Recent zbMATH articles in MSC 26Ahttps://www.zbmath.org/atom/cc/26A2021-04-16T16:22:00+00:00WerkzeugApplication of Sumudu transform in generalized fractional reaction-diffusion equation.https://www.zbmath.org/1456.352092021-04-16T16:22:00+00:00"Alkahtani, Badr"https://www.zbmath.org/authors/?q=ai:alkahtani.badr-saad-t"Gulati, Vartika"https://www.zbmath.org/authors/?q=ai:gulati.vartika"Kılıçman, Adem"https://www.zbmath.org/authors/?q=ai:kilicman.ademSummary: In this paper we investigate the solution of a generalized nonlinear fractional reaction diffusion equation by the application of Sumudu Transform.Symmetry classification and exact solutions of \((3+1)\)-dimensional fractional nonlinear incompressible non-hydrostatic coupled Boussinesq equations.https://www.zbmath.org/1456.760242021-04-16T16:22:00+00:00"Singla, Komal"https://www.zbmath.org/authors/?q=ai:singla.komal"Gupta, R. K."https://www.zbmath.org/authors/?q=ai:gupta.rajeev-kumar|gupta.rahul-k|gupta.raj-kumar|gupta.rajesh-kumar|gupta.rabindra-kumar|gupta.radhey-k|gupta.rajendra-k|gupta.rohit-k|gupta.ram-kumarSummary: The symmetry classifications of two fractional higher dimensional nonlinear systems, namely, \((3+1)\)-dimensional incompressible non-hydrostatic Boussinesq equations and \((3+1)\)-dimensional Boussinesq equations with viscosity, are discussed. Both the fractional Boussinesq equations are considered to have derivatives with respect to all variables of fractional type, and some exact solutions are reported along with graphical illustrations.
{\copyright 2021 American Institute of Physics}Fractional Klein-Kramers dynamics for subdiffusion and Itô formula.https://www.zbmath.org/1456.825662021-04-16T16:22:00+00:00"Orzeł, Sebastian"https://www.zbmath.org/authors/?q=ai:orzel.sebastian"Weron, Aleksander"https://www.zbmath.org/authors/?q=ai:weron.aleksanderAdaptive synchronization of Julia sets generated by Mittag-Leffler function.https://www.zbmath.org/1456.370452021-04-16T16:22:00+00:00"Wang, Yupin"https://www.zbmath.org/authors/?q=ai:wang.yupin"Liu, Shutang"https://www.zbmath.org/authors/?q=ai:liu.shutang"Li, Hui"https://www.zbmath.org/authors/?q=ai:li.hui.2|li.hui.5|li.hui.1|li.hui.3|li.hui.4|li.huiSummary: This paper investigates Julia sets for a class of complex uncertain discrete systems involving Mittag-Leffler functions. Julia sets of some classical maps are generalized, and the influence of parameter \(\alpha\) on the characteristics of these fractal sets is discussed. An adaptive control strategy is proposed to synchronize Julia sets of two systems with different parameters and identify their unknown parameters. An example is presented to further verify the correctness and effectiveness of main theoretical results.Mean value theorems and convexity: an example of cross-fertilization of two mathematical items.https://www.zbmath.org/1456.260052021-04-16T16:22:00+00:00"Hiriart-Urruty, Jean-Baptiste"https://www.zbmath.org/authors/?q=ai:hiriart-urruty.jean-baptisteIn this interesting paper, the author shows on particular examples how mean value theorems of differential calculus and results of convexity theory can have impact on each other. First, he considers the real-value case and then the vectorial case is studied. A new result in the first part is related to the mean-value points set in the Lagrange mean value theorem, by showing that the fact that this set is an interval has equivalent formulations related to the level sets of derivative function, to the monotonicity of derivative function or to the convexity or concavity of the function.
Reviewer: József Sándor (Cluj-Napoca)Certain fractional integral and fractional derivative formulae with their image formulae involving generalized multi-index Mittag-Leffler function.https://www.zbmath.org/1456.260072021-04-16T16:22:00+00:00"Chand, Mehar"https://www.zbmath.org/authors/?q=ai:chand.mehar"Kasmaei, Hamed Daei"https://www.zbmath.org/authors/?q=ai:kasmaei.hamed-daei"Senol, Mehmet"https://www.zbmath.org/authors/?q=ai:senol.mehmetSummary: The main objective of this paper is to establish some image formulas by applying the Riemann-Liouville fractional derivative and integral operators to the product of generalized multiindex Mittag-Leffler function \(E^{\gamma,q}_{(\alpha_j,\beta_j)_m}(.)\). Some more image formulas are derived by applying integral transforms. The results obtained here are quite general in nature and capable of yielding a very large number of known and (presumably) new results.Generalized inverses of increasing functions and Lebesgue decomposition.https://www.zbmath.org/1456.600482021-04-16T16:22:00+00:00"de la Fortelle, Arnaud"https://www.zbmath.org/authors/?q=ai:de-la-fortelle.arnaudSummary: The reader should be aware of the explanatory nature of this article. Its main goal is to introduce to a broader vision of a topic than a more focused research paper, demonstrating some new results but mainly starting from some general consideration to build an overview of a theme with links to connected problems.
