Recent zbMATH articles in MSC 34Bhttps://zbmath.org/atom/cc/34B2024-06-14T15:52:26.737412ZWerkzeugIntroductory differential equationshttps://zbmath.org/1534.340012024-06-14T15:52:26.737412Z"Abell, Martha L."https://zbmath.org/authors/?q=ai:abell.martha-l"Braselton, James P."https://zbmath.org/authors/?q=ai:braselton.james-pPublisher's description: Introductory Differential Equations, Sixth Edition provides the foundations to assist students in learning not only how to read and understand differential equations, but also how to read technical material in more advanced texts as they progress through their studies. The book's accessible explanations and many robust sample problems are appropriate for a first semester course in introductory ordinary differential equations (including Laplace transforms), for a second course in Fourier series and boundary value problems, and for students with no background on the subject.
See the review of the 3rd edition in [Zbl 1206.34001]. For the 4th and 5th editions see [Zbl 1297.34001; Zbl 1390.34001].Explicit solutions of atmospheric Ekman flow with some cases of eddy viscositieshttps://zbmath.org/1534.340032024-06-14T15:52:26.737412Z"Guan, Yi"https://zbmath.org/authors/?q=ai:guan.yi"Wang, JinRong"https://zbmath.org/authors/?q=ai:wang.jinrongSummary: We investigate the classical problem of the wind in the steady atmospheric Ekman layer. For three cases of eddy viscosities which are the related fractional and integer powers functions with respect to height, we construct the explicit solutions, and write the formulas for the surface deflection angle, respectively. Our results extend the corresponding results in [\textit{L. Roberti}, Appl. Anal. 101, No. 15, 5528--5536 (2022; Zbl 1510.34038)] and [\textit{Y. Guan} et al., Appl. Anal. 102, No. 11, 2925--2938 (2023; Zbl 1517.86002)].Existence results for multi-term fractional differential equations with nonlocal boundary conditions involving Atangana-Baleanu derivativehttps://zbmath.org/1534.340062024-06-14T15:52:26.737412Z"Abbas, Ahsan"https://zbmath.org/authors/?q=ai:abbas.ahsan"Mehmood, Nayyar"https://zbmath.org/authors/?q=ai:mehmood.nayyar"Akgül, Ali"https://zbmath.org/authors/?q=ai:akgul.ali"Abdeljawad, Thabet"https://zbmath.org/authors/?q=ai:abdeljawad.thabet"Alqudah, Manar A."https://zbmath.org/authors/?q=ai:alqudah.manar-a(no abstract)Lyapunov-type inequalities for a nonlinear sequential fractional BVP in the frame of generalized Hilfer derivativeshttps://zbmath.org/1534.340072024-06-14T15:52:26.737412Z"Dien, Nguyen Minh"https://zbmath.org/authors/?q=ai:dien.nguyen-minh"Nieto, Juan J."https://zbmath.org/authors/?q=ai:nieto.juan-joseIn this paper, a Lyapunov-type inequality is obtained for a nonlinear sequential fractional boundary value problem involving generalized \(\Psi\)-Hilfer fractional derivatives,
\[
\left( ^H D^{\alpha_1,\beta_1,\Psi}_{a^+} D_{a^+}^{\alpha_2,\beta_2,\Psi}u\right)(t)+f(t,u(t))=0,\quad a<t<b,
\]
\[
u(a)=^H D_{a^+}^{\alpha_3,\beta_3,\Psi}u(b)=0.
\]
This is accomplished by first constructing a Green's function such that the boundary value problem can be written as an equivalent integral equation. From there, the maximum value of the absolute value of the Green's function is found. This allows the authors to obtain a Lyapunov-type inequality. The inequality is used to give a lower bound for the eigenvalues to the corresponding eigenvalue problem. Finally, properties of the Green's function and the nonlinear Leray-Schauder alternative fixed point theorem are used to show the existence of at least one mild solution.
