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The Riemann-Liouville formula for fractional derivatives and integrals (differintegration) is used to derive formulae for matrix order derivatives and integrals. That is, the parameter for integration and differentiation is allowed to…
We give a proposal to generalize the concept of the differential equations on time scales, such that they can be more appropriate for the analysis of real world problems, and give more opportunities to increase the theoretical depth of…
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…
In recent years, the theory for Leibniz integral rule in the fractional sense has not been able to get substantial development. As an urgent problem to be solved, we study a Leibniz integral rule for Riemann-Liouville and Caputo type…
A fractional diffusion equation based on Riemann-Liouville fractional derivatives is solved exactly. The initial values are given as fractional integrals. The solution is obtained in terms of $H$-functions. It differs from the known…
In this work we study the solutions to some fractional higher-order equations. Special cases in which time-fractional derivatives take integer values are also examined and the explicit solutions are presented. Such solutions can be…
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…
Historically the fractional calculus concept works an extended idea based on the question asked by Guillaume de L'H\^opital to Gottfried Wilhelm Leibniz in 1695 about the notation ${d^nf}/{dx^n}$ for the derivative operator "What if…
We survey methods and results of fractional differential equations in which an unknown function is under the operation of integration and/or differentiation of fractional order. As an illustrative example, we review results on fractional…
Inequalities play an important role in pure and applied mathematics. In particular, Opial inequality plays a main role in the study of the existence and uniqueness of initial and boundary value problems for differential equations. It has…
A possibility to represent the standard model of fundamental particles covariant derivatives by means of approximate generalized fractional Riemann-Liouville derivatives of multifractal time and space model is shown.
The goal of this communication is to propose a generalized notion of the "traditional derivative". This generalization includes the fractional derivatives such as the Riemann-Liouville, Gruenwald-Letnikov, Weyl, Riesz, Caputo, Marchaud…
While it is known that one can consider the Cauchy problem for evolution equations with Caputo derivatives, the situation for the initial value problems for the Riemann-Liouville derivatives is less understood. In this paper we propose new…
We describe a general operational method that can be used in the analysis of fractional initial and boundary value problems with additional analytic conditions. As an example, we derive analytic solutions of some fractional generalisation…
What forms will have an equations of modern physics if the dimensions of our time and space are fractional? The generalized equations enumerated by title are presented by help the generalized fractional derivatives of Riemann-Liouville.
A natural consequence of the fractional calculus is its extension to a matrix order of differentiation and integration. A matrix-order derivative definition and a matrix-order integration arise from the generalization of the gamma function…
We introduce new fractional operators of variable order on isolated time scales with Mittag-Leffler kernels. This allows a general formulation of a class of fractional variational problems involving variable-order difference operators. Main…
We begin with a brief overview of the most commonly used fractional derivatives, namely the Caputo and Riemann-Liouville derivatives. We then focus on the study of the fractional time wave equation with the Riemann-Liouville derivative,…
In this paper, we generalize the Riemann-Liouville differential and integral operators on the space of Henstock-Kurzweil integrable distributions, $D_{HK}$. We obtain new fundamental properties of the fractional derivative and integral, a…
We introduce a new version of $\psi$-Hilfer fractional derivative, on an arbitrary time scale. The fundamental properties of the new operator are investigated and, in particular, we prove an integration by parts formula. Using the Laplace…