Our original question was related to the height of random growing trees. When investigating limit processes, we may consider some measures that are defined by increasing functions and their generalized inverses. And this leads to the analysis of Lebesgue decomposition of generalized inverses. Moreover, since the measures that motivated us initially are stochastic, there arises the idea of studying the continuity property of this transform in order to take limits.
When scaling growing processes like trees, time origin and scale can be replaced by another process. This leads us to a clock metaphor, to consider an increasing function as a clock reading from a given timeline. This is nothing more than an explanatory vision, not a mathematical concept, but this is the nature of this paper. So we consider an increasing function as a time change between two timelines; it leads to the idea that an increasing function and its generalized inverse play symmetric roles. We then introduce a universal time that links symmetrically an increasing function and its generalized inverse. We show how both are smoothly defined from this universal time. This allows to describe the Lebesgue decomposition for both an increasing function and its generalized inverse.Equi-Riemann and equi-Riemann type integrable functions with values in a Banach space.https://www.zbmath.org/1456.260112021-04-16T16:22:00+00:00"Mondal, Pratikshan"https://www.zbmath.org/authors/?q=ai:mondal.pratikshan"Dey, Lakshmi Kanta"https://www.zbmath.org/authors/?q=ai:dey.lakshmi-kanta"Jaker Ali, Sk."https://www.zbmath.org/authors/?q=ai:jaker-ali.skSummary: In this paper we study equi-Riemann and equi-Riemann-type integrability of a collection of functions defined on a closed interval of \(\mathbb{R}\) with values in a Banach space. We obtain some properties of such collections and interrelations among them. Moreover we establish equi-integrability of different types of collections of functions. Finally, we obtain relations among equi-Riemann integrability with other properties of a collection of functions.\((q,c)\)-derivative operator and its applications.https://www.zbmath.org/1456.050192021-04-16T16:22:00+00:00"Zhang, Helen W. J."https://www.zbmath.org/authors/?q=ai:zhang.helen-w-jSummary: In this paper, we introduce new concept of \((q,c)\)-derivative operator of an analytic function, which generalizes the ordinary \(q\)-derivative operator. From this definition, we give the concept of \((q,c)\)-Rogers-Szegö polynomials, and obtain the expanded theorem involving \((q,c)\)-Rogers-Szegö polynomials. In addition, we construct two kinds \((q,c)\)-exponential operators, apply them to \((q,c)\)-exponential functions, and establish some new identities. At last, some properties of \((q,c)\)-Rogers-Szegö polynomials are discussed.Exact solutions and conservation laws of multi Kaup-Boussinesq system with fractional order.https://www.zbmath.org/1456.352212021-04-16T16:22:00+00:00"Singla, Komal"https://www.zbmath.org/authors/?q=ai:singla.komal"Rana, M."https://www.zbmath.org/authors/?q=ai:rana.mehwish|rana.meenakshiSummary: The purpose of the present work is to investigate exact solutions of the fractional order multi Kaup-Boussinesq system with \(l=2\) by using the group invariance approach and power series expansion method. Due to the significance of conserved vectors in terms of integrability and behaviour of nonlinear systems, the conservation laws are also derived by testing the nonlinear self-adjointness.A note on the multiple fractional integrals defined on the product of nonhomogeneous measure spaces.https://www.zbmath.org/1456.260092021-04-16T16:22:00+00:00"Kokilashvili, Vakhtang"https://www.zbmath.org/authors/?q=ai:kokilashvili.vakhtang-m"Tsanava, Tsira"https://www.zbmath.org/authors/?q=ai:tsanava.tsiraSummary: In this note we present a trace type inequality in the mixed-norm Lebesgue spaces for multiple fractional integrals defined on an arbitrary measure quasi-metric space.Time fractional and space nonlocal stochastic Boussinesq equations driven by Gaussian white noise.https://www.zbmath.org/1456.370852021-04-16T16:22:00+00:00"Shen, Tianlong"https://www.zbmath.org/authors/?q=ai:shen.tianlong"Huang, Jianhua"https://www.zbmath.org/authors/?q=ai:huang.jianhua"Zeng, Caibin"https://www.zbmath.org/authors/?q=ai:zeng.caibinSummary: We present the time-spatial regularity of the nonlocal stochastic convolution for Caputo-type time fractional nonlocal Ornstein-Uhlenbeck equations, and establish the existence and uniqueness of mild solutions for time fractional and space nonlocal stochastic Boussinesq equations driven by Gaussian white noise.Riesz potential versus fractional Laplacian.https://www.zbmath.org/1456.352192021-04-16T16:22:00+00:00"Ortigueira, Manuel D."https://www.zbmath.org/authors/?q=ai:ortigueira.manuel-d"Laleg-Kirati, Taous-Meriem"https://www.zbmath.org/authors/?q=ai:laleg-kirati.taous-meriem"Machado, J. A. Tenreiro"https://www.zbmath.org/authors/?q=ai:machado.jose-antonio-tenreiroOn fractional and fractal Einstein's field equations.https://www.zbmath.org/1456.830062021-04-16T16:22:00+00:00"El-Nabulsi, Rami Ahmad"https://www.zbmath.org/authors/?q=ai:el-nabulsi.rami-ahmad"Golmankhaneh, Alireza