Related papers: Fractional Operators in the Matrix Variate Case
A review of fundamentals and physical applications of fractional quantum mechanics has been presented. Fundamentals cover fractional Schr\"odinger equation, quantum Riesz fractional derivative, path integral approach to fractional quantum…
In this paper, we investigate the fractional powers of block operator matrices, with a particular focus on their applications to partial differential equations (PDEs). We develop a comprehensive theoretical framework for defining and…
Invariant conditions for conformable fractional problems of the calculus of variations under the presence of external forces in the dynamics are studied. Depending on the type of transformations considered, different necessary conditions of…
In this paper we introduce fractional powers of quaternionic operators. Their definition is based on the theory of slice-hyperholomorphic functions and on the $S$-resolvent operators of the quaternionic functional calculus. The integral…
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)…
A necessary condition and a sufficient condition for one weight norm inequalities on Morrey spaces to hold are given for the fractional maximal operator and the fractional integral operator. We clarify the difference between the behavior of…
Recent results on the construction and applications of the transmutation (transformation) operators are discussed. Three new representations for solutions of the one-dimensional Schr\"odinger equation are considered. Due to the fact that…
We determine the scaling properties of geometric operators such as lengths, areas, and volumes in models of higher derivative quantum gravity by renormalizing appropriate composite operators. We use these results to deduce the fractal…
We introduce a fractional calculus on time scales using the theory of delta (or nabla) dynamic equations. The basic notions of fractional order integral and fractional order derivative on an arbitrary time scale are proposed, using the…
This paper defines a new class of fractional differential operators alongside a family of random variables whose density functions solve fractional differential equations equipped with these operators. These equations can be further used to…
We obtain necessary optimality conditions for variational problems with a Lagrangian depending on a Caputo fractional derivative, a fractional and an indefinite integral. Main results give fractional Euler-Lagrange type equations and…
Most of the special functions of mathematical physics are connected with the representation of Lie groups. The action of elements $D$ of the associated Lie algebras as linear differential operators gives relations among the functions in a…
We introduce a fractional magnetic pseudorelativistic operator for a general fractional order $s\in(0,1)$. First we define a suitable functional setting and we prove some fundamental properties. Then we show the behavior of the operator as…
Fractional processes have gained popularity in financial modeling due to the dependence structure of their increments and the roughness of their sample paths. The non-Markovianity of these processes gives, however, rise to conceptual and…
Recent innovations on the differential calculus for functions of non-commuting variables, begun for a quaternionic variable, are now extended to the case of a general matrix over the complex numbers. The expansion of F(X+Delta) is given to…
In this work we focus on substantial fractional integral and differential operators which play an important role in modeling anomalous diffusion. We introduce a new generalized substantial fractional integral. Generalizations of fractional…
In [1], an operator was introduced which acts parallel to the Riemann-Liouville differintegral on a transformation of the space of real analytic functions and commutes with itself. This paper aims to extend the technique - and its defining…
In this study, we explore the field of physics through the lens of fractional dimensionality. We propose that space is not confined to integer dimensions alone, but can also be understood as a superposition of spaces that exist between…
The theory of derivatives and integrals of non-integer order goes back to Leibniz, Liouville, Grunwald, Letnikov and Riemann. The history of fractional vector calculus (FVC) has only 10 years. The main approaches to formulate a FVC, which…
Partition functions of eigenvalue matrix models possess a number of very different descriptions: as matrix integrals, as solutions to linear and non-linear equations, as tau-functions of integrable hierarchies and as special-geometry…