Recent zbMATH articles in MSC 34Bhttps://zbmath.org/atom/cc/34B2023-05-31T16:32:50.898670ZWerkzeugMore indefinite integrals from Riccati equationshttps://zbmath.org/1508.330052023-05-31T16:32:50.898670Z"Conway, John T."https://zbmath.org/authors/?q=ai:conway.john-thomasSummary: Two new methods for obtaining indefinite integrals of a special function using Riccati equations are presented. One method uses quadratic fragments of the Riccati equation, the solutions of which are given by simple quadratic equations. This method is applied to cylinder functions, parabolic cylinder functions and the general solution of the Mathieu equation. No such indefinite integrals for general Mathieu functions seem to have been presented previously. The second method obtains indefinite integrals by assuming simple algebraic forms involving constants for the Riccati variable \(u(x)\) and then choosing the values of these constants to give simple and interesting integrals. This method is illustrated here for cylinder functions and Associated Legendre functions. All integrals obtained have been checked using Mathematica.On the identification of solutions to Riccati equation and the other polynomial systems of differential equationshttps://zbmath.org/1508.340042023-05-31T16:32:50.898670Z"Zaĭtsev, Maksim Leonidovich"https://zbmath.org/authors/?q=ai:zaitsev.maksim-leonidovich"Akkerman, Vyacheslav Borisovich"https://zbmath.org/authors/?q=ai:akkerman.vyacheslav-borisovichSummary: The authors previously proposed a general method for finding particular solutions for overdetermined PDE systems, where the number of equations is greater than the number of unknown functions. The essence of the method is to reduce the PDE to systems of PDE of a lower dimension, in particular, to ODEs by overdetermining them by additional constraint equations. Reduction of some PDE systems generates overdetermined systems of polynomial ODEs, which are studied in this paper. A method for transforming polynomial ODE systems to linear ODE systems is proposed. The result is interesting from a theoretical point of view if these systems of polynomial ODEs are with constant coefficients. The solution of such nonlinear systems using our method can be represented as a sum of a very large but finite number of oscillations. The amplitudes of these oscillations depend on the initial data nonlinearly. The Navier-Stokes equations and unified PDE systems obtained by the authors earlier can be transformed to such systems. The Riccati equation is also investigated. New special cases are indicated when it is possible to find its solution. Numerical estimates of the complexity of this method for practical implementation are presented.Study on the existence and nonexistence of solutions for a class of nonlinear Erdélyi-Kober type fractional differential equation on unbounded domainhttps://zbmath.org/1508.340052023-05-31T16:32:50.898670Z"Bouteraa, N."https://zbmath.org/authors/?q=ai:bouteraa.noureddine"Inc, Mustafa"https://zbmath.org/authors/?q=ai:inc.mustafa"Hashemi, M. S."https://zbmath.org/authors/?q=ai:hashemi.mir-sajjad"Benaicha, S."https://zbmath.org/authors/?q=ai:benaicha.slimaneThis paper considers a nonlinear Erdélyi-Kober (EK) fractional-order differential equation subject to an EK fractional integral boundary conditions on an unbounded domain. A fixed point theorem due to \textit{A. Benmezaï} and \textit{S. Chentout} [Differ. Equ. Appl. 11, No. 4, 463--480 (2019; Zbl 1436.34004)] is used to establish the existence and nonexistence of positive solutions in a cone.
The authors define terms that are used in the work and also state theorems that are relevant to the work. In Theorem 2.9, conditions for the existence of a fixed point are stated while in Theorem 2.10 conditions for non-existence of fixed points are stated.
To prove existence and nonexistence of positive solutions, the EK fractional integral-order differential equation is first converted into an integral form and the sign properties of the Green functions are stated.
An operator \(L_m\) is defined, then shown to be compact and positive in Lemma 3.5, while in Lemma 3.6, it is shown to be a strongly positive-like operator which is lower bounded in the cone.
The authors also define an operator \(T\) in a Banach space \(\mathcal{B}=C(0,\infty)\) with a norm \(\Vert w \Vert = \sup _{t>0} |w(t)|\), \(w \in \mathcal{B}\). In Lemma 3.7, \(T: B^+ \to B\) is proved by the authors to be compact with the aid of the Arzela-Axcoli theorem.
Finally, the authors state and prove conditions for nonexistence of positive solutions in Theorem 3.9 and conditions for existence of positive solutions in Theorem 3.10.
Reviewer: Ogbu F. Imaga (Ota)Quasilinearization technique for solving nonlinear Riemann-Liouville fractional-order problemshttps://zbmath.org/1508.340092023-05-31T16:32:50.898670Z"Su, Guangwang"https://zbmath.org/authors/?q=ai:su.guangwang"Lu, Liang"https://zbmath.org/authors/?q=ai:lu.liang"Tang, Bo"https://zbmath.org/authors/?q=ai:tang.bo"Liu, Zhenhai"https://zbmath.org/authors/?q=ai:liu.zhenhaiSummary: In this work, we deal with the quasilinearization technique for a class of nonlinear Riemann-Liouville fractional-order two-point boundary value problems. Using quasilinearization technique, we construct a monotone sequence of approximate solutions which has quadratic convergence to the unique solution of the original problem, and establish the corresponding convergence estimates. Moreover, the performance of the technique is examined through a numerical example, which shows that our regularization method is available and stable.Existence of solutions for fractional instantaneous and non-instantaneous impulsive differential equations with perturbation and Dirichlet boundary valuehttps://zbmath.org/1508.340102023-05-31T16:32:50.898670Z"Wang, Yinuo"https://zbmath.org/authors/?q=ai:wang.yinuo"Li, Chuandong"https://zbmath.org/authors/?q=ai:li.chuandong"Wu, Hongjuan"https://zbmath.org/authors/?q=ai:wu.hongjuan"Deng, Hao"https://zbmath.org/authors/?q=ai:deng.haoIn this paper, the authors study a boundary value problem for fractional instantaneous and non-instantaneous impulsive differential equations with perturbation subject to Dirichlet boundary conditions. By using the Weierstrass theorem the existence of classical solutions is proved. An example illustrating the obtained results is also presented.