Khalili"https://www.zbmath.org/authors/?q=ai:golmankhaneh.alireza-khaliliOn initial and terminal value problems for fractional nonclassical diffusion equations.https://www.zbmath.org/1456.352222021-04-16T16:22:00+00:00"Tuan, Nguyen Huy"https://www.zbmath.org/authors/?q=ai:nguyen-huy-tuan."Caraballo, Tomás"https://www.zbmath.org/authors/?q=ai:caraballo.tomasSummary: In this paper, we consider fractional nonclassical diffusion equations under two forms: initial value problem and terminal value problem. For an initial value problem, we study local existence, uniqueness, and continuous dependence of the mild solution. We also present a result on unique continuation and a blow-up alternative for mild solutions of fractional pseudo-parabolic equations. For the terminal value problem, we show the well-posedness of our problem in the case \(0<\alpha\leq 1\) and show the ill-posedness in the sense of Hadamard in the case \(\alpha>1\). Then, under the a priori assumption on the exact solution belonging to a Gevrey space, we propose the Fourier truncation method for stabilizing the ill-posed problem. A stability estimate of logarithmic-type in \(L^q\) norm is first established.Memory-dependent derivative versus fractional derivative. I: Difference in temporal modeling.https://www.zbmath.org/1456.260102021-04-16T16:22:00+00:00"Wang, Jin-Liang"https://www.zbmath.org/authors/?q=ai:wang.jinliang.2"Li, Hui-Feng"https://www.zbmath.org/authors/?q=ai:li.huifengSummary: Since the memory-dependent derivative (MDD) was developed in 2011, it has become a new branch of Fractional Calculus which is still in the ascendant nowadays. How to understand MDD and fractional derivative (FD)? What are the advantages and disadvantages for them? How do they behave in Modeling? These questions guide going deep into the illustration of memory effect. Though the FD is defined on an interval, it mainly reflects the local change. Relative to the FD, the physical meaning of MDD is much clearer. The time-delay reflects the duration of memory effect, and the kernel function reflects the dependent weight. The results show that the MDD is more suitable for temporal modeling. In addition, a numerical algorithm for MDD is also developed here.Multilinear Fefferman-Stein inequality and its generalizations.https://www.zbmath.org/1456.420212021-04-16T16:22:00+00:00"Imerlishvili, Giorgi"https://www.zbmath.org/authors/?q=ai:imerlishvili.giorgi"Meskhi, Alexander"https://www.zbmath.org/authors/?q=ai:meskhi.alexander"Xue, Qingying"https://www.zbmath.org/authors/?q=ai:xue.qingyingSummary: The Fefferman-Stein type inequalities are established for multilinear fractional maximal operators with a variable parameter defined with respect to the basis \(\mathcal{B}\) on \(\mathbb{R}^n\) which may be both either \(\mathcal{Q}\) or \(\mathcal{R}\), where \(\mathcal{Q}\) (resp., \(\mathcal{R}\)) consists of all cubes (resp., of \(n\)-dimensional intervals) with sides parallel to the coordinate axes. Some related two-weight boundedness problems are also investigated.A new material identification pattern for the fractional Kelvin-Zener model describing biomaterials and human tissues.https://www.zbmath.org/1456.741252021-04-16T16:22:00+00:00"Spasic, Dragan T."https://www.zbmath.org/authors/?q=ai:spasic.dragan-t"Kovincic, Nemanja I."https://www.zbmath.org/authors/?q=ai:kovincic.nemanja-i"Dankuc, Dragan V."https://www.zbmath.org/authors/?q=ai:dankuc.dragan-vSummary: The aim of this study is to describe several biomaterials and tissues using a simple material identification pattern applied to the fractional Kelvin-Zener model of viscoelastic body and standard mechanical tests. Each of the descriptions comprises the order of fractional derivative of stress and strain, modulus of elasticity, and stress and strain relaxation constants that obey restrictions imposed by the Clausius-Duhem inequality. These four parameters are obtained by use of the Laplace transform, Post's inversion formula and Newton's method. The suggested approach can serve as an alternative to quasilinear viscoelasticity providing a physically uniform quantitative measure for biomaterials/tissues comparison and can be applied to real data. It works for nonsmooth inputs too. Regarding biomaterials the comparison between an etched poly lactic-co-glycolic acid membrane and the corresponding composite scaffold was made. With respect to human tissues the tympanic membrane, the stapedial tendon, and the stapedial annular ligament were described. The obtained mechanical response for examined cases is in agreement with the experimentally recorded one.A. Kharazishvili's some results of on the structure of pathological functions.https://www.zbmath.org/1456.260032021-04-16T16:22:00+00:00"Kirtadze, Aleks"https://www.zbmath.org/authors/?q=ai:kirtadze.aleks-p"Pantsulaia, Gogi"https://www.zbmath.org/authors/?q=ai:pantsulaia.gogi-rauliThe authors present a brief survey of A. Kharazishvili's works devoted to real-valued functions with strange, pathological and paradoxical structural properties, e.g. absolutely non-measurable functions, Sierpiński-Zygmund functions, sup-measurable and weakly sup-measurable functions of two real variables, and non-measurable functions of two real variables for which both iterated integrals exist.
