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We consider some possible approaches to the fractional-order generalization of definition of variation (functional) derivative. Some problems of formulation of a fractional-order variational derivative are discussed. To give a consistent…
Diffusive representations of fractional differential and integral operators can provide a convenient means to construct efficient numerical algorithms for their approximate evaluation. In the current literature, many different variants of…
Integral transformations are used to estimate high order derivatives of various special functions. Applications are given to numerical integration, where estimates of high order derivatives of the integrand are needed to achieve bounds on…
We define the concept of completely regular ordinary differential operators and give various criteria for operators to belong to this class. We give also criteria for Birkhof regularity of ordinary differential operators in terms of the…
We introduce basic aspects of new operator method, which is very suitable for practical solving differential equations of various types. The main advantage of the method is revealed in opportunity to find compact exact operator solutions of…
Fractional derivatives are generalization to classical integer-order derivatives. The rules which are true for classical derivative need not hold for the fractional derivatives, for example, we cannot simply add the fractional orders…
Identification of fractional order systems is considered from an algebraic point of view. It allows for a simultaneous estimation of model parameters and fractional (or integer) orders from input and output data. It is exact in that no…
We obtain approximation formulas for fractional integrals and derivatives of Riemann-Liouville and Marchaud types with a variable fractional order. The approximations involve integer-order derivatives only. An estimation for the error is…
We give an algorithm to write down all conformally invariant differential operators acting between scalar functions on Minkowski space. All operators of order k are nonlinear and are functions on a finite family of functionally independent…
A classification of commutative integral domains consisting of ordinary differential operators with matrix coefficients is established in terms of morphisms between algebraic curves.
In their work on differential operators in positive characteristic, Smith and Van den Bergh define and study the derived functors of differential operators; they arise naturally as obstructions to differential operators reducing to positive…
Many possible definitions have been proposed for fractional derivatives and integrals, starting from the classical Riemann-Liouville formula and its generalisations and modifying it by replacing the power function kernel with other kernel…
The paper deals with a fractional derivative introduced by means of the Fourier transform. The explicit form of the kernel of general derivative operator acting on the functions analytic on a curve in complex plane is deduced and the…
In this paper we establish some (presumably new) interesting expressions for the composition of some well known fractional integral operators $ I^{\mu}_{a+}, D^{\mu}_{a+} $,$ I^{\gamma , \mu}_{a+}$ and also derive an integral operator…
In differential geometry, the notation d^n f along with the corresponding formalism has fallen into disuse since the birth of exterior calculus. However, differentials of higher order are useful objects that can be interpreted in terms of…
We introduce more general concepts of Riemann-Liouville fractional integral and derivative on time scales, of a function with respect to another function. Sufficient conditions for existence and uniqueness of solution to an initial value…
In this work we show that it is possible to calculate the fractional integrals and derivatives of order $\alpha$ (using the Riemann-Liouville formulation) of power functions $\left( t-\ast\right) ^{\beta}$ with $\beta$ being any real value,…
We investigate the properties of some recently developed variable-order differential operators involving order transition functions of exponential type. Since the characterisation of such operators is performed in the Laplace domain it is…
We introduce a new fractional derivative that generalizes the so-called alternative fractional derivative recently proposed by Katugampola. We denote this new differential operator by $\mathscr{D}_{M}^{\alpha,\beta }$, where the parameter…
This work further develops the properties of fractional differential forms. In particular, finite dimensional subspaces of fractional form spaces are considered. An inner product, Hodge dual, and covariant derivative are defined. Coordinate…