Fractional (non-integer) derivatives and integrals play an important role in theory and applications. The fractional calculus of variations is a rather recent subject with the first results from 1996. This paper presents necessary and sufficient optimality conditions for fractional problems of the calculus of variations with a Lagrangian density depending on the free end-points. The fractional ...

We study the existence and multiplicity of positive solutions a Riemann-Liouville fractional differential equation with r-Laplacian operator singular nonnegative nonlinearity dependent on integrals, subject to nonlocal boundary conditions containing various derivatives Riemann-Stieltjes integrals. use Guo–Krasnosel’skii fixed point theorem in proof our main results.

Fractional systems with Riemann-Liouville derivatives are considered. The initial memory value problem is posed and studied. We obtain explicit steering laws with respect to the values of the fractional integrals of the state variables. The Gramian is generalized and steering functions between memory values are characterized.

in this paper an algorithm is presented for the regularization of singular integrals with any degrees of singularity, which may be employed in all three-dimensional problems analyzed by boundary elements. the integrals in boundary integrals equations are inherently singular. for example, one can mention the integrals confronted in potential problems to evaluate the flow or the gradient of the f...

in this paper we apply hybrid functions of general block-pulse‎ ‎functions and legendre polynomials for solving linear and‎ ‎nonlinear multi-order fractional differential equations (fdes)‎. ‎our approach is based on incorporating operational matrices of‎ ‎fdes with hybrid functions that reduces the fdes problems to‎ ‎the solution of algebraic systems‎. ‎error estimate that verifies a‎ ‎converge...

In this paper, the circulatory integral and Routh?s equations of Lagrange systems are established with Riemann-Liouville fractional derivatives, is obtained by making use relationship between integrals derivatives. Thereafter, given based on integral. Two examples presented to illustrate application results.

The general fractional integrals and derivatives considered so far in the Fractional Calculus literature have been defined for functions on real positive semi-axis. main contribution of this paper is introducing a finite interval. As case Riemann–Liouville interval, we define both left- right-sided operators investigate their interconnections. results presented are 1st 2nd fundamental theorems ...

The multiindex Mittag-Leffler (M-L) function and the multiindex Dzrbashjan-Gelfond-Leontiev (D-G-L) differentiation and integration play a very pivotal role in the theory and applications of generalized fractional calculus. The object of this paper is to investigate the relations that exist between the Riemann-Liouville fractional calculus and multiindex Dzrbashjan-Gelfond-Leontiev differentiat...

In this paper, a new approach to stability for fractional order control system is proposed. Here a dynamic system whose behavior can be modeled by means of differential equations involving fractional derivatives. Applying Laplace transforms to such equations, and assuming zero initial conditions, causes transfer functions with no integer powers of the Laplace transform variable s to appear. In ...

In this article we study a new class of boundary value problems for fractional differential equations and inclusions with multiple orders of fractional derivatives and integrals, in both fractional differential equation and boundary conditions. The Sadovski’s fixed point theorem is applied in the single-valued case while, in multi-valued case, the nonlinear alternative for contractive maps is u...