Reviewer: Sotiris K. Ntouyas (Ioannina)Renormalized oscillation theory for regular linear non-Hamiltonian systemshttps://zbmath.org/1508.340112023-05-31T16:32:50.898670Z"Howard, Peter"https://zbmath.org/authors/?q=ai:howard.peterSummary: In recent work, Baird et al. have generalized the definition of the Maslov index to paths of Grassmannian subspaces that are not necessarily contained in the Lagrangian Grassmannian. Such an extension opens up the possibility of applications to non-Hamiltonian systems of ODE, and Baird and his collaborators have taken advantage of this observation to establish oscillation-type results for obtaining lower bounds on eigenvalue counts in this generalized setting. In the current analysis, the author shows that renormalized oscillation theory, appropriately defined in this generalized setting, can be applied in a natural way, and that it has the advantage, as in the traditional setting of linear Hamiltonian systems, of ensuring monotonicity of crossing points as the independent variable increases for a wide range of system/boundary-condition combinations. This seems to mark the first effort to extend the renormalized oscillation approach to the non-Hamiltonian setting.Landesman-Lazer type conditions for scalar one-sided superlinear nonlinearities with Neumann boundary conditionshttps://zbmath.org/1508.340132023-05-31T16:32:50.898670Z"Marvulli, Rossella"https://zbmath.org/authors/?q=ai:marvulli.rossella"Sfecci, Andrea"https://zbmath.org/authors/?q=ai:sfecci.andreaThe authors prove an existence result for a Neumann problem associated with the second order scalar differential equation
\[
x''+f(t,x)=0.
\]
The nonlinear function \(f\) is assumed to be continuous, and it is controlled at \(-\infty\) and \(+\infty\) by the so called Fučík spectrum. More precisely, they assume a one-sided superlinear behaviour (e.g., when \(x\) tends to \(-\infty\)), and on the other side the nonlinearity stays between the asymptotes of two consecutive Fučík curves. They also allow ``resonance'' with the one or both asymptotes, in which case they need to assume some type of Landesman-Lazer conditions. In my opinion, the most interesting part of these results is the choice of the test functions for these Landesman-Lazer conditions, which are related to those needed for an impact oscillator.
Reviewer: Alessandro Fonda (Trieste)On the local solvability and stability for the inverse spectral problem of the generalized Dirichlet-Regge problemhttps://zbmath.org/1508.340142023-05-31T16:32:50.898670Z"Xu, Xiao Chuan"https://zbmath.org/authors/?q=ai:xu.xiaochuan"Bondarenko, Natalia Pavlovna"https://zbmath.org/authors/?q=ai:bondarenko.natalia-pThis paper is concerned with the theory of inverse spectral problems, which consist in recovering operators from their spectral characteristics. In this paper, the authors consider the generalized Dirichlet-Regge problem \(L(q,\alpha,\beta)\):
\[
-y''(x)+q(x)y(x)=\lambda^{2}y(x), \ \ \ 0<x<a,
\]
\[
y(0)=0, \ \ \ y'(a)+(i\lambda\alpha+\beta)y(a)=0,
\]
where \(\lambda\) is the spectral parameter, the complex-valued potential \(q\in L^{2}(0,a),\) \(\beta\in\mathbb{C}\) and \(\alpha>0\) and study the inverse spectral problem for \(L(q,\alpha,\beta).\) The authors describe the method for solving the inverse problem and this method is based on the reduction of the inverse Dirichlet-Regge problem to the inverse problem by the so-called Cauchy data. For this reduction, a special exponential system being a Riesz basis is constructed. The advantage of this method is that it allows us to overcome difficulties caused by multiple eigenvalues. Under a small perturbation of the spectrum, multiple eigenvalues of the problem \(L(q,\alpha,\beta)\) can split into smaller groups, and this effect is taken into account. Thus, the local solvability and stability in the most general form are obtained without any restrictions on the spectrum. Moreover, in this paper, two uniqueness theorems are formulated. The formulation and the proofs of our main result on local solvability and stability of the inverse problem are given.