Reviewer: Jeffrey Neugebauer (Richmond)Existence study of semilinear fractional differential equations and inclusions for multi-term problem under Riemann-Liouville operatorshttps://zbmath.org/1534.340082024-06-14T15:52:26.737412Z"Hadjadj, Elhabib"https://zbmath.org/authors/?q=ai:hadjadj.elhabib"Meftah, Safia"https://zbmath.org/authors/?q=ai:meftah.safia"Bensayah, Abdallah"https://zbmath.org/authors/?q=ai:bensayah.abdallah"Tellab, Brahim"https://zbmath.org/authors/?q=ai:tellab.brahimIn this paper, the authors study boundary value problems for fractional differential equations and inclusions involving Riemann-Liouville fractional operators subject to multi-term fractional boundary conditions. In the single-valued case the existence and uniqueness result was established via Banach's fixed point theorem. In the multi-valued case both cases, convex and non-convex values multi-valued maps are considered. In the first case, an existence result is obtained by using the Leray-Schauder nonlinear alternative for multi-valued maps, while in the second case, the Covitz-Nadler fixed point theorem for multi-valued contractions was applied. Examples illustrating the obtained results are also presented.
Reviewer: Sotiris K. Ntouyas (Ioannina)Lyapunov-type inequalities for fractional Langevin differential equationshttps://zbmath.org/1534.340112024-06-14T15:52:26.737412Z"Laadjal, Zaid"https://zbmath.org/authors/?q=ai:laadjal.zaid"Ma, Qinghua"https://zbmath.org/authors/?q=ai:ma.qinghuaThis paper establishes some new Lyapunov-type inequalities for fractional Langevin equations with two classes of two-point boundary conditions. Some related results in the literature are generalized.
Reviewer: Lingju Kong (Chattanooga)Solvability for a Hadamard-type fractional integral boundary value problemhttps://zbmath.org/1534.340142024-06-14T15:52:26.737412Z"Xu, Jiafa"https://zbmath.org/authors/?q=ai:xu.jiafa"Liu, Jie"https://zbmath.org/authors/?q=ai:liu.jie.29|liu.jie.7"O'Regan, Donal"https://zbmath.org/authors/?q=ai:oregan.donalUsing fixed point theory and upper and lower solutions, some existence theorems of positive solutions for integral boundary value problems of nonlinear Hadamard fractional differential equations are investigated. Furthermore, the multiplicity of positive solutions of the addressed problems is established. At last, an example is presented to illustrate the main results.
Reviewer: Abdelghani Ouahab (Sidi bel Abbès)Solvability for a higher order implicit fractional multi-point boundary value problems at resonancehttps://zbmath.org/1534.340152024-06-14T15:52:26.737412Z"Zhang, Wei"https://zbmath.org/authors/?q=ai:zhang.wei.26"Fu, Xinyu"https://zbmath.org/authors/?q=ai:fu.xinyuSummary: In this work, we show the existence of solutions for a higher order implicit fractional differential equations with multi-point boundary conditions at resonance. Our results are obtained by utilizing the (Kuratovski) measure of non-compactness and abstract continuation theorem for \(k\)-set contractions. Finally, an example is given to illustrate the applicability of theoretical result.Adomian decomposition method in the theory of nonlinear boundary-value problemshttps://zbmath.org/1534.340272024-06-14T15:52:26.737412Z"Chuiko, S. M."https://zbmath.org/authors/?q=ai:chuiko.sergei-m"Chuiko, O. S."https://zbmath.org/authors/?q=ai:chuiko.o-s"Popov, M. V."https://zbmath.org/authors/?q=ai:popov.mikhail-vyacheslavovichSummary: For the nonlinear periodic boundary-value problem posed for an ordinary differential equation in the critical and noncritical cases, we obtain constructive conditions of its solvability and propose a scheme for finding its solutions by using the Adomian decomposition method.A half-inverse problem for singular diffusion operator with certain boundary conditionshttps://zbmath.org/1534.340282024-06-14T15:52:26.737412Z"Ergün, Abdullah"https://zbmath.org/authors/?q=ai:ergun.abdullah"Amirov, Rauf"https://zbmath.org/authors/?q=ai:amirov.rauf-khThe paper deals with the Sturm-Liouville problem
\[
-y'' + [2 \lambda p(x) + q(x)] y = \lambda^2 \delta(x) y, \quad x \in (0, \pi), \tag{1}
\]
\[
y'(0) = 0, \quad y(\pi) = 0, \tag{2}
\]
which has a quadratic dependence on the spectral parameter \(\lambda\) and a step-like weight function \(\delta(x)\). More precisely, \(p \in W_1^1[0,\pi]\), \(q \in L_2[0,\pi]\), and
\[
\delta(x) = \begin{cases} 1, & x \in (0, a_1), \\
\alpha^2, & x \in (a_1, a_2), \\
\beta^2, & x \in (a_2, \pi), \end{cases}, \quad 0 < a_1 < \frac{\pi}{2} < a_2 < \pi.