Reviewer: George Stoica (Saint John)A variable exponent Sobolev theorem for fractional integrals on quasimetric measure spaces.https://www.zbmath.org/1456.460362021-04-16T16:22:00+00:00"Samko, Stefan"https://www.zbmath.org/authors/?q=ai:samko.stefan-gSummary: We show that the fractional operator \(I^{\alpha(\cdot)}\) of variable order on a bounded open set \(\Omega\) in a quasimetric measure space \((X,d\mu)\) with the growth condition the measure \(\mu\), is bounded from the variable exponent Lebesgue space \(L^{p(\cdot)}(\Omega)\) into \(L^{q(\cdot)}(\Omega)\) in the case \(\inf_{x\in\Omega}[n(x)-\alpha(x)p(x)] > 0\), where \(\frac{1}{q(x)}=\frac{1}{p(x)}-\frac{\alpha(x)}{n(x)}\) and \(n(x)\) comes from the growth condition under the log-continuity condition on \(p(x)\).Fractional characteristic times and dissipated energy in fractional linear viscoelasticity.https://www.zbmath.org/1456.740252021-04-16T16:22:00+00:00"Colinas-Armijo, Natalia"https://www.zbmath.org/authors/?q=ai:colinas-armijo.natalia"Di Paola, Mario"https://www.zbmath.org/authors/?q=ai:di-paola.mario"Pinnola, Francesco P."https://www.zbmath.org/authors/?q=ai:pinnola.francesco-paoloSummary: In fractional viscoelasticity the stress-strain relation is a differential equation with non-integer operators (derivative or integral). Such constitutive law is able to describe the mechanical behavior of several materials, but when fractional operators appear, the elastic and the viscous contribution are inseparable and the characteristic times (relaxation and retardation time) cannot be defined. This paper aims to provide an approach to separate the elastic and the viscous phase in the fractional stress-strain relation with the aid of an equivalent classical model (Kelvin-Voigt or Maxwell). For such equivalent model the parameters are selected by an optimization procedure. Once the parameters of the equivalent model are defined, characteristic times of fractional viscoelasticity are readily defined as ratio between viscosity and stiffness.
In the numerical applications, three kinds of different excitations are considered, that is, harmonic, periodic, and pseudo-stochastic. It is shown that, for any periodic excitation, the equivalent models have some important features: (i) the dissipated energy per cycle at steady-state coincides with the Staverman-Schwarzl formulation of the fractional model, (ii) the elastic and the viscous coefficients of the equivalent model are strictly related to the storage and the loss modulus, respectively.All the generalized characteristics for the solution to a Hamilton-Jacobi equation with the initial data of the Takagi function.https://www.zbmath.org/1456.350822021-04-16T16:22:00+00:00"Fujita, Yasuhiro"https://www.zbmath.org/authors/?q=ai:fujita.yasuhiro"Hamamuki, Nao"https://www.zbmath.org/authors/?q=ai:hamamuki.nao"Yamaguchi, Norikazu"https://www.zbmath.org/authors/?q=ai:yamaguchi.norikazuSummary: We determine all the generalized characteristics for the solution to a Hamilton-Jacobi equation with the initial data of the Takagi function, which is everywhere continuous and nowhere differentiable. This result clarifies how singularities of the solution propagate along generalized characteristics. Moreover it turns out that the Takagi function still keeps the validity of the recent results in [\textit{P. Albano} et al., J. Differ. Equations 268, No. 4, 1412--1426 (2020; Zbl 1437.35153)], in which locally Lipschitz continuous initial data are handled.On a generalization of the class of Jensen convex functions.https://www.zbmath.org/1456.260122021-04-16T16:22:00+00:00"Lara, Teodoro"https://www.zbmath.org/authors/?q=ai:lara.teodoro"Quintero, Roy"https://www.zbmath.org/authors/?q=ai:quintero.roy"Rosales, Edgar"https://www.zbmath.org/authors/?q=ai:rosales.edgar"Sánchez, José Luis"https://www.zbmath.org/authors/?q=ai:sanchez.jose-luisSummary: The main objective of this article is to introduce a new class of real valued functions that include the well-known class of \({m}\)-convex functions introduced by Toader (1984). The members of this collection are called Jensen \({m}\)-convex and are defined, for \({m \in (0,1]}\), via the functional inequality
\[
f\left(\frac{x+y}{c_m} \right)\leq \frac{f(x)+f(y)}{c_m} \quad (x,y \in [0,b]),
\]