Reviewer: Ozge Akcay (Tunceli)Multiplicity of solutions for non-homogeneous Dirichlet problem with one-dimension Minkowski-curvature operatorhttps://zbmath.org/1508.340162023-05-31T16:32:50.898670Z"Lu, Yanqiong"https://zbmath.org/authors/?q=ai:lu.yanqiong"Li, Zhiqiang"https://zbmath.org/authors/?q=ai:li.zhiqiang|li.zhiqiang.1"Chen, Tianlan"https://zbmath.org/authors/?q=ai:chen.tianlanThis paper explores the multiplicity of solutions for the following boundary value problem with one-dimensional Minkowski-curvature operator
\[
\Biggl(\frac{u'}{\sqrt{1-\kappa u'^2}}\Biggr)'+f(u)=0,\;t\in(0,1),
\]
\[
u(0)=sA,\;\;u(1)=sB,
\]
where the constant \(\kappa\) is positive, \(A\) and \(B\) are constants, \(s\in\mathbb{R}\) is a parameter with \(|s(B-A)|<\frac{1}{\sqrt{\kappa}}\) and \(f\in C(\mathbb{R}, \mathbb{R})\) is such that \(f(s)s>0\) for \(s\ne0.\)
In one of the main results, assuming in addition that there exists a positive integer \(j\) such that \(j^2\pi^2<f_0<(j+1)^2\pi^2,\) where \(f_0=\lim_{u\to 0}\frac{\sqrt{1-\kappa u^2}f(u)}{u},\) the authors establish that for every \(B>A>0\) there exist \(s_k\in(0,\frac{1}{\sqrt{\kappa}(B-A)}), k=0,1,...,2N-2,\) such that for every \(s\in(s _{2i-1}, s_{2i}), i=0,1,...,N-1,\) with \(s_{-1}=0,\) the considered problem has at least \(4(N-i)\) nontrivial solutions, where \(N\) is the number of positive integers \(m\) such that \(2m\in[2, j-\frac{3}{2}-\frac{\sqrt{2}}{2}\sqrt{\frac{A}{B-A}}\frac{1}{\pi}].\)
Reviewer: Petio S. Kelevedjiev (Sliven)Multiple solutions for some elliptic boundary value problem with jumping nonlinearitieshttps://zbmath.org/1508.340172023-05-31T16:32:50.898670Z"Choi, Q-Heung"https://zbmath.org/authors/?q=ai:choi.q-heung"Jung, Tacksun"https://zbmath.org/authors/?q=ai:jung.tacksunThis paper is devoted to the multiplicity of solutions in the Lebesgue-Sobolev space \(W^{1,p}(\Omega, \mathbb{R})\) to the following \(p\)-Laplacian boundary value problem with jumping nonlinearities
\[
-(|u'|^{p-2}u')'=b^{(p)}|u|^{p-2}u^{+} - a^{(p)}|u|^{p-2}u^{-}+s|\phi_1^{{(p)}^{p-2}}|\phi_1^{(p)}\;\mbox{in}\;\Omega,\tag{1}
\]
\[
u=0\;\mbox{on}\;\partial\Omega,\tag{2}
\]
where \(1<p<\infty,\) \(u^{+}=\max\{u,0\}, u^{-}=-\{u,0\},\) \(s\in \mathbb{R}, a^{(p)}\) and \(b^{(p)}\) are real numbers depending on \(p\) such that \(a^{(p)}< b^{(p)},\) \(\Omega =(c,d)\) is an open interval, and \(\phi_1^{(p)}\) is the first eigenfunction, depending on \(p\), of the elliptic eigenvalue problem
\[
-(|u'|^{p-2}u')'=\lambda^{(p)}|u|^{p-2}u\;\mbox{in}\;\Omega,\] \[u=0\;\mbox{on}\;\partial\Omega.
\]
The authors write: ``We obtain three theorems depending on the source terms when nonlinearities cross some eigenvalues. We obtain the first theorem and the second one by eigenvalues and the corresponding normalized eigenfunctions of the eigenvalue problem, and the contraction mapping principle on \(p\)-Lebesgue space. We obtain the third result by Leray-Schauder degree theory.'' In particular, in the third theorem it is assumed that the eigenvalues \(\lambda_i^{(p)}, i=1,2,3,\) of the eigenvalue problem are such that \(-\infty<a^{(p)}<\lambda_1^{(p)}, \lambda_2^{(p)}<b^{(p)}<\lambda_3^{(p)}.\) This theorem includes the following assertions:
(i) If \(1<p<\infty, s>0\) and \(\phi_1^{(p)}>0,\) then (1),(2) has no solution.
(ii) If \(1<p<\infty, s<0\) and \(\phi_1^{(p)}<0,\) then (1),(2) has no solution.
(iii) If \(1<p<\infty, s=0,\) then (1),(2) has exactly one solution \(u=0\) for any case \(\phi_1^{(p)}>0\) or \(\phi_1^{(p)}<0.\)
(iv) If \(2\leq p<\infty\) and \(\phi_1^{(p)}>0,\) then there exists \(s_1^{(p)}<0\) such that for any \(s\) with \(s_1^{(p)}\leq s<0,\) (1),(2) has at least three solutions.
(v) If \(2\leq p<\infty\) and \(\phi_1^{(p)}<0,\) then there exists \(s_2^{(p)}>0\) such that for any \(s\) with \(0<s\leq s_2^{(p)},\) (1),(2) has at least three solutions.