\]
The authors obtain the Hochstadt-Lieberman-type uniqueness theorem. Namely, it is shown that the functions \(p(x)\) and \(q(x)\) are uniquely specified by the eigenvalues \(\{ \lambda_n \}\) of the problem (1)-(2), while \(p(x)\) and \(q(x)\) are known a priori on the half-interval \((\frac{\pi}{2}, \pi)\). The method is based on integral representations for solutions of equation (1).
Reviewer: Natalia Bondarenko (Saratov)The local Borg-Marchenko uniqueness theorem of matrix-valued Dirac-type operators for coefficients locally smooth at the right endpointhttps://zbmath.org/1534.340292024-06-14T15:52:26.737412Z"Wei, Guangsheng"https://zbmath.org/authors/?q=ai:wei.guangsheng"Zhang, Zhongfang"https://zbmath.org/authors/?q=ai:zhang.zhongfangThe paper deals with a self-adjoint matrix-valued Dirac-type operator
\[
L(y)(x):= JY'(x)-B(x)Y(x),
\]
subject to separate boundary conditions, defined on \([0,b)\) for \(0<b\le\infty\), where
\[
J=:J_{2m}=\begin{pmatrix} 0&-I_m\\
I_m&0 \end{pmatrix},\quad B(x)=\begin{pmatrix} B_{11}(x)& B_{12}(x)\\
B_{12}(x)&-B_{11}(x) \end{pmatrix},
\]
with \(I_m\) being the identity matrix in \(\mathbb{C}^m\) for \(m\in \mathbb{N}\) and \(B_{11}(x),B_{12}(x)\) two self-adjoint \(m\times m\) matrices for a.c. \(x\in[0,b)\). The authors provide an expression of the Weyl-Titchmarsh matrix associated with the considered operator and prove a local Borg-Marchenko uniqueness theorem by the additional assumption that the potential of the operator is sufficiently smooth in a right neighborhood of a fixed point.
Reviewer: A. S. Makin (Moskva)Existence and nonexistence of solutions for a class of \(p\)-Laplacian-type fractional four-point boundary-value problems with a parameterhttps://zbmath.org/1534.340312024-06-14T15:52:26.737412Z"Pan, Xinyuan"https://zbmath.org/authors/?q=ai:pan.xinyuan"He, Xiaofei"https://zbmath.org/authors/?q=ai:he.xiaofei"Hu, Aimin"https://zbmath.org/authors/?q=ai:hu.aiminIn this paper, a class of four point boundary value problems of \(p\)-Laplacian operator Caputo fractional differential equation with a parameter is investigated. In the boundary conditions, the \(p\)-Laplacian operator and multiple fractional derivative terms are include too. The existence and nonexistence results for the solutions of the boundary value problems are obtained by using the Schauder fixed-point theorem and the Guo-Krasnoselskii fixed-point theorem on cones. Finally, some examples are given out to illustrate the main results.
