Recent zbMATH articles in MSC 34Ahttps://www.zbmath.org/atom/cc/34A2021-04-16T16:22:00+00:00WerkzeugOn conditional stability of Inverse scattering problem on a Lasso-shaped graph.https://www.zbmath.org/1456.340812021-04-16T16:22:00+00:00"Mochizuki, Kiyoshi"https://www.zbmath.org/authors/?q=ai:mochizuki.kiyoshi"Trooshin, Igor"https://www.zbmath.org/authors/?q=ai:trooshin.igor-yuSummary: We investigate the conditional stability of the inverse scattering problem on a lasso-shaped graph using the fundamental equation of inverse scattering theory.
For the entire collection see [Zbl 1415.35004].Integrability analysis of the Shimizu-Morioka system.https://www.zbmath.org/1456.340022021-04-16T16:22:00+00:00"Huang, Kaiyin"https://www.zbmath.org/authors/?q=ai:huang.kaiyin"Shi, Shaoyun"https://www.zbmath.org/authors/?q=ai:shi.shaoyun"Li, Wenlei"https://www.zbmath.org/authors/?q=ai:li.wenleiSummary: The aim of this paper is to give some new insights into the Shimizu-Morioka system
\[
\dot{x}=y,\quad \dot{y}=x-\lambda y-xz,\quad \dot{z}=-\alpha z+x^2,
\]
from the integrability point of view. Firstly, we propose a linear scaling in time and coordinates which converts the Shimizu-Morioka system into a special case of the Rucklidge system when \(\alpha\neq 0\) and discuss the relationship between Shimizu-Morioka system and Rucklidge system. Based on this observation, Darboux integrability of the Shimizu-Morioka system with \(\alpha\neq 0\) is trivially derived from the corresponding results on the Rucklidge system. When \(\alpha=0\), we investigate Darboux integrability of the Shimizu-Morioka system by the Gröbner basis in algebraic geometry. Secondly, we use the stability of the singular points and periodic orbits to study the nonexistence of global \(C^1\) first integrals of the Shimizu-Morioka system. Finally, in the case \(\alpha\neq 0\), we prove it is not rationally integrable for almost all parameter values by an extended Morales-Ramis theory, and in the case \(\alpha=0\), we show that it is not algebraically integrable by quasi-homogeneous decompositions and Kowalevski exponents. Our results are in accord with the fact that this system admits chaotic behaviors for a large range of its parameters.Implicit differential inclusions with acyclic right-hand sides: an essential fixed points approach.https://www.zbmath.org/1456.340152021-04-16T16:22:00+00:00"Andres, Jan"https://www.zbmath.org/authors/?q=ai:andres.jan"Górniewicz, Lech"https://www.zbmath.org/authors/?q=ai:gorniewicz.lechSummary: Effective criteria are given for the solvability of initial as well as boundary valueproblems to implicit ordinary differential inclusions whose right-hand sides are governed by compactacyclic maps. Cauchy and periodic implicit problems are also considered on proximate retracts. Ournew approach is based on the application of the topological essential fixed point theory. Implicitproblems for partial differential inclusions are only indicated.Spectral expansion for singular conformable fractional Sturm-Liouville problem.https://www.zbmath.org/1456.340032021-04-16T16:22:00+00:00"Allahverdiev, Bilender P."https://www.zbmath.org/authors/?q=ai:allahverdiev.bilender-pasaoglu"Tuna, Hüseyin"https://www.zbmath.org/authors/?q=ai:tuna.huseyin"Yalçinkaya, Yüksel"https://www.zbmath.org/authors/?q=ai:yalcinkaya.yukselSummary: With this study, the spectral function for singular conformable fractional Sturm-Lioville problem is demonstrated. Further, we establish a Parseval equality and spectral expansion formula by terms of the spectral function.Response to: ``Comment on `The asymptotic iteration method revisited'''.https://www.zbmath.org/1456.340132021-04-16T16:22:00+00:00"Ismail, Mourad E. H."https://www.zbmath.org/authors/?q=ai:ismail.mourad-el-houssieny"Saad, Nasser"https://www.zbmath.org/authors/?q=ai:saad.nasserSummary: This response concers the Comment by \textit{F. M. Fernández} [J. Math. Phys. 61, No. 6, 064101, 2 p. (2020; Zbl 1456.34012)].
{\copyright 2020 American Institute of Physics}Comment on: ``The asymptotic iteration method revisited'''.https://www.zbmath.org/1456.340122021-04-16T16:22:00+00:00"Fernández, Francisco M."https://www.zbmath.org/authors/?q=ai:fernandez.francisco-mSummary: This comment concerns [\textit{M. E. H. Ismail} and \textit{N. Saad}, ibid. 61, No. 3, 033501, 12 p. (2020; Zbl 1443.34021)]. In this comment, we show that the eigenvalues of a quartic anharmonic oscillator obtained recently by means of the asymptotic iteration method may not be as accurate as the authors claim them to be.