Reviewer: Petio S. Kelevedjiev (Sliven)Global and asymptotic behaviors of bifurcation curves of one-dimensional nonlocal elliptic equationshttps://zbmath.org/1508.340182023-05-31T16:32:50.898670Z"Shibata, Tetsutaro"https://zbmath.org/authors/?q=ai:shibata.tetsutaroSummary: We study the one-dimensional nonlocal elliptic equation
\[
\begin{aligned}
- \left(\int_0^1 | u (x) |^p d x + b\right)^q u''(x) &= \lambda u (x)^p, \,\, x \in I : = (0, 1), \,\, u(x) > 0, \,\, x \in I,\\
u(0) &= u(1) = 0,
\end{aligned}
\]
where \(b \geq 0\), \(p \geq 1\), \(q > 1 - \frac{1}{p}\) are given constants and \(\lambda > 0\) is a bifurcation parameter. We establish the global behavior of bifurcation curves and precise asymptotic formulas for \(u_\lambda(x)\) as \(\lambda \to \infty\).Bifurcation diagrams of one-dimensional Kirchhoff-type equationshttps://zbmath.org/1508.340192023-05-31T16:32:50.898670Z"Shibata, Tetsutaro"https://zbmath.org/authors/?q=ai:shibata.tetsutaroIn this paper, the author considers a nonlocal elliptic problem, which includes the one-dimensional Kirchhoff-type equation,
\[
-\left( b+a\left\Vert u^{\prime}\right\Vert ^{2}\right) u^{\prime\prime}(x)=\lambda u(x)^{p},\quad x\in I:=\left( -1,1\right),
\]
\[
u(x)>0,
\]
\[
u\left( \pm1\right) =0,
\]
where \(\left\Vert u^{\prime}\right\Vert =\left( \int_{I}u^{\prime}(x)^{2}dx\right) ^{1/2}\), \(\alpha>0, b>0, p>0\) are given constants and \(\lambda>0\) is a bifurcation parameter. For the cases \(0<p<1\) or \(3<p<\infty\) and \(1\leq p\leq3\), the author gives and proves some theorems to establish the exact solution \(u_{\lambda}(x)\) of the problem, and the complete shape of the bifurcation curves \(\lambda=\lambda(\xi)\), where \(\xi=\left\Vert u_{\lambda}\right\Vert_{\infty}\). The author also explicitly obtains the problem's first eigenvalue and eigenfunction using a simple time map method.
Reviewer: Fatma Hıra (Atakum)The problem with non-separated multipoint-integral conditions for high-order differential equations and a new general solutionhttps://zbmath.org/1508.340202023-05-31T16:32:50.898670Z"Assanova, Anar T."https://zbmath.org/authors/?q=ai:assanova.anar-turmaganbetkyzy"Imanchiyev, Askarbek E."https://zbmath.org/authors/?q=ai:imanchiev.askarbek-ermekovichSummary: The problem with non-separated multipoint-integral conditions for high-order differential equations is considered. An interval is divided into \(m\) parts, the values of a solution at the beginning points of the subintervals are considered as additional parameters, and the high-order differential equations are reduced to the Cauchy problems on the subintervals for system of differential equations with parameters. Using the solutions to these problems, new general solutions to high-order differential equations are introduced and their properties are established. Based on the general solution, non-separated multipoint-integral conditions, and continuity conditions of a solution at the interior points of the partition, the linear system of algebraic equations with respect to parameters is composed. Algorithms of the parametrization method are constructed and their convergence is proved. Sufficient conditions for the unique solvability of considered problem are set. It is shown that the solvability of boundary value problems is equivalent to the solvability of systems composed. Methods for solving boundary value problems are proposed, which are based on the construction and solving these systems.Existence of positive and symmetric solutions for some elliptic equation on spherical cap of \(S^3\)https://zbmath.org/1508.340212023-05-31T16:32:50.898670Z"Hirose, Munemitsu"https://zbmath.org/authors/?q=ai:hirose.munemitsuThis paper considers the existence of positive and symmetric solutions for the following boundary value problems on a spherical cap of the unit ball in four-dimensional space of the form
\[
\left\{ \begin{array}{ll} \triangle_{S^3}w+\lambda w+w^p=0 & \hbox{in}\ D, \\
w>0 & \hbox{in}\ D, \\
w=0 & \hbox{on}\ \partial D, \end{array} \right.
\]
where \(\triangle_{S^3}\) is the Laplace-Beltrami operator, \(\lambda\in \mathbb{R}\) and \(p>1\) are parameters, \(D\) is a spherical cap including \((0,0,0,1)\) on \(\textbf{S}^3\) whose geodesic radius is denoted by \(\beta\in(0,\pi)\), and the boundary of \(D\) is defined by
\[
\partial D=\{(x_1,x_2,x_3,x_4)\in \mathbb{R}^4: x_4=\cos \beta,\quad x_1^2+x_2^2+x_3^2=\sin^2\beta\}.
\]
To obtain the symmetric solutions for the above boundary value problems, at first the author uses the transformation
\[
u(\theta)=w(x)=w(x_1,x_2,x_3,x_4),\quad x_4=\cos\theta
\]
to transform the above problem into the following second-order two-point boundary value problem
\[
\left\{ \begin{array}{ll} u''+2\cot\theta u'+\lambda u+u^p=0 & \hbox{in}\ (0,\beta), \\
u(\theta)>0 & \hbox{in}\ [0,\beta), \\
u(\beta)=0, &\\
u(0)~ \hbox{is finite and positive},\\
u'(0)=0. \end{array} \right.
\]
And then, using the existence results of solutions to second order initial value problems by Kabeya, Y., Yanagida, E. and Yotsutani, S., for the subcritical case, the author establishes the existence and non-existence results for such solution completely if the spherical cap is contained in the hemisphere. In addition, the author obtains some partial results if the spherical cap contains the hemisphere.