Reviewer: Xiping Liu (Shanghai)Quadrature method for equations with nonlinear boundary conditions arising in a thermal explosion theoryhttps://zbmath.org/1534.340322024-06-14T15:52:26.737412Z"Ko, Eunkyung"https://zbmath.org/authors/?q=ai:ko.eunkyungSummary: We consider a 1-dimensional reaction diffusion equation with the following boundary conditions arising in a theory of the thermal explosion
\[
\begin{cases}
-u^{\prime \prime} = \lambda f (u(t)), \, t \in (0,l),\\
-u^\prime (0) + C(0)u(0)=0,\\
u^\prime (l) + C(l) u(l)=0,
\end{cases}
\]
where \(C : [0, \infty) \to (0, \infty)\) is a continuous and nondecreasing function, \(\lambda > 0\) is a parameter and \(f : [0, \infty) \to (0, \infty)\) is a continuous function. We establish the extension of Quadrature method introduced in [8]. Using this extension, we provide numerical results for models with a typical function of \(f\) and \(C\) in a thermal explosion theory, which verify the existence, uniqueness and multiplicity results proved in [\textit{P. V. Gordon} et al., Nonlinear Anal., Real World Appl. 15, 51--57 (2014; Zbl 1302.35151)].Reproducing kernel representation of the solution of second order linear three-point boundary value problemhttps://zbmath.org/1534.340332024-06-14T15:52:26.737412Z"Bai, Hongfang"https://zbmath.org/authors/?q=ai:bai.hongfang"Leong, Ieng Tak"https://zbmath.org/authors/?q=ai:leong.ieng-tak|leong.iengtak"Dang, Pei"https://zbmath.org/authors/?q=ai:dang.pei(no abstract)Three general uniqueness theorems for BVPs of ODEs with applicationshttps://zbmath.org/1534.340342024-06-14T15:52:26.737412Z"Vidossich, Giovanni"https://zbmath.org/authors/?q=ai:vidossich.giovanniSummary: We prove three uniqueness theorems for BVPs of ODEs whose unusual assumptions allow to prove existence results for BVPs related to various types of linear and nonlinear boundary conditions.Fundamental solution of a singular Bessel differential operator with a negative parameterhttps://zbmath.org/1534.340352024-06-14T15:52:26.737412Z"Lyakhov, L. N."https://zbmath.org/authors/?q=ai:lyakhov.l-n"Sanina, E. L."https://zbmath.org/authors/?q=ai:sanina.e-l"Roshchupkin, S. A."https://zbmath.org/authors/?q=ai:roshchupkin.s-a"Bulatov, Yu. N."https://zbmath.org/authors/?q=ai:bulatov.yu-nSummary: The singular differential Bessel operator \({{B}_{{ - \gamma }}}\) with negative parameter \(- \gamma < 0\) is considered. Solutions to the singular differential Bessel equation \({{B}_{{ - \gamma }}}u + {{\lambda }^2}u = 0\) are represented by linearly independent functions \({{\mathbb{J}}_{\mu }}\) and \({{\mathbb{J}}_{{ - \mu }}}\), \(\mu = \frac{{\gamma + 1}}{2} \). Some properties of the functions \({{\mathbb{J}}_{\mu }} \), which are expressed in terms of the properties of the Bessel-Levitan \(j\)-function, are studied. Direct and inverse Bessel \({{\mathbb{J}}_{\mu }} \)-transforms are introduced. Based on the \(\mathbb{T} \)-pseudoshift operator introduced earlier, a generalized \(\mathbb{T} \)-shift operator belonging to the Levitan class of generalized shifts commuting with the Bessel operator \({{B}_{{ - \gamma }}}\) is constructed. A fundamental solution is found for the singular differential operator \({{B}_{{ - \gamma }}}\) with a singularity at an arbitrary point on the semiaxis \([0,\infty )\).The multiplicity of positive solutions for a certain logistic problemhttps://zbmath.org/1534.340362024-06-14T15:52:26.737412Z"Huang, Shao-Yuan"https://zbmath.org/authors/?q=ai:huang.shaoyuan"Lee, Wei-Hsun"https://zbmath.org/authors/?q=ai:lee.wei-hsun"Ho, Ming-Hsuan"https://zbmath.org/authors/?q=ai:ho.ming-hsuanSummary: We study the exact multiplicity and bifurcation curves of positive solutions for the logistic problem \[\begin{cases} -u''=\lambda \bigg(\frac{u}{1+u}\bigg)^p, \quad\text{in}\,\, (-1,1), \\ u(-1)=u(1)=0, \end{cases}\] where \(\lambda, p >0\). We prove that the bifurcation curve is monotone increasing for \(0< p\leq 1\) and \(\subset\)-shaped for \(p >1\).Computation of eigenvalues of fractional Sturm-Liouville problemshttps://zbmath.org/1534.340372024-06-14T15:52:26.737412Z"Maralani, E. M."https://zbmath.org/authors/?q=ai:maralani.e-m"Saei, F. D."https://zbmath.org/authors/?q=ai:saei.farhad-dastmalchi"Akbarfam, A. A. J."https://zbmath.org/authors/?q=ai:akbarfam.a-a-j"Ghanbari, K."https://zbmath.org/authors/?q=ai:ghanbari.kazemIn this paper, the authors study eigenvalues of the fractional-order Sturm-Liouville equation \[-^CD^\alpha_{0^+}\circ D^\alpha_{0^+}y(t)+q(t)y(t)=\lambda y(t),\quad 0<\alpha\le 1, \quad t\in[0,1],\] satisfying the boundary conditions \[I^{1-\alpha}_{0^+}y(t)|_{t=0}=0\quad\text{and}\quad I^{1-\alpha}_{0^+}y(t)|_{t=1}=0,\] where \(q\in L^2(0,1)\) is a real-valued potential function. The authors use an iterative approach to approximate eigenvalues when \(q(s)=1\) and when \(q(s)=s^\beta\). They find that the eigenvalues are zeros of equations involving generalized Mittag-Leffler functions. As \(\alpha\to1\), the authors obtain the classical results. Tables of the eigenvalues and illustrative graphs of the eigenfunctions are given. Finally, the authors give an analysis of their iterative method.