where \({c_{m} := \frac{m+1}{m}}\). These functions generate a new kind of functional convexity that is studied in terms of its behavior with respect to basic algebraic operations such as sums, products, compositions, etc. in this paper. In particular, it is proved that any starshaped Jensen convex function is Jensen \({m}\)-convex. At the same time an interesting example (Example 3) shows how the classes of Jensen \({m}\)-convex functions depend on \({m}\). All the techniques employed come from traditional basic calculus and most of the calculations have been done with Mathematica 8.0.0 and validated with Maple 15 as well as all the figures included.On fractional differentiation.https://www.zbmath.org/1456.260082021-04-16T16:22:00+00:00"Gladkov, S. O."https://www.zbmath.org/authors/?q=ai:gladkov.sergei-oktyabrinovich"Bogdanova, S. B."https://www.zbmath.org/authors/?q=ai:bogdanova.sofya-borisovnaSummary: Due to the operation of fractional differentiation introduced with the help of Fourier integral, the results of calculating fractional derivatives for certain types of functions are given. Using the numerical method of integration, the values of fractional derivatives for arbitrary dimensionality \(\varepsilon \), (where \(\varepsilon\) is any number greater than zero) are calculated. It is proved that for integer values of \(\varepsilon\) we obtain ordinary derivatives of the first, second and more high orders. As an example it was considered heat conduction equation of Fourier, where spatial derivation was realized with the use of fractional derivatives. Its solution is given by Fourier integral. Mmoreover, it was shown that integral went into the required results in special case of the whole \(\varepsilon\) obtained in \(n\)-dimensional case, where \(n = 1, 2\dots \), etc.Finite difference approximation of a generalized time-fractional telegraph equation.https://www.zbmath.org/1456.650612021-04-16T16:22:00+00:00"Delić, Aleksandra"https://www.zbmath.org/authors/?q=ai:delic.aleksandra"Jovanović, Boško S."https://www.zbmath.org/authors/?q=ai:jovanovic.bosko-s"Živanović, Sandra"https://www.zbmath.org/authors/?q=ai:zivanovic.sandra.1Summary: We consider a class of a generalized time-fractional telegraph equations. The existence of a weak solution of the corresponding initial-boundary value problem has been proved. A finite difference scheme approximating the problem is proposed, and its stability is proved. An estimate for the rate of convergence, in special discrete energetic Sobolev's norm, is obtained. The theoretical results are confirmed by numerical examples.Some boundary-value problems of fractional-differential mobile-immobile migration dynamics in a profile filtration flow.https://www.zbmath.org/1456.761262021-04-16T16:22:00+00:00"Bulavatsky, V. M."https://www.zbmath.org/authors/?q=ai:bulavatskyi.volodymyr-m"Bohaienko, V. O."https://www.zbmath.org/authors/?q=ai:bohaienko.v-oSummary: Within the framework of fractional-differential mathematical model, the formulation of boundary-value problems of convective diffusion of soluble substances with regard to immobilization under the conditions of stationary profile filtration of groundwater from reservoir to drainage is performed. In case of averaging the filtration rate over the complex potential region, closed solutions of boundary-value problems, corresponding to classical and nonlocal boundary conditions are obtained. In the general case of the variable filtration velocity, a technique is developed for the numerical solution to the boundary-value problem of convective diffusion in a fractional-differential formulation, the problems of parallelizing computations are covered, and the results of the computer experiments are presented.Separating sets by functions and by sets.https://www.zbmath.org/1456.260042021-04-16T16:22:00+00:00"Szyszkowska, Paulina"https://www.zbmath.org/authors/?q=ai:szyszkowska.paulinaThe topic of characterising sets that can be separated with a function from a given class \(\mathcal F\) (of generalised continuous functions), motivated by Urysohn's lemma, has originated with a paper by \textit{A. Maliszewski} [Fundam. Math. 175, No. 3, 271--283 (2002; Zbl 1017.26002)] and has been followed by a number of works related to Maliszewski's colleagues and students; for more references on this topic see the bibliography of the article under review.
This article is a follow-up to [the author, Topology Appl. 206, 46--57 (2016; Zbl 1345.26007)] and focuses, mainly, on the problem of characterising sets that can be separated with Darboux quasi-continuous and upper semicontinuous (\(\mathscr{D\!Q}\textup{usc}\)) functions.