{\copyright 2020 American Institute of Physics}Stochastic resonance in a fractional oscillator with random damping strength and random spring stiffness.https://www.zbmath.org/1456.700412021-04-16T16:22:00+00:00"He, Guitian"https://www.zbmath.org/authors/?q=ai:he.guitian"Tian, Yan"https://www.zbmath.org/authors/?q=ai:tian.yan"Wang, Yan"https://www.zbmath.org/authors/?q=ai:wang.yan.6|wang.yan|wang.yan.2|wang.yan.1|wang.yan.5|wang.yan.4|wang.yan.3Solutions for a singular Hadamard-type fractional differential equation by the spectral construct analysis.https://www.zbmath.org/1456.340082021-04-16T16:22:00+00:00"Zhang, Xinguang"https://www.zbmath.org/authors/?q=ai:zhang.xinguang"Yu, Lixin"https://www.zbmath.org/authors/?q=ai:yu.lixin"Jiang, Jiqiang"https://www.zbmath.org/authors/?q=ai:jiang.jiqiang"Wu, Yonghong"https://www.zbmath.org/authors/?q=ai:wu.yonghong.1"Cui, Yujun"https://www.zbmath.org/authors/?q=ai:cui.yujunSummary: In this paper, we consider the existence of positive solutions for a Hadamard-type fractional differential equation with singular nonlinearity. By using the spectral construct analysis for the corresponding linear operator and calculating the fixed point index of the nonlinear operator, the criteria of the existence of positive solutions for equation considered are established. The interesting point is that the nonlinear term possesses singularity at the time and space variables.Bendixson criterion in impulsive systems.https://www.zbmath.org/1456.340112021-04-16T16:22:00+00:00"Ding, Changming"https://www.zbmath.org/authors/?q=ai:ding.changming"Duan, Zhipeng"https://www.zbmath.org/authors/?q=ai:duan.zhipeng"Pan, Shiyao"https://www.zbmath.org/authors/?q=ai:pan.shiyaoConsider the sufficiently smooth planar vector field \(F(x,y)\). Under the assumption that the divergence of \(F\) has constant sign in some simply connected region \(U\subset\mathbb{R}^2\) and is not identically zero in any subregion of \(U\), then the criterion of Bendixson says that there is no closed orbit of \(F\) entirely located in \(U\). The authors recall basic definitions and properties of planar impulsive systems. They present a simple example showing that the Bendixson criterion does not hold for planar impulsive systems in its original form. Introducing the concept of a periodic orbit of order \(k\), where \(k\) is related to the number of jumps, they prove an extended Bendixson criterion which excludes the existence of periodic orbits of order \(k \leq 2\).
Reviewer: Klaus R. Schneider (Berlin)Expressions of meromorphic solutions of a certain type of nonlinear complex differential equations.https://www.zbmath.org/1456.300572021-04-16T16:22:00+00:00"Chen, Jun-Fan"https://www.zbmath.org/authors/?q=ai:chen.junfan"Lian, Gui"https://www.zbmath.org/authors/?q=ai:lian.guiSummary: In this paper, the expressions of meromorphic solutions of the following nonlinear complex differential equation of the form \[f^n+Q_d(z,f)=\sum_{i=1}^3p_i(z)e^{\alpha_i(z)}\] are studied by using Nevanlinna theory, where \(n\geq5\) is an integer, \(Q_d(z,f)\) is a differential polynomial in \(f\) of degree \(d\leq n-4\)~with rational functions as its coefficients, \(p_1(z), p_2(z), p_3(z)\)~are non-vanishing rational functions, and \(\alpha_1(z), \alpha_2(z), \alpha_3(z)\) are nonconstant polynomials such that \(\alpha_1'(z), \alpha_2'(z), \alpha_3'(z)\) are distinct each other. Moreover, examples are given to illustrate the accuracy of the condition.Forced oscillation of sublinear impulsive differential equations via nonprincipal solution.https://www.zbmath.org/1456.340292021-04-16T16:22:00+00:00"Mostepha, Naceri"https://www.zbmath.org/authors/?q=ai:mostepha.naceri"Özbekler, Abdullah"https://www.zbmath.org/authors/?q=ai:ozbekler.abdullahSummary: In this paper, we give new oscillation criteria for forced sublinear impulsive differential equations of the form
\[
\begin{cases}
(r(t)x^\prime)^\prime + q(t)|x|^{\gamma - 1} x = f(t), &t \neq \theta_i;\\
\Delta r(t) x^\prime + q_i|x|^{\gamma-1}x = f_i, &t= \theta_i,
\end{cases}
\]
where \(\gamma \in (0,1)\), under the assumption that associated homogenous linear equation
\[
\begin{cases}
(r(t)z^\prime)^\prime + q(t)z = 0, &t \neq \theta_i;\\
\Delta r (t)z^\prime + q_i z = 0, &t = \theta_i
\end{cases}
\]
is nonoscillatory.Coupled systems of linear differential-algebraic and kinetic equations with application to the mathematical modelling of muscle tissue.https://www.zbmath.org/1456.340512021-04-16T16:22:00+00:00"Plunder, Steffen"https://www.zbmath.org/authors/?q=ai:plunder.steffen"Simeon, Bernd"https://www.zbmath.org/authors/?q=ai:simeon.berndSummary: We consider a coupled system composed of a linear differential-algebraic equation (DAE) and a linear large-scale system of ordinary differential equations where the latter stands for the dynamics of numerous identical particles. Replacing the discrete particles by a kinetic equation for a particle density, we obtain in the mean-field limit the new class of partially kinetic systems.
We investigate the influence of constraints on the kinetic theory of those systems and present necessary adjustments. We adapt the mean-field limit to the DAE model and show that index reduction and the mean-field limit commute. As a main result, we prove Dobrushin's stability estimate for linear systems. The estimate implies convergence of the mean-field limit and provides a rigorous link between the particle dynamics and their kinetic description.
