Related papers: Fractional Differential Forms II
The f-invariant is a higher version of the e-invariant that takes values in the divided congruences between modular forms; in the situation of a cartesian product of two framed manifolds, the f-invariant can actually be computed from the…
In this paper we present three types of Caputo-Hadamard derivatives of variable fractional order, and study the relations between them. An approximation formula for each fractional operator, using integer-order derivatives only, is…
Nonlinear fractional dynamics with scale invariance in continuous and discrete time approaches are described. We use non-integer-order integro-differential operators that can be interpreted as generalizations of scaling (dilation)…
We elaborate on the interpretation of some mixed finite element spaces in terms of differential forms. First we develop a framework in which we show how tools from algebraic topology can be applied to the study of their cohomological…
An elementary system leading to the notions of fractional integrals and derivatives is considered. Various physical situations whose description is associated with fractional differential equations of motion are discussed.
Fractional derivatives and integrations of non-integers orders was introduced more than three centuries ago but only recently gained more attention due to its application on nonlocal phenomenas. In this context, several formulations of…
There has recently been considerable interest in using a nonstandard piecewise approximation to formulate fractional order differential equations as difference equations that describe the same dynamical behaviour and are more amenable to a…
The modulating functions method has been used for the identification of linear and nonlinear systems. In this paper, we generalize this method to the on-line identification of fractional order systems based on the Riemann-Liouville…
We show how the high order finite element spaces of differential forms due to Raviart-Thomas-N\'edelec-Hiptmair fit into the framework of finite element systems, in an elaboration of the finite element exterior calculus of…
A uniform gradient for functions u which satisfy a system of N second-order partial differential inequalities is given in this paper. Some structure conditions are given for the coefficients of the matrices of second-order terms and of…
We discuss some of the mathematical properties of the fractional derivative defined by means of Fourier transforms. We first consider its action on the set of test functions $\Sc(\mathbb R)$, and then we extend it to its dual set,…
The function spaces of continuously differentiable functions are extensively studied and appear in various mathematical settings. In this context, we investigate the spaces of continuously fractional differentiable functions of order…
We study expansions of Drinfeld modular forms of rank \(r \geq 2\) along the boundary of moduli varieties. Product formulas for the discriminant forms \(\Delta_{\mathfrak{n}}\) are developed, which are analogous with Jacobi's formula for…
In this paper we provide a definition of fractional gradient operators, related to directional derivatives. We develop a fractional vector calculus, providing a probabilistic interpretation and mathematical tools to treat multidimensional…
In this article we study solutions to second order linear difference equations with variable coefficients. Under mild conditions we provide closed form solutions using finite continued fraction representations. The proof of the results are…
In a recent paper, Cohl and Costas-Santos derived a number of interesting multi-derivative and multi-integral relations for associated Legendre and Ferrers functions in which the orders of those functions are changed in integral steps.…
We develop a family of finite element spaces of differential forms defined on cubical meshes in any number of dimensions. The family contains elements of all polynomial degrees and all form degrees. In two dimensions, these include the…
Recently, fractional differential equations have been investigated via the famous variational iteration method. However, all the previous works avoid the term of fractional derivative and handle them as a restricted variation. In order to…
Derivatives and integration operators are well-studied examples of linear operators that commute with scaling up to a fixed multiplicative factor; i.e., they are scale-invariant. Fractional order derivatives (integration operators) also…
In this paper, as the second in our series of papers on differential geometry of microlinear Frolicher spaces, we study differenital forms. The principal result is that the exterior differentiation is uniquely determined geometrically, just…