Two sets \(A_0,A_1\subset\mathbb R\) are said to be (classically exactly) separated with a function \(f\colon\mathbb R\to\mathbb R\) if \(f^{-1}(0)=A_0\) and \(f^{-1}(1)=A_1\). A characterisation of pairs \((A_0,A_1)\) which can be separated with a \(\mathscr{D\!Q}\textup{usc}\) function has been provided by the author first in [loc. cit.] and sounds there (Theorem~3.7) as follows: there are semi-closed \(\mathscr F_\sigma\) sets \(A_0,A_1\) such that (i)~\(\operatorname{cl}A_1\cap A_0=\emptyset\) and (ii)~\(\mathbb R\setminus A_0\) and \(\mathbb R\setminus(A_1\cup G)\) are both bilaterally dense in themselves; here \(G\) is the union of all intervals contiguous to \(\operatorname{cl}A_1\) that meet \(A_0\). In the present work, the author points out an error in that characterisation (Example~3.2). Then, along the same lines as in [Szczuka, loc. cit.] she proves that the previously claimed characterisation becomes complete if one adds the condition that also \(A_1\cup G\) is semi-closed (the main result, Theorem 3.3). Some other modes of separation with \(\mathscr{D\!Q}\textup{usc}\) functions are also considered.
Reviewer: Piotr Sworowski (Bydgoszcz)Wave propagation in a fractional viscoelastic Andrade medium: diffusive approximation and numerical modeling.https://www.zbmath.org/1456.740242021-04-16T16:22:00+00:00"Ben Jazia, A."https://www.zbmath.org/authors/?q=ai:ben-jazia.a"Lombard, B."https://www.zbmath.org/authors/?q=ai:lombard.bruno"Bellis, C."https://www.zbmath.org/authors/?q=ai:bellis.cedricSummary: This study focuses on the numerical modeling of wave propagation in fractionally-dissipative media. These viscoelastic models are such that the attenuation is frequency-dependent and follows a power law with non-integer exponent within certain frequency regimes. As a prototypical example, the Andrade model is chosen for its simplicity and its satisfactory fits of experimental flow laws in rocks and metals. The corresponding constitutive equation features a fractional derivative in time, a non-local-in-time term that can be expressed as a convolution product whose direct implementation bears substantial memory cost. To circumvent this limitation, a diffusive representation approach is deployed, replacing the convolution product by an integral of a function satisfying a local time-domain ordinary differential equation. An associated quadrature formula yields a local-in-time system of partial differential equations, which is then proven to be well-posed. The properties of the resulting model are also compared to those of the Andrade model. The quadrature scheme associated with the diffusive approximation, and constructed either from a classical polynomial approach or from a constrained optimization method, is investigated. Finally, the benefits of using the latter approach are highlighted as it allows to minimize the discrepancy with the original model. Wave propagation simulations in homogeneous domains are performed within a split formulation framework that yields an optimal stability condition and which features a joint fourth-order time-marching scheme coupled with an exact integration step. A set of numerical experiments is presented to assess the overall approach. Therefore, in this study, the diffusive approximation is demonstrated to provide an efficient framework for the theoretical and numerical investigations of the wave propagation problem associated with the fractional viscoelastic medium considered.Perturbing the mean value theorem: implicit functions, the Morse lemma, and beyond.https://www.zbmath.org/1456.260062021-04-16T16:22:00+00:00"Lowry-Duda, David"https://www.zbmath.org/authors/?q=ai:lowry-duda.david"Wheeler, Miles H."https://www.zbmath.org/authors/?q=ai:wheeler.miles-hOne can begin with the authors' abstract:
``The mean value theorem of calculus states that, given a differentiable function \(f\) on an interval \([a, b]\), there exists at least one mean value abscissa \(c\) such that the slope of the tangent line at \((c, f (c))\) is equal to the slope of the secant line through \((a, f (a))\) and \((b, f (b))\). In this article, we study how the choices of \(c\) relate to varying the right endpoint \(b\). In particular, we ask: When we can write \(c\) as a continuous function of \(b\) in some interval? As we explore this question, we touch on the implicit function theorem, a simplified version of the Morse lemma, and the theory of analytic functions.''
It is noted that the mean value theorem is one of the truly fundamental theorems of calculus. This theorem, auxiliary notions, and the main statements of the problem of the present research are briefly explained. Some examples are considered.
Several statements are proven with explanations. Some main results of this investigation are summarized in one theorem. Also, a special attention is given to further generalizations. It includes some explanations and proving certain statements.
Finally, some open questions related to the present research are given and discussed.