Our research is inspired by mathematical models for muscle tissue where the macroscopic behaviour is governed by the equations of continuum mechanics, often discretised by the finite element method, and the microscopic muscle contraction process is described by Huxley's sliding filament theory. The latter represents a kinetic equation that characterises the state of the actin-myosin bindings in the muscle filaments. Linear partially kinetic systems are a simplified version of such models, with focus on the constraints.
For the entire collection see [Zbl 1445.34004].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.Study of a boundary value problem for fractional order \(\psi\)-Hilfer fractional derivative.https://www.zbmath.org/1456.340052021-04-16T16:22:00+00:00"Harikrishnan, S."https://www.zbmath.org/authors/?q=ai:harikrishnan.sugumaran"Shah, Kamal"https://www.zbmath.org/authors/?q=ai:shah.kamal"Kanagarajan, K."https://www.zbmath.org/authors/?q=ai:kanagarajan.kana|kanagarajan.kuppusamySummary: This manuscript is devoted to the existence theory of a class of random fractional differential equations (RFDEs) involving boundary condition (BCs). Here we take the corresponding derivative of arbitrary order in \(\psi\)-Hilfer sense. By utilizing classical fixed point theory and nonlinear analysis we establish some basic results of the qualitative theory such as existence, uniqueness and stability of solutions to the considered boundary value problem of RFDEs. Further, for the justification of our analysis we provide two examples.Remarks on the slow relaxation for the fractional Kuramoto model for synchronization.https://www.zbmath.org/1456.340612021-04-16T16:22:00+00:00"Ha, Seung-Yeal"https://www.zbmath.org/authors/?q=ai:ha.seung-yeal"Jung, Jinwook"https://www.zbmath.org/authors/?q=ai:jung.jinwookSummary: The collective behavior of an oscillatory system is ubiquitous in our nature, and one interesting issue in the dynamics of many-body oscillatory systems is the relaxation dynamics toward relative equilibria such as phase-locked states. For the Kuramoto model, relaxation dynamics occurs exponentially fast for generic initial data. However, some synchronization phenomena observed in our nature exhibit a slow subexponential relaxation. Thus, as one of the possible attempts for such slow relaxation, a second-order inertia term was added to the Kuramoto model in the previous literature so that the resulting second-order model can exhibit a slow relaxation dynamics for some range of inertia and coupling strength. In this paper, we present another Kuramoto type model exhibiting a slow algebraic relaxation. More precisely, our proposed model replaces the classical derivative by the Caputo fractional derivative in the original Kuramoto model. For this new model, we present several sufficient frameworks for fractional complete synchronization and practical synchronization.{
\copyright 2018 American Institute of Physics}Boundary value problems for Caputo fractional differential equations with nonlocal and fractional integral boundary conditions.https://www.zbmath.org/1456.340042021-04-16T16:22:00+00:00"Derbazi, Choukri"https://www.zbmath.org/authors/?q=ai:derbazi.choukri"Hammouche, Hadda"https://www.zbmath.org/authors/?q=ai:hammouche.haddaSummary: In this paper, we study the existence and uniqueness of solutions for fractional differential equations with nonlocal and fractional integral boundary conditions. New existence and uniqueness results are established using the Banach contraction principle. Other existence results are obtained using O'Regan fixed point theorem and Burton and Kirk fixed point. In addition, an example is given to demonstrate the application of our main results.Dynamics and optimal control of a Monod-Haldane predator-prey system with mixed harvesting.https://www.zbmath.org/1456.921222021-04-16T16:22:00+00:00"Liu, Xinxin"https://www.zbmath.org/authors/?q=ai:liu.xinxin"Huang, Qingdao"https://www.zbmath.org/authors/?q=ai:huang.qingdaoImplicit parametrizations and applications in optimization and control.https://www.zbmath.org/1456.490082021-04-16T16:22:00+00:00"Tiba, Dan"https://www.zbmath.org/authors/?q=ai:tiba.danThe subject is the characterization (with numerical applications in mind) of the manifold \(V\) of solutions of the nonlinear system
\[
F_j(x_1, x_2, \dots, x_d) = 0 \quad (1 \le j \le l )\quad l \le d - 1 \tag{1}
\]
in the vicinity of \(x^0 = (x^0_1, x^0_2 , \dots ,x^0_d) \in V,\) under the Jacobian assumption
\[
\frac{\partial (F_1, F_2, \dots, F_l)}{\partial (x_1, x_2, \dots, x_l)} \ne 0
\quad \hbox{in} \ x^0 = (x^0_1, x^0_2 , \dots ,x^0_d) \, .