Reviewer: Minghe Pei (Jilin)Principal solutions revisitedhttps://zbmath.org/1508.340222023-05-31T16:32:50.898670Z"Clark, Stephen"https://zbmath.org/authors/?q=ai:clark.stephen-l|clark.stephen-s|clark.stephen-r|clark.stephen-a|clark.stephen-p-h|clark.stephen-j|clark.stephen-d"Gesztesy, Fritz"https://zbmath.org/authors/?q=ai:gesztesy.fritz"Nichols, Roger"https://zbmath.org/authors/?q=ai:nichols.rogerAuthors' abstract: The main objective of this paper is to identify principal solutions associated with Sturm-Liouville operators on arbitrary open intervals \((a,b) \subseteq \mathbb{R}\), as introduced by \textit{M. Morse} and \textit{W. Leighton} in the scalar context in [Trans. Am. Math. Soc. 40, 252--286 (1936; JFM 62.0577.02)] and by \textit{P. Hartman} in the matrix-valued situation in [Duke Math. J. 24, 25--35 (1957; Zbl 0077.08701)], with Weyl-Titchmarsh solutions, as long as the underlying Sturm-Liouville differential expression is nonoscillatory (resp., disconjugate or bounded from below near an endpoint) and in the limit point case at the endpoint in question. In addition, we derive an explicit formula for Weyl-Titchmarsh functions in this case (the latter appears to be new in the matrix-valued context).
For the entire collection see [Zbl 1350.60004].
Reviewer: Hüseyin Tuna (Burdur)New results on the sign of the Green function of a two-point \(n\)-th order linear boundary value problemhttps://zbmath.org/1508.340232023-05-31T16:32:50.898670Z"Almenar, Pedro"https://zbmath.org/authors/?q=ai:almenar.pedro"Jódar, Lucas"https://zbmath.org/authors/?q=ai:jodar-sanchez.lucas-aIn this paper, there is studied the sign of the derivatives of the Green function of the problem
\begin{align*}
Ly&=0,\quad x\in [a, b],\\
y^{(i)}(a)&= y^{(j)}(b)=0,\quad i\in \alpha, \quad j\in \beta,
\end{align*}
where \(L: \mathcal{C}^n([a, b])\to \mathcal{C}([a, b])\) is given by
\[
Ly=y^{(n)}+a_{n-1}(x)y^{(n-1)}+\cdots+a_0(x) y, \quad x\in [a, b],
\]
\(a_j \in \mathcal{C}([a, b])\), \(j\in \{0, \ldots, n-1\}\), \(\alpha=\{\alpha_1, \ldots, \alpha_k\}\), \(\beta=\{\beta_1, \ldots, \beta_{n-k}\}\), \(k\in \{1, \ldots, n-1\}\), \(\alpha_1, \beta_1\geq 0\), \(\alpha_k, \beta_{n-k}<n\). The authors analyze the sign of the derivatives of the solutions of related two-points \(n\)-th order boundary value problems subject to \(n-1\) boundary conditions. In the paper, is introduced the concept of hyperdisfocality.
Reviewer: Svetlin Georgiev (Sofia)Lyapunov-type inequalities for fractional differential operators with non-singular kernelshttps://zbmath.org/1508.340242023-05-31T16:32:50.898670Z"Basua, Debananda"https://zbmath.org/authors/?q=ai:debananda.basua"Jonnalagadda, Jagan Mohan"https://zbmath.org/authors/?q=ai:jonnalagadda.jaganmohanIn this paper, the authors first study the properties of Green's functions for conjugate and anti-periodic boundary value problems involving Caputo-Fabrizio and Atangana-Baleanu-Caputo fractional derivatives of order \(\rho\in (0, 1]\). The obtained properties are then applied to derive Lyapunov-type inequalities for related fractional boundary value problems. The lower bounds for the eigenvalues of fractional eigenvalue problems are also discussed in the paper.
For the entire collection see [Zbl 1491.65006].
Reviewer: Lingju Kong (Chattanooga)Methods for solving problems on thermal conductivity of multilayer media in the presence of heat sourceshttps://zbmath.org/1508.340252023-05-31T16:32:50.898670Z"Afanasenkova, Yu. V."https://zbmath.org/authors/?q=ai:afanasenkova.yu-v"Gladyshev, Yu. A."https://zbmath.org/authors/?q=ai:gladyshev.yu-a"Kalmanovich, V. V."https://zbmath.org/authors/?q=ai:kalmanovich.v-vSummary: The solution of problems on phase transitions in multilayer media when they are heated is of practical interest owing to the increasing use of multilayer materials in engineering and construction under various temperature conditions. In this paper, we assume that the medium contains distributed heat sources caused by physical or chemical processes, which can lead to phase transitions. We describe the method of generalized Bers powers and the matrix method for solving the heat conduction problem in multilayer media and for the search for boundaries of phase transitions.Coexistence of limit cycles in a septic planar differential system enclosing a non-elementary singular point, using Riccati equationhttps://zbmath.org/1508.340262023-05-31T16:32:50.898670Z"Allaoua, R."https://zbmath.org/authors/?q=ai:allaoua.r"Cheurfa, R."https://zbmath.org/authors/?q=ai:cheurfa.rachid"Bendjeddou, A."https://zbmath.org/authors/?q=ai:bendjeddou.ahmedSummary: We give a new family of planar polynomial differential systems of degree seven with a non elementary point at the origin. We show the integrability of this family by transforming it into a Riccati equation. We determine sufficient conditions for the coexistence of algebraic and non-algebraic limit cycle surrounding this non elementary point. Moreover these limits cycles are explicitly given. An example is given and its phase portrait is drawn as an illustration of our result.On the uniqueness and expression of limit cycles in planar polynomial differential system via monotone iterative techniquehttps://zbmath.org/1508.340282023-05-31T16:32:50.898670Z"Liang, Haihua"https://zbmath.org/authors/?q=ai:liang.haihua"Huang, Jianfeng"https://zbmath.org/authors/?q=ai:huang.jianfengThe paper concerns an investigation of the limit cycles for the planar differential system
\[
\frac{dx}{dt} = \lambda x - y+P_n(x,y), \quad \frac{dy}{dt} = x+\lambda y+Q_n(x,y),
\]
where \(\lambda \in \mathbb{R}\) and \(P_n(x,y)\), \(Q_n(x,y)\) are homogeneous polynomials of degree \(n\). Applying the method of upper and lower solutions coupled with monotone iterative method to study the periodic boundary value problems for the classical Abel equation the authors prove the uniqueness of limit cycle of this system under some suitable conditions and provide the expression describing the shape of the mentioned limit cycle.
