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2,158
result(s) for
"Calculus, Operational."
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Uniqueness of fat-tailed self-similar profiles to Smoluchowski’s coagulation equation for a perturbation of the constant kernel
This article is concerned with the question of uniqueness of self-similar profiles for Smoluchowski’s coagulation equation which
exhibit algebraic decay (fat tails) at infinity. More precisely, we consider a rate kernel
Establishing uniqueness of self-similar
profiles for Smoluchowski’s coagulation equation is generally considered to be a difficult problem which is still essentially open.
Concerning fat-tailed self-similar profiles this article actually gives the first uniqueness statement for a non-solvable kernel.
eBook
Spectral expansions of non-self-adjoint generalized Laguerre semigroups
We provide the spectral expansion in a weighted Hilbert space of a substantial class of invariant non-self-adjoint and non-local
Markov operators which appear in limit theorems for positive-valued Markov processes. We show that this class is in bijection with a
subset of negative definite functions and we name it the class of generalized Laguerre semigroups. Our approach, which goes beyond the
framework of perturbation theory, is based on an in-depth and original analysis of an intertwining relation that we establish between
this class and a self-adjoint Markov semigroup, whose spectral expansion is expressed in terms of the classical Laguerre polynomials. As
a by-product, we derive smoothness properties for the solution to the associated Cauchy problem as well as for the heat kernel. Our
methodology also reveals a variety of possible decays, including the hypocoercivity type phenomena, for the speed of convergence to
equilibrium for this class and enables us to provide an interpretation of these in terms of the rate of growth of the weighted Hilbert
space norms of the spectral projections. Depending on the analytic properties of the aforementioned negative definite functions, we are
led to implement several strategies, which require new developments in a variety of contexts, to derive precise upper bounds for these
norms.
eBook
Abstract Algebraic Construction in Fractional Calculus: Parametrised Families with Semigroup Properties
What structure can be placed on the burgeoning field of fractional calculus with assorted kernel functions? This question has been addressed by the introduction of various general kernels, none of which has both a fractional order parameter and a clear inversion relation. Here, we use ideas from abstract algebra to construct families of fractional integral and derivative operators, parametrised by a real or complex variable playing the role of the order. These have the typical behaviour expected of fractional calculus operators, such as semigroup and inversion relations, which allow fractional differential equations to be solved using operational calculus in this general setting, including all types of fractional calculus with semigroup properties as special cases.
Journal Article
Discrete and Continuous Operational Calculus in N-Critical Shocks Reliability Systems with Aging under Delayed Information
We study a reliability system subject to occasional random shocks of random magnitudes W0,W1,W2,… occurring at times τ0,τ1,τ2,…. Any such shock is harmless or critical dependent on Wk≤H or Wk>H, given a fixed threshold H. It takes a total of N critical shocks to knock the system down. In addition, the system ages in accordance with a monotone increasing continuous function δ, so that when δT crosses some sustainability threshold D at time T, the system becomes essentially inoperational. However, it can still function for a while undetected. The most common way to do the checking is at one of the moments τ1,τ2,… when the shocks are registered. Thus, if crossing of D by δ occurs at time T∈τk,τk+1, only at time τk+1, can one identify the system’s failure. The age-related failure is detected with some random delay. The objective is to predict when the system fails, through the Nth critical shock or by the observed aging moment, whichever of the two events comes first. We use and embellish tools of discrete and continuous operational calculus (D-operator and Laplace–Carson transform), combined with first-passage time analysis of random walk processes, to arrive at fully explicit functionals of joint distributions for the observed lifetime of the system and cumulative damage to the system. We discuss various special cases and modifications including the assumption that D is random (and so is T). A number of examples and numerically drawn figures demonstrate the analytic tractability of the results.