\]
The first step involves the underdetermined linear system
\[
v(x) \cdot \nabla F_j(x) = 0 \quad (1 \le j \le l)
\]
which is used to obtain bases \((v_1(x), v_2(x), \dots, v_{d - l}(x))\) for the tangent spaces of \(V.\) Next, the chain of differential equations
\begin{align*}
\frac{\partial y_1(t_1)}{\partial t_1} &= v_1(y_1(t_1)), y_1(0) = x^0
\cr
\frac{\partial y_2(t_1, t_2)}{\partial t_2} &= v_2(y_2(t_1, t_2)),\quad y_2(t_1, 0) = y(t_1)
\cr
& \hskip 2em \dots \dots \dots \dots
\cr
\frac{\partial y_{d - l}(t_1, t_2, \dots, t_{d - l})}{ \partial t_{d - l}}
&= v_{d - l}(y_{d - l}(t_1, t_2, \dots, t_{d - l})) \, ,
\cr
& \hskip 2.7em y_{d - l}(t_1, \dots , t_{d - l - 1}, 0) = y_{d - l - 1}(t_1, t_2, \dots , t_{d - l - 1})
\end{align*}
is set up, thus constructing a parametrization of \(V\) which may be considered as an explicit form of the implicit function theorem. The result is used to construct an algorithm for the solution of the problem of minimizing a function
\(g(x_0, x_2, \dots , x_d)\) subject to (1). Some generalizations are covered, such as the case where (1) includes inequality constraints and/or regularity is relaxed. In the last section the results are applied to the control problem
of minimizing \(l(x(0), x(1))\) among the trajectories of the system
\(x'(t) = f(t, x(t), u(t))\) subject to the state-control constraint \(h(x(t), u(t)) = 0.\) There are several numerical implementation of the algorithms and the author notes that computations can be carried out using standard Matlab routines.
Reviewer: Hector O. Fattorini (Los Angeles)Solvability and stability of the inverse Sturm-Liouville problem with analytical functions in the boundary condition.https://www.zbmath.org/1456.340142021-04-16T16:22:00+00:00"Bondarenko, Natalia P."https://www.zbmath.org/authors/?q=ai:bondarenko.natalia-pThe paper deals with the boundary value problem
\[
-y''(x)+q(x)y(x)=\lambda y(x),\; 0<x<\pi,
\]
\[
y(0)=0,\; f_1(\lambda)y'(\pi)+f_2(\lambda)y(\pi)=0,
\]
where \(f_k(\lambda)\) are entire functions in \(\lambda.\) The author studies the inverse problem of recovering the potential \(q(x)\) from a part of the spectrum. Local and global solvability are established for the solution of this nonlinear inverse problem.
Reviewer: Vjacheslav Yurko (Saratov)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 theorems of existence of solutions for fractional hybrid \(q\)-difference inclusion.https://www.zbmath.org/1456.340062021-04-16T16:22:00+00:00"Samei, Mohammad Esmael"https://www.zbmath.org/authors/?q=ai:samei.mohammad-esmael"Ranjbar, Ghorban Khalilzadeh"https://www.zbmath.org/authors/?q=ai:ranjbar.ghorban-khalilzadehSummary: The purpose of this study is to obtain the existence of solutions for the fractional hybrid \(q\)-differential inclusions with the boundary conditions. Besides, we give the solution set of these \(q\)-differential inclusions with boundary values. Also, we investigate dimension of the results set for second fractional \(q\)-differential inclusions. Lastly, we present an example to elaborate our results and present the obtained results of close to mathematical calculations.A method of solving a nonlinear boundary value problem with a parameter for a loaded differential equation.https://www.zbmath.org/1456.340192021-04-16T16:22:00+00:00"Dzhumabaev, Dulat"https://www.zbmath.org/authors/?q=ai:dzhumabaev.dulat-syzdykbekovich"Bakirova, Elmira"https://www.zbmath.org/authors/?q=ai:bakirova.elmira-a"Mynbayeva, Sandugash"https://www.zbmath.org/authors/?q=ai:mynbayeva.sandugash-tabyldyevnaIn this paper, the authors study a nonlinear loaded differential equation with a parameter on a finite interval
\[
\frac{dx}{dt}=f_0(t,x,\mu)+f_1(t,x(\theta_0),x(\theta_1),\dots,x(\theta_{m-1}),x(\theta_m),\mu),\ t\in(0,T),x\in \mathbb{R}^n,\mu\in \mathbb{R}^l
\]
subjected to the boundary value conditions
\[
g[x(0),x(T),\mu]=0,
\]
where \(f_0:[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^l\to \mathbb{R}^n\), \(f_1:[0,T]\times\mathbb{R}^{n(m+1)}\times\mathbb{R}^l\to \mathbb{R}^n\), and \(g:\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}^l\to \mathbb{R}^{n+l}\) are continuous functions, \(0=\theta_0<\theta_1<\cdots<\theta_{m-1}<\theta_m=T\), \(\|x\|=\max_{i=1,\dots,n}|x_i|\).
The interval is partitioned by the load points, at which the values of the solution to the equation are set as additional parameters. A nonlinear boundary value problem for the considered equation is reduced to a nonlinear multipoint boundary value problem for the system of nonlinear ordinary differential equations with parameters. For fixed parameters, the authors obtain the Cauchy problems for ordinary differential equations on the subintervals. Substituting the values of the solutions to these problems into the boundary condition and continuity conditions at the partition points, they compose a system of nonlinear algebraic equations in parameters. They develop a method of solving the boundary value problem with a parameter for the above loaded differential equation which is based on finding the solution to the system of nonlinear algebraic equations by an iterative process.