Reviewer: Alexander Grin (Grodno)On the qualitative behavior of a class of generalized Liénard planar systemshttps://zbmath.org/1508.340292023-05-31T16:32:50.898670Z"Villari, Gabriele"https://zbmath.org/authors/?q=ai:villari.gabriele"Zanolin, Fabio"https://zbmath.org/authors/?q=ai:zanolin.fabioThis work is devoted to the problem of existence of limit cycles for a class of Liénard generalized differential systems
\[
\frac{dx}{dt} = y - F(x,y), \quad \frac{dy}{dt} = -g(x),
\]
assuming that \(F : \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}\) and \(g : \mathbb{R} \rightarrow \mathbb{R}\) are locally Lipschitz continuous functions, in order to guarantee the uniqueness of the solutions for the associated initial value problems. It is also assumed that \(g(0) = 0\), \(g(x)x > 0 \) for \(x = 0\) and that the origin is the only singular point of the system. At first, the authors discuss some basic facts of the mentioned system related to the use of the energy of the associated Duffing equations as a Lyapunov function. Then they study the case \(F(x, y) = \lambda B(y)A(x)\), where \(A(x)\) satisfies the standard assumptions on \(F(x)\) in the classical case.
Reviewer: Alexander Grin (Grodno)Global bifurcation for a class of nonlinear ODEshttps://zbmath.org/1508.340402023-05-31T16:32:50.898670Z"Bettiol, Renato G."https://zbmath.org/authors/?q=ai:bettiol.renato-g"Piccione, Paolo"https://zbmath.org/authors/?q=ai:piccione.paoloA global bifurcation result for positive periodic solutions (with fixed period) to scalar equations of the form
\[
u''-\mu (u- |u|^{q-1}u)=0
\]
where \(\mu>0\) is the bifurcation parameter, and \(q>1\) is fixed. A brief survey on global bifurcation theory is also presented. As an application of the main result, the authors give a bifurcation-theoretic proof of a classical non-uniqueness result for conformal metrics with constant scalar curvature, that was independently discovered by Kobayashi and Schoen in the 1980s.
Reviewer: Adriana Buică (Cluj-Napoca)Assignment flows for data labeling on graphs: convergence and stabilityhttps://zbmath.org/1508.340472023-05-31T16:32:50.898670Z"Zern, Artjom"https://zbmath.org/authors/?q=ai:zern.artjom"Zeilmann, Alexander"https://zbmath.org/authors/?q=ai:zeilmann.alexander"Schnörr, Christoph"https://zbmath.org/authors/?q=ai:schnorr.christophSummary: The assignment flow recently introduced in the \textit{J. Math. Imaging and Vision} 58/2 (2017) constitutes a high-dimensional dynamical system that evolves on a statistical product manifold and performs contextual labeling (classification) of data given in a metric space. Vertices of an underlying corresponding graph index the data points and define a system of neighborhoods. These neighborhoods together with nonnegative weight parameters define the regularization of the evolution of label assignments to data points, through geometric averaging induced by the affine e-connection of information geometry. From the point of view of evolutionary game dynamics, the assignment flow may be characterized as a large system of replicator equations that are coupled by geometric averaging. This paper establishes conditions on the weight parameters that guarantee convergence of the continuous-time assignment flow to integral assignments (labelings), up to a negligible subset of situations that will not be encountered when working with real data in practice. Furthermore, we classify attractors of the flow and quantify corresponding basins of attraction. This provides convergence guarantees for the assignment flow which are extended to the discrete-time assignment flow that results from applying a Runge-Kutta-Munthe-Kaas scheme for the numerical geometric integration of the assignment flow. Several counter-examples illustrate that violating the conditions may entail unfavorable behavior of the assignment flow regarding contextual data classification.Stability analysis of the boundary value problem modelling a two-layer oceanhttps://zbmath.org/1508.340522023-05-31T16:32:50.898670Z"Marynets, Kateryna"https://zbmath.org/authors/?q=ai:marynets.katerynaIn this paper, the author investigates the stability of solutions of a mathematical model of a two-layer ocean away from the Equator in the case of a piecewise constant eddy viscosity. Detailed analysis of investigated mathematical model bases on a logarithmic matrix norm. Also the paper contains the analysis of three different parameter-dependent eddy viscosity profiles and computations of bounds of solutions to the corresponding problems.
