635 research outputs found
On one-sided Lie nilpotent ideals of associative rings
We prove that a Lie nilpotent one-sided ideal of an associative ring is
contained in a Lie solvable two-sided ideal of . An estimation of derived
length of such Lie solvable ideal is obtained depending on the class of Lie
nilpotency of the Lie nilpotent one-sided ideal of One-sided Lie nilpotent
ideals contained in ideals generated by commutators of the form are also studied.Comment: 5 page
Propagation Speed of the Maximum of the Fundamental Solution to the Fractional Diffusion-Wave Equation
In this paper, the one-dimensional time-fractional diffusion-wave equation
with the fractional derivative of order is revisited. This
equation interpolates between the diffusion and the wave equations that behave
quite differently regarding their response to a localized disturbance: whereas
the diffusion equation describes a process, where a disturbance spreads
infinitely fast, the propagation speed of the disturbance is a constant for the
wave equation. For the time fractional diffusion-wave equation, the propagation
speed of a disturbance is infinite, but its fundamental solution possesses a
maximum that disperses with a finite speed. In this paper, the fundamental
solution of the Cauchy problem for the time-fractional diffusion-wave equation,
its maximum location, maximum value, and other important characteristics are
investigated in detail. To illustrate analytical formulas, results of numerical
calculations and plots are presented. Numerical algorithms and programs used to
produce plots are discussed.Comment: 22 pages 6 figures. This paper has been presented by F. Mainardi at
the International Workshop: Fractional Differentiation and its Applications
(FDA12) Hohai University, Nanjing, China, 14-17 May 201
Operational Rules for a Mixed Operator of the Erdélyi-Kober Type
2000 Mathematics Subject Classification: 26A33 (main), 44A40, 44A35, 33E30, 45J05, 45D05In the paper, the machinery of the Mellin integral transform is applied
to deduce and prove some operational relations for a general operator of the
Erdélyi-Kober type. This integro-differential operator is a composition of
a number of left-hand sided and right-hand sided Erdélyi-Kober derivatives
and integrals. It is referred to in the paper as a mixed operator of the
Erdélyi-Kober type.
For special values of parameters, the operator is reduced to some well
known differential, integro-differential, or integral operators studied earlier
by different authors. The differential operators of hyper-Bessel type, the
Riemann-Liouville fractional derivative, the Caputo fractional derivative,
and the multiple Erdélyi-Kober fractional derivatives and integrals are examples of its particular cases. In the general case however, the constructions
suggested in the paper are new objects not yet well studied in the literature. The initial impulse to consider the operators presented in the paper
arose while the author studied a problem to find scale-invariant solutions of
some partial differential equations of fractional order: It turned out, that
scale-invariant solutions of these partial differential equations of fractional
order are described by ordinary differential equations of fractional order
containing some particular cases of the mixed operator of Erdélyi-Kober
type
Maximum Principle and Its Application for the Time-Fractional Diffusion Equations
MSC 2010: 26A33, 33E12, 35B45, 35B50, 35K99, 45K05 Dedicated to Professor Rudolf Gorenflo
on the occasion of his 80th anniversaryIn the paper, maximum principle for the generalized time-fractional diffusion equations including the multi-term diffusion equation and the diffusion equation of distributed order is formulated and discussed. In these equations, the time-fractional derivative is defined in the Caputo sense. In contrast to the Riemann-Liouville fractional derivative, the Caputo fractional derivative is shown to possess a suitable generalization of the extremum principle well-known for ordinary derivative. As an application, the maximum principle is used to get some a priori estimates for solutions of initial-boundary-value problems for the generalized time-fractional diffusion equations and then to prove uniqueness of their solutions
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