Reviewer: Yanqiong Lu (Lanzhou)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 class of planar differential systems with explicit expression for two limit cycles.https://www.zbmath.org/1456.340282021-04-16T16:22:00+00:00"Hamizi, Saad Eddine"https://www.zbmath.org/authors/?q=ai:hamizi.saad-eddine"Boukoucha, Rachid"https://www.zbmath.org/authors/?q=ai:boukoucha.rachidSummary: The existence of limit cycles is interesting and very important in applications. It is a key to understand the dynamic of polynomial differential systems. The aim of this paper is to investigate a class of a multi-parameter planar polynomial differential systems. Under some suitable conditions, the existence of two limit cycles, these limit cycles are explicitly given. Some examples are presented in order to illustrate the applicability of our results. algebras.Approximate solution of Bagley-Torvik equations with variable coefficients and three-point boundary-value conditions.https://www.zbmath.org/1456.651832021-04-16T16:22:00+00:00"Huang, Q. A."https://www.zbmath.org/authors/?q=ai:huang.qiongao|huang.qiu-an"Zhong, X. C."https://www.zbmath.org/authors/?q=ai:zhong.xichang|zhong.xiaochun|zhong.xian-ci"Guo, B. L."https://www.zbmath.org/authors/?q=ai:guo.baolin|guo.boling|guo.baolongSummary: The fractional Bagley-Torvik equation with variable coefficients is investigated under three-point boundary-value conditions. By using the integration method, the considered problems are transformed into Fredholm integral equations of the second kind. It is found that when the fractional order is \(1< \alpha <2\), the obtained Fredholm integral equation is with a weakly singular kernel. When the fractional order is \(0< \alpha <1\), the given Fredholm integral equation is with a continuous kernel or a weakly singular kernel depending on the applied boundary-value conditions. The uniqueness of solution for the obtained Fredholm integral equation of the second kind with weakly singular kernel is addressed in continuous function spaces. A new numerical method is further proposed to solve Fredholm integral equations of the second kind with weakly singular kernels. The approximate solution is made and its convergence and error estimate are analyzed. Several numerical examples are computed to show the effectiveness of the solution procedures.A computationally efficient iterative scheme for solving fourth-order boundary value problems.https://www.zbmath.org/1456.650532021-04-16T16:22:00+00:00"Tomar, Saurabh"https://www.zbmath.org/authors/?q=ai:tomar.saurabhSummary: A new rapid-converging analytical scheme is introduced to obtain the approximate analytical solutions of nonlinear fourth-order two-point boundary value problems, which appear in various physical phenomena. The idea of the method to obtain the solution of such problems is essentially based on reducing the solution of the main problem to the solution of an integral problem. The introduced technique consists of two steps. First, construct an integral operator by introducing Green's function, and then, the Normal-S iterative scheme is applied to this integral operator to construct the iterative approach for such problems, which yields a simple way to improve the convergence of the iterative solutions to the problem. We also discuss the convergence of the introduced iterative method. To exhibit the performance of the method, we consider some numerical test examples. The obtained results are compared with the existing analytical and numerical approaches to reveal the superiority and computational efficiency of the proposed approach. In fact, it is a direct recursive and computationally cost-effective method for dealing with strong nonlinearity. The numerical simulations signify the applicability and effectiveness of the present work.Relaxation for a class of control systems with unilateral constraints.https://www.zbmath.org/1456.340172021-04-16T16:22:00+00:00"Papageorgiou, Nikolaos S."https://www.zbmath.org/authors/?q=ai:papageorgiou.nikolaos-s"Vetro, Calogero"https://www.zbmath.org/authors/?q=ai:vetro.calogero"Vetro, Francesca"https://www.zbmath.org/authors/?q=ai:vetro.francescaThe paper is concerned with a nonlinear feedback control system of type
\[
\left\{\begin{array}{ll}-x' \in A(x(t))+f(t,x(t))u(t)\\
x(0)=x_0, \quad u(t) \in U(t,x(t))\end{array}\right.
\]
in a time interval \(T=[0,b]\). Here, \(A:D(A)\subset \mathbb{R}^N \to 2^{\mathbb{R}^N}\) is a maximal monotone mapping and the control constraint multifunction \(U:T\times \mathbb{R}^N \to 2^{\mathbb{R}^N} \setminus \{ \emptyset \}\) has nonconvex values. It is assumed that \(U(t, \cdot )\) is lower semicontinuous for a.a. \(t\in T\). The authors introduce a control relaxed system by \(Q\)-regularization (in the sense of Cesari). Then, they show that every original state is a relaxed state and the set of the original states is dense in the set of the relaxed states, which is closed in \(C(T, \mathbb{R}^N)\).
Reviewer: Petru Jebelean (Timişoara)Finite-time stability for differential inclusions with applications to neural networks.https://www.zbmath.org/1456.340162021-04-16T16:22:00+00:00"Matusik, Radosław"https://www.zbmath.org/authors/?q=ai:matusik.radoslaw"Nowakowski, Andrzej"https://www.zbmath.org/authors/?q=ai:nowakowski.andrzej-f"Plaskacz, Sławomir"https://www.zbmath.org/authors/?q=ai:plaskacz.slawomir"Rogowski, Andrzej"https://www.zbmath.org/authors/?q=ai:rogowski.andrzejThe paper studies differential inclusions of the form
\[
x'\in F(t,x),
\]
where \(F:[0,\infty )\times\mathbb{R}^n\to \mathcal{P}(\mathbb{R}^n)\) is a set-valued map with non-empty compact convex values. It is assumed that \(F(t,.)\) is upper semicontinuous, \(F\) satisfies a certain linear growth condition and that the origin is an equilibrium point (i.e, \(0\in F(t,0)\) for almost all \(t\in [0,\infty )\)).
