in this paper, we consider the second-kind chebyshev polynomials (skcps) for the numerical solution of the fractional optimal control problems (focps). firstly, an introduction of the fractional calculus and properties of the shifted skcps are given and then operational matrix of fractional integration is introduced. next, these properties are used together with the legendre-gauss quadrature fo...

This paper discuss the longstanding problems of fractional calculus such as too many definitions while lacking physical or geometrical meanings, and try to extend fractional calculus to any dimension. First, some different definitions of fractional derivatives, such as the Riemann-Liouville derivative, the Caputo derivative, Kolwankar’s local derivative and Jumarie’s modified Riemann-Liouville ...

This article compares conformable fractional Derivative with Riemann-Liouville and Caputo derivative by comparing solutions to ordinary differential equations involving the three derivatives via numerical simulations of solutions. The result shows that can be used as an alternative for order α 1/2<α<1.

We study mixed Riemann-Liouville fractional integration operators and derivative in Marchaud form of function two variables Hölder spaces different orders each variables. The obtained are results generalized to the case with power weight.

where 2 < α ≤ 3, D denotes the Riemann-Liouville fractional derivative, λ is a positive constant, f (t, x) may change sign and be singular at t = 0, t = 1, and x = 0. By means of the Guo-Krasnoselskii fixed point theorem, the eigenvalue intervals of the nonlinear fractional functional differential equation boundary value problem are considered, and some positive solutions are obtained, respecti...

We study the nonlocal boundary value problem for a mixed type equation with Riemann–Liouville fractional partial derivative. In hyperbolic part of domain, functional is solved by iteration method. The reduced to solving differential equation.

In this paper, we introduce the nabla fractional derivative and integral on time scales in Riemann-Liouville sense. We also Gr\"unwald-Letnikov Some of basic properties theorems related to calculus are discussed.

In the present work we discuss the existence of solutions for a system of nonlinear fractional integro-differential equations with initial conditions. This system involving the Caputo fractional derivative and Riemann−Liouville fractional integral. Our results are based on a fixed point theorem of Schauder combined with the diagonalization method.

in this paper, exp-function and (g′/g)expansion methods are presented to derive traveling wave solutions for a class of nonlinear space-time fractional differential equations. as a results, some new exact traveling wave solutions are obtained.