Reviewer: Jeffrey Neugebauer (Richmond)Partially-isospectral Sturm-Liouville boundary value problems on the finite segmenthttps://zbmath.org/1534.340382024-06-14T15:52:26.737412Z"Mirzaev, Olim Èrkinovich"https://zbmath.org/authors/?q=ai:mirzaev.olim-erkinovichSummary: In paper, an algorithm is proposed for constructing isospectral and partially-isospectral Sturm-Liouville boundary value problems on the finite segment.Inverse fractional Sturm-Liouville problem with eigenparameter in the boundary conditionshttps://zbmath.org/1534.340392024-06-14T15:52:26.737412Z"Sa'idu, Auwalu"https://zbmath.org/authors/?q=ai:saidu.auwalu"Koyunbakan, Hikmet"https://zbmath.org/authors/?q=ai:koyunbakan.hikmet(no abstract)Constructing the asymptotics of solutions to differential Sturm-Liouville equations in classes of oscillating coefficientshttps://zbmath.org/1534.340402024-06-14T15:52:26.737412Z"Valeev, N. F."https://zbmath.org/authors/?q=ai:valeev.nurmukhamet-fuatovich|valeev.nur-f"Nazirova, E. A."https://zbmath.org/authors/?q=ai:nazirova.elvira-airatovna"Sultanaev, Ya. T."https://zbmath.org/authors/?q=ai:sultanaev.yaudat-talgatovichSummary: The article is focused on the development of a method allowing one to construct asymptotics for solutions to ODEs of arbitrary order with oscillating coefficients on the semiaxis. The idea of the method is presented on the example of studying the asymptotics of the Sturm-Liouville equation.Existence and multiplicity of solutions for a class of nonlocal elliptic transmission systemshttps://zbmath.org/1534.340412024-06-14T15:52:26.737412Z"Abdelmalek, Brahim"https://zbmath.org/authors/?q=ai:abdelmalek.brahim"Ali, Djellit"https://zbmath.org/authors/?q=ai:ali.djellit"Sameh, Tamrabet"https://zbmath.org/authors/?q=ai:sameh.tamrabetSummary: By using the approach based on variationnel methods and critical point theory, more precisely, the symmetric mountain pass theorem, we study the existence and multiplicity of weak solutions for a class of elliptic transmision system with nonlocal term.Computation of the eigenvalues for the angular and Coulomb spheroidal wave equationhttps://zbmath.org/1534.340422024-06-14T15:52:26.737412Z"Schmid, Harald"https://zbmath.org/authors/?q=ai:schmid.haraldA new method to find the eigenvalues of the Coulomb spheroidal wave equation (CSWE)
\[
\frac d{dx}\left( (1-x^2)\frac d{dx}w(x) \right)+\left( \lambda +\beta x+\gamma ^2(1-x)^2-\frac{\mu ^2}{1-x^2} \right) w(x)=0
\]
is presented. Here \(\mu ,\beta ,\gamma \in \mathbb{C}\) are fixed parameters and \(\lambda \in \mathbb{C}\) is an called eigenvalue if CSWE has a bounded nontrivial solution on \((-1,1)\). Without restriction \(\text{Re}\mu \ge0\) may be assumed.