Reviewer: Tatuana Badokina (Saransk)Some properties of the solution of the nonlinear equation of oscillations in modeling the magnetic separationhttps://zbmath.org/1508.340552023-05-31T16:32:50.898670Z"Petrivskyi, Yaroslav"https://zbmath.org/authors/?q=ai:petrivskyi.yaroslav"Petrivskyi, Volodymyr"https://zbmath.org/authors/?q=ai:petrivskyi.volodymyrSummary: A qualitative analysis of the equation simulating the process of dry enrichment of raw materials with weak magnetic properties on a drum magnetic separator is carried out. The parametric nature of the role of the free term of the equation, which is the bifurcation point for the model, is established. The study of the properties of the singular point made it possible to allow to build a function to characterize a periodic partial solution and an algorithm for calculating the separation angle of the particle from the surface of the drum during enrichment by the dry separation method, which is convenient for practical use. From a physical point of view, in the process of magnetic separation, when there is friction proportional to the square of the angular velocity in the system, with the force acting on the particles of constant magnetic force, the work expended on overcoming the friction forces increases with the square of the angular velocity, while the operation of the external forces remains unchanged.
For the entire collection see [Zbl 1479.34005].On the existence and evaluation of Stokes phenomena in fluid mechanicshttps://zbmath.org/1508.340662023-05-31T16:32:50.898670Z"Alexandrakis, Nik"https://zbmath.org/authors/?q=ai:alexandrakis.nikSummary: A singularly perturbed, high order KdV-type model, which describes localized travelling waves (``solitons'') is being considered. We focus on the \textit{Inner solution}, and detect \textit{Stokes phenomena} that are crucial as to whether we can obtain a suitable solution. We provide a simple proof that the corresponding \textit{Stokes constant} is non-zero. Also, we evaluate this splitting constant numerically by using two methods that are induced by the underlying theory.Spectral decimation of a self-similar version of almost Mathieu-type operatorshttps://zbmath.org/1508.340802023-05-31T16:32:50.898670Z"Mograby, Gamal"https://zbmath.org/authors/?q=ai:mograby.gamal"Balu, Radhakrishnan"https://zbmath.org/authors/?q=ai:balu.radhakrishnan"Okoudjou, Kasso A."https://zbmath.org/authors/?q=ai:okoudjou.kasso-a"Teplyaev, Alexander"https://zbmath.org/authors/?q=ai:teplyaev.alexanderSummary: We introduce and study self-similar versions of the one-dimensional almost Mathieu operators. Our definition is based on a class of self-similar Laplacians \(\{\Delta_p\}_{p\in(0, 1)}\) instead of the standard discrete Laplacian and includes the classical almost Mathieu operators as a particular case, namely, when the Laplacian's parameter is \(p = \frac{1}{2}\). Our main result establishes that the spectra of these self-similar almost Mathieu operators can be described by the spectra of the corresponding self-similar Laplacians through the spectral decimation framework used in the context of spectral analysis on fractals. The spectral-type of the self-similar Laplacians used in our model is singularly continuous when \(p \neq \frac{1}{2}\). In these cases, the self-similar almost Mathieu operators also have singularly continuous spectra despite the periodicity of the potentials. In addition, we derive an explicit formula of the integrated density of states of the self-similar almost Mathieu operators as the weighted pre-images of the balanced invariant measure on a specific Julia set.
{\copyright 2022 American Institute of Physics}The weak eigenfunctions of boundary-value problem with symmetric discontinuitieshttps://zbmath.org/1508.341152023-05-31T16:32:50.898670Z"Olğar, Hayati"https://zbmath.org/authors/?q=ai:olgar.hayati"Mukhtarov, Oktay S."https://zbmath.org/authors/?q=ai:mukhtarov.oktay-sh"Muhtarov, Fahreddin S."https://zbmath.org/authors/?q=ai:muhtarov.fahreddin-s"Aydemir, Kadriye"https://zbmath.org/authors/?q=ai:aydemir.kadriyeSummary: The main goal of this study is the investigation of discontinuous boundary-value problems for second-order differential operators with symmetric transmission conditions. We introduce the new notion of weak functions for such type of discontinuous boundary-value problems and develop an operator-theoretic method for the investigation of the spectrum and completeness property of the weak eigenfunction systems. In particular, we define some self-adjoint compact operators in suitable Sobolev spaces such that the considered problem can be reduced to an operator-pencil equation. The main result of this paper is that the spectrum is discrete and the set of eigenfunctions forms a Riesz basis of the suitable Hilbert space.Positive solutions for fractional \((p, q)\)-difference boundary value problemshttps://zbmath.org/1508.390092023-05-31T16:32:50.898670Z"Qin, Zhongyun"https://zbmath.org/authors/?q=ai:qin.zhongyun"Sun, Shurong"https://zbmath.org/authors/?q=ai:sun.shurongSummary: In this paper, we investigate the boundary value problem of a class of fractional \((p, q)\)-difference equations involving the Riemann-Liouville fractional derivative. Based on the generalization of Banach contraction principle, we obtain a sufficient condition for existence and uniqueness of solutions of the problem. By applying a fixed point theorem in cones, we establish a sufficient condition for the existence of at least one positive solution of the problem. As an application, some examples are presented to illustrate the main results.The Landau-Kolmogorov problem on a finite interval in the Taikov casehttps://zbmath.org/1508.410042023-05-31T16:32:50.898670Z"Skorokhodov, Dmytro"https://zbmath.org/authors/?q=ai:skorokhodov.dmytro-sLet \(n\in\mathbb{N}\), \(\sigma>0\) and let \(\|\cdot\|_2\) denote the \(L^2\)-norm on \([-1,1]\). The author find an exact bound for \(|f^{(k)}(t)|\), \(k < n\), under constraints \(\|f\|_2\leq 1\) and \(\|f^{(n)}\|_2\leq \sigma\), where \(t\in [-1,1]\) is fixed. Then, for \(n=1\) and \(n=2\), the Landau-Kolmogorov problem on the interval \([-1,1]\) in the Taikov case is solved by proving the Karlin-type conjecture. Furthermore, the smallest possible constant \(A>0\) and the smallest possible constant \(B=B(A)\) in the inequality \(\|f^{(k)}\|_{\infty}\leq A\|f\|_2 + B\|f^{(n)}\|_2\) for \(k\in \{n - 2,n - 1\}\) are found.