By using a nonsmooth Lyapunov function, sufficient conditions for weak and strong finite-time stability are obtained in terms of contingent epiderivatives and hypoderivatives of the Lyapunov function.
An application to a class of Hopfield neural networks is also provided.
Reviewer: Aurelian Cernea (Bucharest)On the behavior of solutions with positive initial data to higher-order differential equations with general power-law nonlinearity.https://www.zbmath.org/1456.340102021-04-16T16:22:00+00:00"Korchemkina, Tatiana"https://www.zbmath.org/authors/?q=ai:korchemkina.tatianaSummary: Higher-order differential equation with general power-law nonlinearity are considered. In particular, solutions with positive initial data are studied depending on the values of nonlinearity exponents. It is proven that if the sum of nonlinearity exponents is greater than one, then any considered solution has a finite right domain boundary. In the case of a constant potential solutions with power-law behavior are found.
For the entire collection see [Zbl 1445.34003].Generalized stochastic resonance in a linear fractional system with a random delay.https://www.zbmath.org/1456.827672021-04-16T16:22:00+00:00"Gao, Shi-Long"https://www.zbmath.org/authors/?q=ai:gao.shilongKink solitary solutions to a hepatitis C evolution model.https://www.zbmath.org/1456.340562021-04-16T16:22:00+00:00"Telksnys, Tadas"https://www.zbmath.org/authors/?q=ai:telksnys.tadas"Navickas, Zenonas"https://www.zbmath.org/authors/?q=ai:navickas.zenonas"Sanjuán, Miguel A. F."https://www.zbmath.org/authors/?q=ai:sanjuan.miguel-a-f"Marcinkevicius, Romas"https://www.zbmath.org/authors/?q=ai:marcinkevicius.romas"Ragulskis, Minvydas"https://www.zbmath.org/authors/?q=ai:ragulskis.minvydasSummary: The standard nonlinear hepatitis C evolution model described in (Reluga et al. 2009) is considered in this paper. The generalized differential operator technique is used to construct analytical kink solitary solutions to the governing equations coupled with multiplicative and diffusive terms. Conditions for the existence of kink solitary solutions are derived. It appears that kink solitary solutions are either in a linear or in a hyperbolic relationship. Thus, a large perturbation in the population of hepatitis infected cells does not necessarily lead to a large change in uninfected cells. Computational experiments are used to illustrate the evolution of transient solitary solutions in the hepatitis C model.A generalized fractional-order Chebyshev wavelet method for two-dimensional distributed-order fractional differential equations.https://www.zbmath.org/1456.651302021-04-16T16:22:00+00:00"Do, Quan H."https://www.zbmath.org/authors/?q=ai:do.quan-h"Ngo, Hoa T. B."https://www.zbmath.org/authors/?q=ai:ngo.hoa-t-b"Razzaghi, Mohsen"https://www.zbmath.org/authors/?q=ai:razzaghi.mohsenSummary: We provide a new effective method for the two-dimensional distributed-order fractional differential equations (DOFDEs). The technique is based on fractional-order Chebyshev wavelets. An exact formula involving regularized beta functions for determining the Riemann-Liouville fractional integral operator of these wavelets is given. The given wavelets and this formula are utilized to find the solutions of the given two-dimensional DOFDEs. The method gives very accurate results. The given numerical examples support this claim.Fractional differential equations involving Hadamard fractional derivatives with nonlocal multi-point boundary conditions.https://www.zbmath.org/1456.340072021-04-16T16:22:00+00:00"Subramanian, Muthaiah"https://www.zbmath.org/authors/?q=ai:subramanian.muthaiah"Manigandan, Murugesan"https://www.zbmath.org/authors/?q=ai:manigandan.murugesan"Gopal, Thangaraj Nandha"https://www.zbmath.org/authors/?q=ai:gopal.thangaraj-nandhaSummary: In this paper, we investigate the existence and uniqueness of solutions for the Hadamard fractional boundary value problems with nonlocal multipoint boundary conditions. By using Leray-Schauder nonlinear alternative, Leray Schauder degree theory, Krasnoselskii fixed point theorem, Schaefer fixed point theorem, Banach fixed point theorem, Nonlinear Contractions, the existence and uniqueness of solutions are obtained. As an application, two examples are given to demonstrate our results.The KW equations and the Nahm pole boundary condition with knots.https://www.zbmath.org/1456.813112021-04-16T16:22:00+00:00"Mazzeo, Rafe"https://www.zbmath.org/authors/?q=ai:mazzeo.rafe-r"Witten, Edward"https://www.zbmath.org/authors/?q=ai:witten.edwardIn this detailed technical paper the authors extend further their previous analysis of the Kapustin-Witten (KW) equations with Nahm pole boundary condition now adapted to general 4-manifolds-with-boundary such that the boundary-3-manifold contains a knot or more generally a link.
The motivation is a conjecture of the second author that the coefficients of the Laurent expansion of the Jones polynomial of a link \(L\subset {\mathbb R}^3\) arise by counting solutions of the KW equations on the half-space \({\mathbb R}^4_+\) obeying a generalized Nahm pole boundary condition on \(\partial {\mathbb R}^4_+={\mathbb R}^3\supset L\) i.e. the Nahm pole boundary condition generalized to be compatible with the extra information of containing a link on the boundary. Roughly this means to prescribe further singularities in the Higgs field part of the KW pair along each link component while the connection part is continuous up to the boundary as before. The conjecture is important because it is well-known that computing the Jones polynomial of a link is an exponentially difficult problem in terms of e.g. the crossing number of any plane diagram of the link.