Reviewer: Symon Serbenyuk (Kyïv)Lipschitz estimates for functions of Dirac and Schrödinger operators.https://www.zbmath.org/1456.811742021-04-16T16:22:00+00:00"Skripka, A."https://www.zbmath.org/authors/?q=ai:skripka.a-n|skripka.annaSummary: We establish new Lipschitz-type bounds for functions of operators with noncompact perturbations that produce Schatten class resolvent differences. The results apply to suitable perturbations of Dirac and Schrödinger operators, including some long-range and random potentials, and to important classes of test functions. The key feature of these bounds is an explicit dependence on the Lipschitz seminorm and decay parameters of the respective scalar functions and, in the case of Dirac and Schrödinger operators, on the \(L^p\)- or \(\mathcal{l}^p(L^2)\)-norm of the potential.
{\copyright 2021 American Institute of Physics}Properties of field functionals and characterization of local functionals.https://www.zbmath.org/1456.812962021-04-16T16:22:00+00:00"Brouder, Christian"https://www.zbmath.org/authors/?q=ai:brouder.christian"Dang, Nguyen Viet"https://www.zbmath.org/authors/?q=ai:dang.nguyen-viet"Laurent-Gengoux, Camille"https://www.zbmath.org/authors/?q=ai:laurent-gengoux.camille"Rejzner, Kasia"https://www.zbmath.org/authors/?q=ai:rejzner.kasiaSummary: Functionals (i.e., functions of functions) are widely used in quantum field theory and solid-state physics. In this paper, functionals are given a rigorous mathematical framework and their main properties are described. The choice of the proper space of test functions (smooth functions) and of the relevant concept of differential (Bastiani differential) are discussed. The relation between the multiple derivatives of a functional and the corresponding distributions is described in detail. It is proved that, in a neighborhood of every test function, the support of a smooth functional is uniformly compactly supported and the order of the corresponding distribution is uniformly bounded. Relying on a recent work by Dabrowski, several spaces of functionals are furnished with a complete and nuclear topology. In view of physical applications, it is shown that most formal manipulations can be given a rigorous meaning. A new concept of local functionals is proposed and two characterizations of them are given: the first one uses the additivity (or Hammerstein) property, the second one is a variant of Peetre's theorem. Finally, the first step of a cohomological approach to quantum field theory is carried out by proving a global Poincaré lemma and defining multi-vector fields and graded functionals within our framework.{
\copyright 2018 American Institute of Physics}Hamiltonian model and dynamic analyses for a hydro-turbine governing system with fractional item and time-lag.https://www.zbmath.org/1456.740382021-04-16T16:22:00+00:00"Xu, Beibei"https://www.zbmath.org/authors/?q=ai:xu.beibei"Chen, Diyi"https://www.zbmath.org/authors/?q=ai:chen.diyi"Zhang, Hao"https://www.zbmath.org/authors/?q=ai:zhang.hao.3|zhang.hao.4|zhang.hao.2|zhang.hao.1|zhang.hao"Wang, Feifei"https://www.zbmath.org/authors/?q=ai:wang.feifei"Zhang, Xinguang"https://www.zbmath.org/authors/?q=ai:zhang.xinguang"Wu, Yonghong"https://www.zbmath.org/authors/?q=ai:wu.yonghong.1Summary: This paper focus on a Hamiltonian mathematical modeling for a hydro-turbine governing system including fractional item and time-lag. With regards to hydraulic pressure servo system, a universal dynamical model is proposed, taking into account the viscoelastic properties and low-temperature impact toughness of constitutive materials as well as the occurrence of time-lag in the signal transmissions. The Hamiltonian model of the hydro-turbine governing system is presented using the method of orthogonal decomposition. Furthermore, a novel Hamiltonian function that provides more detailed energy information is presented, since the choice of the Hamiltonian function is the key issue by putting the whole dynamical system to the theory framework of the generalized Hamiltonian system. From the numerical experiments based on a real large hydropower station, we prove that the Hamiltonian function can describe the energy variation of the hydro-turbine suitably during operation. Moreover, the effect of the ractional \(\alpha\) and the time-lag \(\tau\) on the dynamic variables of the hydro-turbine governing system are explored and their change laws identified, respectively. The physical meaning between fractional calculus and time-lag are also discussed in nature. All of the above theories and numerical results are expected to provide a robust background for the safe operation and control of large hydropower stations.On the weak computability of continuous real functions.https://www.zbmath.org/1456.030642021-04-16T16:22:00+00:00"Bauer, Matthew S."https://www.zbmath.org/authors/?q=ai:bauer.matthew-steven"Zheng, Xizhong"https://www.zbmath.org/authors/?q=ai:zheng.xizhongSummary: In computable analysis, sequences of rational numbers which effectively converge to a real number \(x\) are used as the (\(\rho\text{-})\) names of \(x\). A real number \(x\) is computable if it has a computable name, and a real function \(f\) is computable if there is a Turing machine \(M\) which computes \(f\) in the sense that, \(M\) accepts any \(\rho\)-name of \(x\) as input and outputs a \(\rho\)-name of \(f(x)\) for any \(x\) in the domain of \(f\). By weakening the effectiveness requirement of the convergence and classifying the converging speeds of rational sequences, several interesting classes of real numbers of weak computability have been introduced in literature, e.g., in addition to the class of computable real numbers (EC), we have the classes of semi-computable (SC), weakly computable (WC), divergence bounded computable (DBC) and computably approximable real numbers (CA). In this paper, we are interested in the weak computability of continuous real functions and try to introduce an analogous classification of weakly computable real functions. We present definitions of these functions by Turing machines as well as by sequences of rational polygons and prove these two definitions are not equivalent. Furthermore, we explore the properties of these functions, and among others, show their closure properties under arithmetic operations and composition.
