. The fractional derivatives are described in the Caputo sense. The present results can be implemented on the numerical solutions of the fractional diffusion-wave equations. Numerical results show that HPM is effectiv...

Fractional control systems with the Caputo derivative are considered. The modified controllability Gramian and the minimum energy optimal control problem are investigated. Construction of minimizing steering controls for the modified energy functional are proposed.

In this article, by using Schaefer fixed point theorem, we establish sufficient conditions for the existence and uniqueness of solutions for a class of impulsive integro-differential equations with nonlocal conditions involving the Caputo fractional derivative.