Reviewer: Yuri A. Farkov (Moskva)Spectral expansion of Sturm-Liouville problems with eigenvalue-dependent boundary conditionshttps://zbmath.org/1508.471022023-05-31T16:32:50.898670Z"Yokuş, Nihal"https://zbmath.org/authors/?q=ai:yokus.nihal"Arpat, Esra Kir"https://zbmath.org/authors/?q=ai:arpat.esra-kirSummary: In this paper, we consider the operator \(L\) generated in \(L_2(\mathbb{R}_+)\) by the differential expression \[l(y) = y''+ q(x)y,\ x\in\mathbb{R}_+ := [0;1)\] and the boundary condition \[\frac{y'(0)}{y(0)}= \alpha_0+\alpha_1\lambda +\alpha_2\lambda^2\] where \(q\) is a complex valued function \(\alpha_i(\mathbb{C})\), \(i = 0, 1, 2\) with \(\alpha_2\ne 0\). We have proved that spectral expansion of \(L\) in terms of the principal functions under the condition \[q\in AC(\mathbb{R}_+),\quad\lim_{x\to\infty}q(x) = 0,\quad\sup_{x\in\mathbb{R}_+}[e^{\varepsilon\sqrt{x}}| q'(x)|]<\infty,\quad \varepsilon> 0,\] taking into account the spectral singularities. We have also proved the convergence of the spectral expansion.A discontinuous finite element approximation to singular Lane-Emden type equationshttps://zbmath.org/1508.650802023-05-31T16:32:50.898670Z"Izadi, Mohammad"https://zbmath.org/authors/?q=ai:izadi.mohammad|izadi.mohammad-aSummary: In this article, we develop the local discontinuous Galerkin finite element method for the numerical approximations of a class of singular second-order ordinary differential equations known as the Lane-Emden type equations equipped with initial or boundary conditions. These equations have been considered via different models that naturally appear for example in several phenomena in astrophysical science. By converting the governing equations into a first-order systems of differential equations, the approximate solution is sought in a piecewise discontinuous polynomial space while the natural upwind fluxes are used at element interfaces. The existence-uniqueness of the weak formulation is provided and the numerical stability of the method in the \(L^\infty\) norm is established. Five illustrative test problems are given to demonstrate the applicability and validity of the scheme. Comparisons between the numerical results of the proposed method with existing results are carried out in order to show that the new approximation algorithm provides accurate solutions even near the singular point.Multi-scale orthogonal basis method for nonlinear fractional equations with fractional integral boundary value conditionshttps://zbmath.org/1508.650912023-05-31T16:32:50.898670Z"Jiang, Wei"https://zbmath.org/authors/?q=ai:jiang.wei.2"Chen, Zhong"https://zbmath.org/authors/?q=ai:chen.zhong"Hu, Ning"https://zbmath.org/authors/?q=ai:hu.ning.1"Song, Haiyang"https://zbmath.org/authors/?q=ai:song.haiyang"Yang, Zhaohong"https://zbmath.org/authors/?q=ai:yang.zhaohongSummary: In this paper, we investigate the multi-scale orthogonal basis method for fractional integral boundary value problems. We apply the Newton iteration method to linearize the nonlinear problems and employees the idea of collocation method to determine the coefficients of multi-scale orthogonal basis, then the approximation solution is obtained. The error estimation and stable analysis are presented in detailed. The final numerical experiments verify that the accuracy of our method.Torsional instability and sensitivity analysis in a suspension bridge model related to the Melan equationhttps://zbmath.org/1508.740392023-05-31T16:32:50.898670Z"Falocchi, Alessio"https://zbmath.org/authors/?q=ai:falocchi.alessioSummary: Inspired by the Melan equation we propose a model for suspension bridges with two cables linked to a deck, through inextensible hangers. We write the energy of the system and we derive from variational principles two nonlinear and nonlocal hyperbolic partial differential equations, involving the vertical displacement and the torsional rotation of the deck. We prove existence and uniqueness of a weak solution and we perform some numerical experiments on the isolated system; moreover we propose a sensitivity analysis of the system by mechanical parameters in terms of torsional instability. Our results display that there are specific thresholds of torsional instability with respect to the initial amplitude of the longitudinal mode excited.Singularly perturbed problems with multi-tempo fast variableshttps://zbmath.org/1508.932102023-05-31T16:32:50.898670Z"Kurina, G. A."https://zbmath.org/authors/?q=ai:kurina.galina-a"Kalashnikova, M. A."https://zbmath.org/authors/?q=ai:kalashnikova.margarita-aSummary: The article contains a survey of publications studying problems characterized by the presence of fast variables with various rates of change (time scales). We consider the passage to the limit from the solution of a perturbed problem to the solution of a degenerate one, asymptotic solutions of initial and boundary value problems, stability and controllability, asymptotic solutions of optimal control problems, and problems with ``hidden'' multi-tempo variables. In addition, problems with control constraints, game problems, and stochastic systems are given. The last section presents practical problems with multi-tempo fast motions.