Journal Article
Dependent Competing Failure Processes in Reliability Systems
This paper deals with a reliability system hit by three types of shocks ranked as harmless, critical, or extreme, depending on their magnitudes, being below H1, between H1 and H2, and above H2, respectively. The system’s failure is caused by a single extreme shock or by a total of N critical shocks. In addition, the system fails under occurrences of M pairs of shocks with lags less than some δ (δ-shocks) in any order. Thus, the system fails when one of the three named cumulative damages occurs first. Thus, it fails due to the competition of the three associated shock processes. We obtain a closed-form joint distribution of the time-to-failure, shock count upon failure, δ-shock count, and cumulative damage to the system on failure, to name a few. In particular, the reliability function directly follows from the marginal distribution of the failure time. In a modified system, we restrict δ-shocks to those with small lags between consecutive harmful shocks. We treat the system as a generalized random walk process and use an embellished variant of discrete operational calculus developed in our earlier work. We demonstrate analytical tractability of our formulas which are also validated, through Monte Carlo simulation.
Journal Article
Towards a Comprehensive Stability Theory for Feynman’s Operational Calculus: The Time Independent Setting
Via a general construction, we are able to establish a quite general and comprehensive stability theory for Feynman’s operational calculus in the time independent setting. In particular, we are able to establish stability of the operational calculus with respect to general types of the time-ordering measures. While the domain of the operational calculus is somewhat restricted as compared to the “standard” version of the operational calculus (established by Jefferies and Johnson (Russ. J. Math. Phys. 8:153–171,
2001
; Mat. Zametki 70:815–838,
2001
; Adv. Appl. Clifford Algebras 11:239–264,
2001
; Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5:171–199,
2002
), the advantages of this relatively minor domain restriction are significant in that the stability theory (with respect to the time-ordering measures) as it stands to this time is contained, essentially in its entirety, in the principle result of this paper, Theorem
2
. Moreover, Theorem
2
allows immediate, and rather far-reaching extensions of the stability theory that, until now, have not been possible.
Journal Article
Towards a Comprehensive Stability Theory for Feynman’s Operational Calculus: The Time-Dependent Setting
We establish a comprehensive stability theory for Feynman’s operational calculus (informally, the forming of functions of several noncommuting operators) in the time-dependent setting. Indeed, the main theorem, Theorem
2
, contains many of the current stability theorems for the operational calculus and allows the stability theory to be significantly extended. The assumptions needed for the main theorem, Theorem
2
, are rather mild and fit in nicely with the current abstract theory of the operational calculus in the time-dependent setting. Moreover, Theorem
2
allows the use of arbitrary time-ordering measures, as long as the discrete parts of these measures are finitely supported.
Journal Article
A Practical Guide to Prabhakar Fractional Calculus
The Mittag–Leffler function is universally acclaimed as the Queen function of fractional calculus. The aim of this work is to survey the key results and applications emerging from the three-parameter generalization of this function, known as the Prabhakar function. Specifically, after reviewing key historical events that led to the discovery and modern development of this peculiar function, we discuss how the latter allows one to introduce an enhanced scheme for fractional calculus. Then, we summarize the progress in the application of this new general framework to physics and renewal processes. We also provide a collection of results on the numerical evaluation of the Prabhakar function.
Journal Article
Why Fractional Derivatives with Nonsingular Kernels Should Not Be Used
In recent years, many papers discuss the theory and applications of new fractional-order derivatives that are constructed by replacing the singular kernel of the Caputo or Riemann-Liouville derivative by a non-singular (i.e., bounded) kernel. It will be shown here, through rigorous mathematical reasoning, that these non-singular kernel derivatives suffer from several drawbacks which should forbid their use. They fail to satisfy the fundamental theorem of fractional calculus since they do not admit the existence of a corresponding convolution integral of which the derivative is the left-inverse; and the value of the derivative at the initial time
t
= 0 is always zero, which imposes an unnatural restriction on the differential equations and models where these derivatives can be used. For the particular cases of the so-called Caputo-Fabrizio and Atangana-Baleanu derivatives, it is shown that when this restriction holds the derivative can be simply expressed in terms of integer derivatives and standard Caputo fractional derivatives, thus demonstrating that these derivatives contain nothing new.
Journal Article