To determine the eigenvalues, polynomials \(\Theta _k=\Theta _k(t)=\delta ^{\textsf{T}}d_k\) are defined for \(k=1,2,\dots\), where \(\delta ^{\textsf{T}}=\left(1,-\dfrac{\beta +t}{\mu +1}\right)\) and
\begin{align*}
u_k=\begin{pmatrix} 0&\dfrac{t-\beta }k-\dfrac{2t}{k+\mu+1} \\
0& -\dfrac{\mu +1}k\end{pmatrix}d_{k-1} -\begin{pmatrix} \dfrac{t-\beta }{k(k+\mu +1)}&\dfrac{4\gamma ^2}{k+\mu +1}\\
-\dfrac 1k&0 \end{pmatrix}u_{k-1},
\end{align*}
\(d_k=d_{k-1}+u_k\) for \(k=1,2,3\) with \(u_0=d_0=\left( \dfrac{\beta -t}{\mu +1},1 \right)^{\textsf{ T}}\).
Then \(\Theta (t)=\lim_{k\to \infty }\Theta _k(t)\) defines an entire function \(\Theta \), and \(\lambda \) is an eigenvalue of CSWE if and only if \(t=\lambda -\mu (\mu +1)\) is a zero of \(\Theta \). An eigenfunction can be expressed as a series in terms of \(\mu \) and \(d_k(t)\).
Several examples are provided, for \(\gamma ^2=-25\) and \(\mu =2\) as well as \(\mu =2+0.05i\). For the case \(\mu =2\), the numerical values are compared with results in earlier publications.
The procedure and its proof in the case of the generalized spheroidal wave equation is outlined, where \(\mu^2\) is replaced by the more general term \(\mu^2+\alpha ^2+2\alpha \mu x\).
Reviewer: Manfred Möller (Johannesburg)Existence of two infinite families of solutions for singular superlinear equations on exterior domainshttps://zbmath.org/1534.340432024-06-14T15:52:26.737412Z"Iaia, Joseph"https://zbmath.org/authors/?q=ai:iaia.joseph-aSummary: In this article we study radial solutions of \(\Delta u + K(|x|) f(u) =0\) in the exterior of the ball of radius \(R>0\) in \(\mathbb{R}^N\) with \(N>2\) where \(f\) grows superlinearly at infinity and is singular at \(0\) with \(f(u) \sim \frac{1}{|u|^{q-1}u}\) and \(0<q<1\) for small \(u\). We assume \(K(|x|)\sim|x|^{-\alpha}\) for large \(|x|\) and establish existence of two infinite families of sign-changing solutions when \(N+q(N-2)<\alpha<2(N-1)\).A quintic \(\mathbb{Z}_2\)-equivariant Liénard system arising from the complex Ginzburg-Landau equationhttps://zbmath.org/1534.340462024-06-14T15:52:26.737412Z"Chen, Hebai"https://zbmath.org/authors/?q=ai:chen.hebai"Chen, Xingwu"https://zbmath.org/authors/?q=ai:chen.xingwu"Jia, Man"https://zbmath.org/authors/?q=ai:jia.man"Tang, Yilei"https://zbmath.org/authors/?q=ai:tang.yileiIn this paper, a quintic Liénard system associated with the famous Ginzburg-Landau equation is investigated for the corresponding global phase portraits in the Poincaré disc. The system presents complicated dynamics, which not only involves various local and global bifurcations, but also exhibits infinitely many bifurcation surfaces of saddle connection.
Although only the case \(a_2<0\), namely the sum of indices of equilibria is \(-1\), is considered, the dynamics of this system cannot be studied via counting the isolate zeros of abelian integrals as usual, and the work is meaningful both in theory and in practice.
