in this paper, operational matrices of riemann-liouville fractional integration and caputo fractional differentiation for shifted jacobi polynomials are considered. using the given initial conditions, we transform the fractional differential equation (fde) into a modified fractional differential equation with zero initial conditions. next, all the existing functions in modified differential equ...

In this paper, boundary value problems of fractional order are converted into an optimal control problems. Then an approximate solution is constructed from translations and dilations of a B-spline function such that the exact boundary conditions are satisfied. The fractional differential operators are taken in the Riemann-Liouville and Caputo sense. Several example are given and the optimal err...

in this article we implement an operational matrix of fractional integration for legendre polynomials. we proposed an algorithm to obtain an approximation solution for fractional differential equations, described in riemann-liouville sense, based on shifted legendre polynomials. this method was applied to solve linear multi-order fractional differential equation with initial conditions, and the...

The author (Appl. Math. Comput. 218(3):860-865, 2011) introduced a new fractional integral operator given by, ( I a+f ) (x) = ρ1−α Γ(α) ∫ x a τρ−1f(τ) (xρ − τρ)1−α dτ, which generalizes the well-known Riemann-Liouville and the Hadamard fractional integrals. In this paper we present a new fractional derivative which generalizes the familiar Riemann-Liouville and the Hadamard fractional derivativ...

We state, prove, and discuss new general inequality for convex and increasing functions. As a special case of that general result, we obtain new fractional inequalities involving fractional integrals and derivatives of Riemann-Liouville type. Consequently, we get the inequality of H. G. Hardy from 1918. We also obtain new results involving fractional derivatives of Canavati and Caputo types as ...

in this paper, boundary value problems of fractional order are converted into an optimal control problems. then an approximate solution is constructed from translations and dilations of a b-spline function such that the exact boundary conditions are satisfied. the fractional differential operators are taken in the riemann-liouville and caputo sense. several example are given and the optimal err...

and Applied Analysis 3 where n α 1, α denotes the integer part of number α, provided that the right side is pointwise defined on 0, ∞ . Definition 2.2 see 20 . The Riemann-Liouville fractional integral of order α > 0 of a function f : 0, ∞ → R is given by I 0 f t 1 Γ α ∫ t 0 t − s α−1f s ds, 2.2 provided that the right side is pointwise defined on 0, ∞ . From the definition of the Riemann-Liouv...

In this paper, we rigorously construct 2d Liouville Quantum Field Theory on the Riemann sphere introduced in the 1981 seminal work by Polyakov Quantum Geometry of bosonic strings. We also establish some of its fundamental properties like conformal covariance under PSL2(C)-action, Seiberg bounds, KPZ scaling laws, KPZ formula and the Weyl anomaly (Polyakov-Ray-Singer) formula for Liouville Quant...

We obtain nonperturbative results in the framework of continuous Liouville theory. In particular, we express the specific heat Z of pure gravity in terms of an expansion of integrals on moduli spaces of punctured Riemann spheres. The integrands are written in terms of the Liouville action. We show that Z satisfies the Painlevé I. Partly supported by the European Community Research Programme Gau...

in this paper, under appropriate oscillating behaviours of the nonlinear term, we prove some multiplicity results for a class of nonlinear fractional equations. these problems have a variational structure and we find three solutions for them by exploiting an abstract result for smooth functionals defined on a reflexive banach space. to make the nonlinear methods work, some careful analysis of t...