Fractional master equation: non-standard analysis and Liouville–Riemann derivative

Fractional master equations may be defined either by means of Liouville–Riemann (L–R) fractional derivative or via non-standard analysis. The first approach describes processes with long-range dependence whilst the second approach deals with processes involving independent increments. The present papers put in evidence some of the differences between these two modellings, and to this end it especially considers more fractional Poisson processes.     

Degenerate Linear Evolution Equations with the Riemann–Liouville Fractional Derivative

We study the unique solvability of the Cauchy and Schowalter–Sidorov type problems in a Banach space for an evolution equation with a degenerate operator at the fractional derivative under the assumption that the operator acting on the unknown function in the equation is p-bounded with respect to the operator at the fractional derivative. The conditions are found ensuring existence of a unique solution representable by means of the Mittag-Leffler type functions. Some abstract results are illustrated by an example of a finite-dimensional degenerate system of equations of a fractional order and employed in the study of unique solvability of an initial-boundary value problem for the linearized Scott-Blair system of dynamics of a medium.     

Fractional equations of Volterra type involving a Riemann–Liouville derivative

In this paper, we will discuss the existence of solutions of fractional equations of Volterra type with the Riemann–Liouville derivative. Existence results are obtained by using a Banach fixed point theorem with weighted norms and by a monotone iterative method too. An example illustrates the results.     

Optimal Controls for Riemann–Liouville Fractional Evolution Systems without Lipschitz Assumption

In this paper, an evolution system with a Riemann–Liouville fractional derivative is proposed and analyzed. With the help of a resolvent technique, a suitable concept of solutions to this system is formulated and the corresponding existence of solutions is demonstrated. Furthermore, without the Lipschitz continuity of the nonlinear term, the optimal control result is derived by setting up minimizing sequences twice. Our work essentially generalizes previous results on optimal controls of all evolution systems. Finally, a simple example is presented to illustrate our theoretical results.     

Some Remarks on One-dimensional Functions and Their Riemann–Liouville Fractional Calculus

A one-dimensional continuous function of unbounded variation on [0,1] has been constructed.The length of its graph is infnite,while part of this function displays fractal features.The Box dimension of its Riemann–Liouville fractional integral has been calculated.     

Boundary Value Problems for Riemann–Liouville Fractional Differential Inclusions with Nonlocal Hadamard Fractional Integral Conditions

In this paper, we study a new class of boundary value problems from a fractional differential inclusion of Riemann–Liouville type and nonlocal Hadamard fractional integral boundary conditions. Some new existence results for convex as well as non-convex multi-valued maps are obtained using standard fixed point theorems. The obtained results are illustrated by examples.     

Complex Grünwald–Letnikov,Liouville, Riemann–Liouville,and Caputo derivatives for analytic functions

The well-known Liouville, Riemann–Liouville and Caputo derivatives are extended to the complex functions space, in a natural way, and it is established interesting connections between them and the Grünwald–Letnikov derivative. Particularly, starting from a complex formulation of the Grünwald–Letnikov derivative we establishes a bridge with existing integral formulations and obtained regularised integrals for Liouville, Riemann–Liouville, and Caputo derivatives. Moreover, it is shown that we can combine the procedures followed in the computation of Riemann–Liouville and Caputo derivatives with the Grünwald–Letnikov to obtain a new way of computing them. The theory we present here will surely open a new way into the fractional derivatives computation.     

Monotone Iterative Technique for Riemann–Liouville Fractional Integro-Differential Equations with Advanced Arguments

In this paper, we consider existence and uniqueness of solutions for nonlinear boundary value problems involving Riemann–Liouville fractional integro-differential equations with advanced arguments. By establishing a new comparison theorem and applying the monotone iterative technique, we show the existence of extremal solutions.     