Reviewer: Gabor Etesi (Budapest)A Green's function iterative approach for the solution of a class of fractional BVPs arising in physical models.https://www.zbmath.org/1456.650522021-04-16T16:22:00+00:00"Hadid, S."https://www.zbmath.org/authors/?q=ai:hadid.samir-b"Khuri, S. A."https://www.zbmath.org/authors/?q=ai:khuri.suheil-a"Sayfy, A."https://www.zbmath.org/authors/?q=ai:sayfy.ali-m-sSummary: The aim of this study is to present an alternative approach for the numerical solution of a wide class of fractional boundary value problems (FBVPs) that arise in various physical applications. Examples of such FBVP include but not limited to Bagley-Torvik, Riccati, Bratu, and Troesch's problems that appear in applied mathematics and mechanics. The method is based on first constructing an integral operator that is given in terms of the Green's function associated with the linear differential term of the fractional differential equation. Fixed point iterative procedures, such as Picard's and Mann's, are then applied to the operator to generate an iterative scheme that yields a convergent semi-analytical solution. Numerical examples are reported to confirm the efficiency, reliability, accuracy and fast convergence of the scheme.Bistable labyrinth-like structures and chimera states in a 2D lattice of van der Pol oscillators.https://www.zbmath.org/1456.340352021-04-16T16:22:00+00:00"Shepelev, Igor A."https://www.zbmath.org/authors/?q=ai:shepelev.igor-aleksandrovich"Anishchenko, V. S."https://www.zbmath.org/authors/?q=ai:anishchenko.vadim-semenovichSummary: The present work is devoted to the numerical analysis of the dynamics of a 2D lattice of coupled van der Pol oscillators in the regime of relaxation oscillations. It is shown that the influence of coupling leads to the shift of effective values of the control parameters of individual oscillators. The strong coupling can even cause the transition to bistable dynamics which is never observed in a single van der Pol oscillator. The numerically constructed phase-parametric diagram that takes into account the shifts of parameters shows that the bistability arises through the pitchfork bifurcation when varying the coupling strength. The lattice dynamics is analyzed when the control and coupling parameters are varied within a wide range, and the regime diagrams are constructed in the planes of system parameters. A new type of spatiotemporal pattern, a so-called ``labyrinth-like structure'', is found and described in detail. We also reveal for the first time and study a spiral wave chimera with a new kind of the incoherence core in the form of ``labyrinth-like structure''.Increased order generalized combination synchronization of non-identical dimensional fractional-order systems by introducing different observable variable functions.https://www.zbmath.org/1456.340622021-04-16T16:22:00+00:00"Kaouache, S."https://www.zbmath.org/authors/?q=ai:kaouache.smail"Hamri, N. E."https://www.zbmath.org/authors/?q=ai:hamri.n-e"Hacinliyan, A. S."https://www.zbmath.org/authors/?q=ai:hacinliyan.avadis-simon"Kandiran, E."https://www.zbmath.org/authors/?q=ai:kandiran.e"Deruni, B."https://www.zbmath.org/authors/?q=ai:deruni.b"Keles, A. C."https://www.zbmath.org/authors/?q=ai:keles.a-cSummary: An increased order generalized combination synchronization (IOGCS) of non-identical dimensional fractional-order systems with suitable different observable variable functions is proposed and analyzed in this paper. This synchronization scheme is applied for the combination of two fractional-order unified drive systems and the fractional-order Liu response system. In view of the stability property of linear fractional-order systems, an effective nonlinear control scheme is designed to achieve the desired synchronization. Theoretical analysis and numerical simulations are shown to demonstrate the effectiveness of the proposed method.New shock-wave and periodic-wave solutions for some physical and engineering models: Vakhnenko-Parkes, GEWB, GRLW and some integrable equations.https://www.zbmath.org/1456.340012021-04-16T16:22:00+00:00"Alquran, Marwan"https://www.zbmath.org/authors/?q=ai:alquran.marwan-taiseer"Jaradat, Imad"https://www.zbmath.org/authors/?q=ai:jaradat.imad"Sivasundaram, Seenith"https://www.zbmath.org/authors/?q=ai:sivasundaram.seenith"Al Shraiedeh, Laila"https://www.zbmath.org/authors/?q=ai:al-shraiedeh.lailaSummary: In this work, the modified unified expansion and the Bernoulli sub-equation methods are implemented to extract new shock-wave and periodic-wave solutions for important physical and engineering models. We study four models; the Vakhnenko-Parkes (VP) equation, the generalized equal width-Burgers (GEWB) equation with \(p \in \{1, 2\}\) and the generalized regularized-long-wave (GRLW) equation by the modified unified method, whereas the first-second fourth-order integrable equations by the Bernoulli sub-equation method. Shock-wave and periodic wave solutions are obtained for these models. All obtained solutions are verified