Abstract. We consider Hadamard fractional derivatives and integrals of variable fractional order. A new type of fractional operator, which we call the Hadamard–Marchaud fractional derivative, is also considered. The objective is to represent these operators as series of terms involving integer-order derivatives only, and then approximate the fractional operators by a finite sum. An upper bound ...

A partial differential equation (PDE) of order m is a relation of the form F (x, u,Du,Du, · · · , Du) = 0. (0.1) Here F is a given function of x ∈ R, ”unknown” function u = u(x), and its derivatives up to order m. We denote Du the set of all the derivatives of u of order k. Using multi-indices l = (l1, · · · , ln), i.e. vectors in R with nonnegative integer components, we can write Du = { Du = ...

The coulomb wavefunctions, originally constructed for real p > 0, real q and integer 2 > 0, are delined for p, n, and 1 all complex. We examine the complex continuation of a variety of series and continued-fraction expansions for the Coulomb functions and their logarithmic derivatives, and then see how these expansions may be selectively combined to calculate both the regular and irregular func...

Abstract The fractional wave equation is presented as a generalization of the wave equation when arbitrary fractional order derivatives are involved. We have considered variable dielectric environments for the wave propagation phenomena. The Jumarie’s modified Riemann-Liouville derivative has been introduced and the solutions of the fractional Riccati differential equation have been applied to ...

In this paper an implicit fully discrete local discontinuous Galerkin (LDG) finite element method is applied to solve the time-fractional seventh-order Korteweg-de Vries (sKdV) equation, which is introduced by replacing the integer-order time derivatives with fractional derivatives. We prove that our scheme is unconditional stable and L error estimate for the linear case with the convergence ra...

Fractional generalization of an exterior derivative for calculus of variations is defined. The Hamilton and Lagrange approaches are considered. Fractional Hamilton and Euler–Lagrange equations are derived. Fractional equations are obtained by fractional variation of Lagrangian and Hamiltonian that have only integer derivatives. PACS numbers: 45.20.−d, 45.20.Jj

Liouville's fractional integration is used to de ne polygamma functions (z) for negative integer n. It's shown that such (z) can be represented in a closed form by means of the rst derivatives of the Hurwitz Zeta function. Relations to the Barnes G-function and generalized Glaisher's constants are also discussed.

Singular system has great relationship with gauge field theory, condensed matter theory and some other research areas. Based on the mixed integer Riemann-Liouville fractional derivatives, singular is studied. Firstly, constrained Hamilton equation inherent constraint are presented. Secondly, Lie symmetry conserved quantity analyzed, including determined equation, limited additional structural e...

This work devotes to developing a systematic and convenient approach based on the celebrated convolution quadrature theory design analyze difference formulas for fractional calculus at an arbitrary shifted point $$x_{n-\theta }$$ . The developed theory, called (SCQ), covers most from aspects of characterizing formation related generating functions which are convergent with integer orders. For s...

The main aim of this article is to generalize the Legendre operational matrix to the fractional derivatives and implemented it to solve the nonlinear multi-order fractional differential equations. In this approach, a truncated Legendre series together with the Legendre operational matrix of fractional derivatives are used. The main characteristic behind the approach using this technique is that...