For the entire collection see [Zbl 1391.03010].Banded operational matrices for Bernstein polynomials and application to the fractional advection-dispersion equation.https://www.zbmath.org/1456.352142021-04-16T16:22:00+00:00"Jani, M."https://www.zbmath.org/authors/?q=ai:jani.mostafa|jani.mahendra"Babolian, E."https://www.zbmath.org/authors/?q=ai:babolian.esmail|babolian.esmaeil"Javadi, S."https://www.zbmath.org/authors/?q=ai:javadi.shahnam|javadi.samaneh|javadi.seyed-mohammad-mahdi|javadi.sonya|javadi.samanech"Bhatta, D."https://www.zbmath.org/authors/?q=ai:bhatta.dambaru-d|bhatta.dilliSummary: In the papers, dealing with derivation and applications of operational matrices of Bernstein polynomials, a basis transformation, commonly a transformation to power basis, is used. The main disadvantage of this method is that the transformation may be ill-conditioned. Moreover, when applied to the numerical simulation of a functional differential equation, it leads to dense operational matrices and so a dense coefficient matrix is obtained. In this paper, we present a new property for Bernstein polynomials. Using this property, we build exact banded operational matrices for derivatives of Bernstein polynomials. Next, as an application, we propose a new numerical method based on a Petrov-Galerkin variational formulation and the new operational matrices utilizing the dual Bernstein basis for the time-fractional advection-dispersion equation. We show that the proposed method leads to a narrow-banded linear system and so less computational effort is required to obtain the desired accuracy for the approximate solution. We also obtain the error estimation for the method. Some numerical examples are provided to demonstrate the efficiency of the method and to support the theoretical claims.On increasing solutions of half-linear delay differential equations.https://www.zbmath.org/1456.340732021-04-16T16:22:00+00:00"Matucci, Serena"https://www.zbmath.org/authors/?q=ai:matucci.serena"Řehák, Pavel"https://www.zbmath.org/authors/?q=ai:rehak.pavelSummary: We establish conditions guaranteeing that all eventually positive increasing solutions of a half-linear delay differential equation are regularly varying and derive precise asymptotic formulae for them. The results presented here are new also in the linear case and some of the observations are original also for non-functional equations. A substantial difference is pointed out between the delayed and nondelayed case for eventually positive decreasing solutions.Lebesgue integral.https://www.zbmath.org/1456.260022021-04-16T16:22:00+00:00"Florescu, Liviu C."https://www.zbmath.org/authors/?q=ai:florescu.liviu-cPublisher's description: This book presents a compact and self-contained introduction to the theory of measure and integration. The introduction into this theory is as necessary (because of its multiple applications) as difficult for the uninitiated. Most measure theory treaties involve a large amount of prerequisites and present crucial theoretical challenges. By taking on another approach, this textbook provides less experienced readers with material that allows an easy access to the definition and main properties of the Lebesgue integral.
The book will be welcomed by upper undergraduate/early graduate students who wish to better understand certain concepts and results of probability theory, statistics, economic equilibrium theory, game theory, etc., where the Lebesgue integral makes its presence felt throughout. The book can also be useful to students in the faculties of mathematics, physics, computer science, engineering, life sciences, as an introduction to a more in-depth study of measure theory.Schur-\(m\) power convexity of Cauchy means and its application.https://www.zbmath.org/1456.260132021-04-16T16:22:00+00:00"Wang, Dong-Sheng"https://www.zbmath.org/authors/?q=ai:wang.dongsheng"Fu, Chunru"https://www.zbmath.org/authors/?q=ai:fu.chunru"Shi, Huannan"https://www.zbmath.org/authors/?q=ai:shi.huannanThe Cauchy mean generalizes many well-known classical means of two variables. Schur convexity, Schur geometric convexity, Schur harmonic convexity and Schur power convexity are important tools in their study. In recent years, the application of majorization theory to the Schur convexity of various means has been very active and many results have been obtained. The authors of this paper obtain necessary and sufficient conditions for the Cauchy mean to be Schur-\(m\) power convex or concave. As applications, Schur-\(m\) power convexity of the exponential mean is discussed and a comparative inequality between the Gini mean and the Stolarsky mean is given.
Reviewer: Ioan Raşa (Cluj-Napoca)