and categorized regarding its physical structures.On existence theorems for generalized abstract measure integrodifferential equations.https://www.zbmath.org/1456.450022021-04-16T16:22:00+00:00"Dhage, Bapurao Chandrabhan"https://www.zbmath.org/authors/?q=ai:dhage.bapurao-chandrabhanSummary: In this paper, an existence and uniqueness results for a nonlinear abstract measure integrodifferential equation are proved via classical fixed point theorems of Schauder (see [\textit{A. Granas} and \textit{J. Dugundji}, Fixed point theory. New York, NY: Springer (2003; Zbl 1025.47002)]) and the author [Electron. J. Qual. Theory Differ. Equ. 2002, Paper No. 6, 9 p. (2002; Zbl 1029.47034)] under weaker Carathéodory condition. The existence for extremal solutions is also proved under certain Chandrabhan condition and using a hybrid fixed point theorem of the author [loc. cit.] in an ordered Banach space. Our existence results presented in this paper include the existence results of \textit{R. R. Sharma} [Proc. Am. Math. Soc. 32, 503--510 (1972; Zbl 0213.36201)], \textit{S. R. Joshi} [J. Math. Phys. Sci. 13, 497--506 (1979; Zbl 0435.34053)], \textit{G. R. Shendge} and \textit{S. R. Joshi} [Acta Math. Hung. 41, 53--59 (1983; Zbl 0536.34040)] and the author [J. Math. Phys. Sci. 20, 367--380 (1986; Zbl 0619.45005); with \textit{P. R. M. Reddy}, Jñānābha 49, No. 2, 82--93 (2019; Zbl 07273233)] on nonlinear abstract measure and abstract measure integrodifferential equations as special cases under weaker continuity condition.A characterization of generalized exponential dichotomy.https://www.zbmath.org/1456.340092021-04-16T16:22:00+00:00"Wang, Liugen"https://www.zbmath.org/authors/?q=ai:wang.liugen"Xia, Yonghui"https://www.zbmath.org/authors/?q=ai:xia.yonghui"Zhao, Ninghong"https://www.zbmath.org/authors/?q=ai:zhao.ninghongGiven the linear system
\[
x'(t) = A(t)x(t) \tag{1}
\]
\((x(t)\) a \(n\)-vector, \(A(t)\) a continuous \(n \times n\) matrix) let \(X(t)\) be a fundamental matrix. The system has an \textit{exponential dichotomy} if there exist a projection \(P\) and two positive constants \(K, \alpha\) with
\begin{align*}
\|X(t)PX^{-1}(s)\| &\le K \exp(-\alpha(t - s)) \quad (t \ge s) \\
\|X(t)(I - P)X^{-1}(s)\| &\le K \exp(\alpha(t - s)) \hskip 0.8em \quad (t \le s)\tag{2}
\end{align*}
(for \(A(t) = A\) existence of an exponential dichotomy means absence of purely imaginary eigenvalues). This definition is generalized by the authors to that of \textit{generalized exponential dichotomy}, where (2) becomes
\[
\begin{aligned}
\|X(t)PX^{-1}(s)\| &\le K \exp\bigg(- \int_s^t \alpha(\tau) d\tau\bigg) \quad (t \ge s) \\
\|X(t)(I - P)X^{-1}(s)\| &\le K \exp \bigg( \int_s^t \alpha(\tau) d\tau \bigg) \hskip 0.8em \quad (t \le s)
\end{aligned}
\]
with \(\alpha(t)\) a nonnegative continuous function satisfying
\[
\int_{-\infty}^0 \alpha(\tau) d\tau = \int_0^\infty \alpha(\tau) d\tau = + \infty \, . \tag{3}
\]
For instance, if \(\alpha(t)\) satisfies (3) the system \(x'(t) = -\alpha(t), y'(t) = \alpha(t)\) has a generalized exponential dichotomy but not necessarily an exponential dichotomy. The system (1) is of \textit{generalized bounded growth} if
\[
\|X(t) X^{-1}(s)\| \le \mu \exp \bigg( \int_s^t \rho(\tau) d\tau \bigg) \quad (t \ge s)
\]
where \(\mu \ge 1\) and the function \(\rho(t)\) is continuous, nonnegative and nonincreasing. The authors prove several results on existence of a generalized exponential dichotomy for a system (1) having generalized bounded growth. The last theorem relates existence of a generalized exponential dichotomy for (1) and existence of nontrivial bounded solutions.
Reviewer: Hector O. Fattorini (Los Angeles)Variable order fractional permanent magnet synchronous motor: dynamical analysis and numerical simulation.https://www.zbmath.org/1456.340582021-04-16T16:22:00+00:00"Zahra, Waheed K."https://www.zbmath.org/authors/?q=ai:zahra.waheed-k"Hikal, M. M."https://www.zbmath.org/authors/?q=ai:hikal.m-mSummary: In this paper, the variable order fractional permanent magnet synchronous motor (VOFPMSM) is investigated. Conditions for existence and uniqueness of the solution of the VOFPMSM are proposed. The stability behavior of the system's equilibrium points along with the variation of the motor parameters and the order of differentiation is discussed. Sufficient conditions that guarantee the asymptotic stability of each of the equilibrium points of the system are established. Also, the required conditions that give the effect of Hopf bifurcation of the system are established in terms of the system parameters and the order of differentiation and consequently the appearance of the chaotic behavior of the VOFPMSM. New numerical techniques based on the modified backward Euler's schemes for continuous and discontinuous variable order fractional model are presented. The obtained numerical results demonstrate the merits of the proposed method and the variable order fractional permanent magnet synchronous motor over the fractional permanent magnet synchronous motor.