Reviewer: Qinlong Wang (Guilin)Impulsive boundary-value problem for the Lyapunov equation with values in Hilbert spacehttps://zbmath.org/1534.340612024-06-14T15:52:26.737412Z"Panasenko, E. V."https://zbmath.org/authors/?q=ai:panasenko.evgeniy-v"Pokutnyi, O. O."https://zbmath.org/authors/?q=ai:pokutnij.olexandr-oSummary: We investigate boundary-value problems for the operator-differential Lyapunov equation with impulsive action at fixed times with values in the Hilbert space. The case where the corresponding generating operator may have a nonclosed set of values is analyzed. The results are illustrated by an example of the problem with diagonal operators.Existence and multiplicity of solutions for boundary value problem of singular two-term fractional differential equation with delay and sign-changing nonlinearityhttps://zbmath.org/1534.340732024-06-14T15:52:26.737412Z"Bai, Rulan"https://zbmath.org/authors/?q=ai:bai.rulan"Zhang, Kemei"https://zbmath.org/authors/?q=ai:zhang.kemei"Xie, Xue-Jun"https://zbmath.org/authors/?q=ai:xie.xuejunSummary: In this paper, we consider the existence of solutions for a boundary value problem of singular two-term fractional differential equation with delay and sign-changing nonlinearity. By means of the Guo-Krasnosel'skii fixed point theorem and the Leray-Schauder nonlinear alternative theorem, we obtain some results on the existence and multiplicity of solutions, respectively.Spectral data asymptotics for fourth-order boundary value problemshttps://zbmath.org/1534.340852024-06-14T15:52:26.737412Z"Bondarenko, Natal'ya Pavlovna"https://zbmath.org/authors/?q=ai:bondarenko.natalya-pavlovnaSummary: In this paper, we derive sharp asymptotics for the spectral data (eigenvalues and weight numbers) of the fourth-order linear differential equation with a distribution coefficient and three types of separated boundary conditions. Our methods rely on the recent results concerning regularization and asymptotic analysis for higher-order differential operators with distribution coefficients. The results of this paper have applications to the theory of inverse spectral problems as well as a separate significance.Stability of a flexible missile and asymptotics of the eigenvalues of fourth order boundary value problemshttps://zbmath.org/1534.340862024-06-14T15:52:26.737412Z"Zinsou, Bertin"https://zbmath.org/authors/?q=ai:zinsou.bertinThe paper studies the eigenvalue problem
\[
y^{(4)}(x)-(g(x)y'(x))' = \lambda^2 y(x)
\]
on the interval \([0,a]\) with certain boundary conditions at the endpoints. These boundary conditions are dependent on the spectral parameter \(\lambda\). The author claims that the problem occurs when investigating elastic rods or the stability of a flexible missile. The asymptotics of the spectral parameter is studied, and the results are in the final section applied to the flexible missile problem.
The paper is not very readable, e.g. the definition of Birkhoff regular problem is missing.
Reviewer: Jiři Lipovský (Hradec Králové)Cauchy problem for the loaded Korteweg-de Vries equation in the class of periodic functionshttps://zbmath.org/1534.353542024-06-14T15:52:26.737412Z"Khasanov, A. B."https://zbmath.org/authors/?q=ai:khasanov.a-b"Khasanov, T. G."https://zbmath.org/authors/?q=ai:khasanov.t-gSummary: The inverse spectral problem method is applied to finding a solution of the Cauchy problem for the loaded Korteweg-de Vries equation in the class of periodic infinite-gap functions. A simple algorithm for constructing a high-order Korteweg-de Vries equation with loaded terms and a derivation of an analog of Dubrovin's system of differential equations are proposed. It is shown that the sum of a uniformly convergent function series constructed by solving the Dubrovin system of equations and the first trace formula actually satisfies the loaded nonlinear Korteweg-de Vries equation. In addition, we prove that if the initial function is a real \(\pi \)-periodic analytic function, then the solution of the Cauchy problem is a real analytic function in the variable \(x\) as well, and also that if the number \({\pi }/{n}\), \(n\in \mathbb{N}\), \(n\ge 2 \), is the period of the initial function, then the number \({\pi }/{n}\) is the period for solving the Cauchy problem with respect to the variable \(x\).Generalized Green's operator of the matrix integral-differential boundary value problem unsolved with respect to the derivativehttps://zbmath.org/1534.450032024-06-14T15:52:26.737412Z"Chuiko, Sergii M."https://zbmath.org/authors/?q=ai:chuiko.sergii-mykhailovych"Nesmelova, Olga V."https://zbmath.org/authors/?q=ai:nesmelova.olga-volodymyrivna"Kuz'mina, Vlada O."https://zbmath.org/authors/?q=ai:kuzmina.vlada-oSummary: Solvability conditions and a structure of the generalized Green's operator of a linear Noether boundary value problem for a matrix