On the Approximation of Riemann–Liouville Integral by Fractional Nabla h-Sum and Applications

First of all, in this paper, we prove the convergence of the nabla h-sum to the Riemann–Liouville integral in the space of continuous functions and in some weighted spaces of continuous functions. The connection with time scales convergence is discussed. Second, the efficiency of this approximation is shown through some Cauchy fractional problems with singularity at the initial value. The fractional Brusselator system is solved as a practical case.     

Probabilistic solutions to nonlinear fractional differential equations of generalized Caputo and Riemann–Liouville type

This paper provides well-posedness and integral representations of the solutions to nonlinear equations involving generalized Caputo and Riemann–Liouville type fractional derivatives. As particular cases, we study the linear equation with non constant coefficients and the generalized composite fractional relaxation equation. Our approach relies on the probabilistic representation of the solution to the generalized linear problem recently obtained by the authors. These results encompass some known cases in the context of classical fractional derivatives, as well as their far reaching extensions including various mixed derivatives.     

Caputo–Hadamard Fractional Derivatives of Variable Order

In this article, 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 obtained and an estimation for the error is given. At the end, we compare the exact fractional derivative of a concrete example with some numerical approximations.     

On the Riemann–Liouville Operators with Variable Limits

Siberian Mathematical Journal -     

Riemann–Liouville abstract fractional Cauchy problem with damping

This paper is concerned with Riemann–Liouville abstract fractional Cauchy Problems with damping. The notion of Riemann–Liouville fractional (α,β,c)$(\alpha ,\beta ,c)$ resolvent is developed, where 0<β<α≤1$0<\beta <\alpha \le 1$. Some of its properties are obtained. By combining such properties with the properties of general Mittag-Leffler functions, existence and uniqueness results of the strong solution of Riemann–Liouville abstract fractional Cauchy Problems with damping are established. As an application, a fractional diffusion equation with damping is presented.     

Local and Nonlocal Boundary Value Problems for Degenerating and Nondegenerating Pseudoparabolic Equations with a Riemann–Liouville Fractional Derivative

We study local and nonlocal boundary value problems for degenerating and nondegenerating third-order pseudoparabolic equations of the general form with variable coefficients and with a Riemann–Liouville fractional derivative. For their solutions, we obtain a priori estimates that imply the uniqueness of the solution and its stability with respect to the right-hand side and the initial data.     

Infinite systems of noncolliding generalized meanders and Riemann–Liouville differintegrals

Yor’s generalized meander is a temporally inhomogeneous modification of the 2(ν + 1)-dimensional Bessel process with ν  >   ? 1, in which the inhomogeneity is indexed by $\kappa \in [0, 2(\nu+1))$ . We introduce the noncolliding particle systems of the generalized meanders and prove that they are Pfaffian processes, in the sense that any multitime correlation function is given by a Pfaffian. In the infinite particle limit, we show that the elements of matrix kernels of the obtained infinite Pfaffian processes are generally expressed by the Riemann–Liouville differintegrals of functions comprising the Bessel functions J _ν used in the fractional calculus, where orders of differintegration are determined by ν ? κ. As special cases of the two parameters (ν, κ), the present infinite systems include the quaternion determinantal processes studied by Forrester, Nagao and Honner and by Nagao, which exhibit the temporal transitions between the universality classes of random matrix theory.     

To Boundary-Value Problems for Degenerating Pseudoparabolic Equations With Gerasimov–Caputo Fractional Derivative

In this paper we consider a boundary-value problems for degenerating pseudoparabolic equation with variable coefficients and with Gerasimov–Caputo fractional derivative. To solve the problem we obtain a priori estimates in differential and difference settings. These a priori estimates imply uniqueness and stability of the solution with respect to the initial data and the right-hand side on the layer, as well as the convergence of the solution of each of the difference problem to the solution of the corresponding differential problem.     

The Schatten–von Neumann class associated with the Gabor–Riemann–Liouville operator

Periodica Mathematica Hungarica - In this paper, we define the localization operator associated with the Riemann–Liouville operator, and show that it is not only bounded, but it is also in...