integral-differential system unsolved with respect to the derivative, which generalizes integral-differential systems of the Fredholm type with a degenerate kernel. To solve the matrix integral-differential boundary value problem unsolved with respect to the derivative, original solvability conditions were used, as well as the structure of the general solution of the Sylvester-type matrix equation.Optimal mass of structure with motion described by Sturm-Liouville operator: design and predesignhttps://zbmath.org/1534.490282024-06-14T15:52:26.737412Z"Belinskiy, Boris P."https://zbmath.org/authors/?q=ai:belinsky.boris-p"Smith, Tanner A."https://zbmath.org/authors/?q=ai:smith.tanner-aSummary: We find an optimal design of a structure described by a Sturm-Liouville (S-L) problem with a spectral parameter in the boundary conditions. Using an approach from calculus of variations, we determine a set of critical points of a corresponding mass functional. However, these critical points -- which we call predesigns -- do not necessarily themselves represent meaningful solutions: it is of course natural to expect a mass to be real and positive. This represents a generalization of previous work on the topic in several ways. First, previous work considered only boundary conditions and S-L coefficients under certain simplifying assumptions. Principally, we do not assume that one of the coefficients vanishes as in the previous work. Finally, we introduce a set of solvability conditions on the S-L problem data, confirming that the corresponding critical points represent meaningful solutions we refer to as designs. Additionally, we present a natural schematic for testing these conditions, as well as suggesting a code and several numerical examples.A class of fractional two-point boundary value problems: an iterative approachhttps://zbmath.org/1534.651072024-06-14T15:52:26.737412Z"Khuri, S. A."https://zbmath.org/authors/?q=ai:khuri.suheil-a"Sayfy, A."https://zbmath.org/authors/?q=ai:sayfy.ali-m-sSummary: The current study's goal is to describe and implement a practical numerical solution for addressing a class of two-point nonlinear fractional boundary value problems (FBVP). The fractional differential equations under investigation are complimented with Dirichlet or mixed boundary conditions. The proposed iterative scheme, known as the Green-Picard or Green-Mann iteration approach, is a newly developed method that embeds Green's function into a customized integral operator before applying either Picard's or Mann's iterative procedures. The method converges quickly and with little CPU time, and the scheme's convergence analysis is addressed. To demonstrate the validity and application of the semi-analytical approach, numerical tests were performed on a variety of FBVPs. The numerical findings indicate that the proposed approach has good performance and precision.One-dimensional run-and-tumble motions with generic boundary conditionshttps://zbmath.org/1534.810022024-06-14T15:52:26.737412Z"Angelani, Luca"https://zbmath.org/authors/?q=ai:angelani.lucaSummary: The motion of run-and-tumble particles in one-dimensional finite domains are analyzed in the presence of generic boundary conditions. These describe accumulation at walls, where particles can either be absorbed at a given rate, or tumble, with a rate that may be, in general, different from that in the bulk. This formulation allows us to treat in a unified way very different boundary conditions (fully and partially absorbing/reflecting, sticky, sticky-reactive and sticky-absorbing boundaries) which can be recovered as appropriate limits of the general case. We report the general expression of the mean exit time, valid for generic boundaries, discussing many case studies, from equal boundaries to more interesting cases of different boundary conditions at the two ends of the domain, resulting in nontrivial expressions of mean exit times.
{{\copyright} 2023 The Author(s). Published by IOP Publishing Ltd}Spectral minimal partitions of unbounded metric graphshttps://zbmath.org/1534.810552024-06-14T15:52:26.737412Z"Hofmann, Matthias"https://zbmath.org/authors/?q=ai:hofmann.matthias"Kennedy, James B."https://zbmath.org/authors/?q=ai:kennedy.james-b"Serio, Andrea"https://zbmath.org/authors/?q=ai:serio.andreaSummary: We investigate the existence or non-existence of spectral minimal partitions of unbounded metric graphs, where the operator applied to each of the partition elements is a Schrödinger operator of the form \(-\Delta +V\) with suitable (electric) potential \(V\), which is taken as a fixed, underlying function on the whole graph.
We show that there is a strong link between spectral minimal partitions and infimal partition energies on the one hand, and the infimum \(\Sigma\) of the essential spectrum of the corresponding Schrödinger operator on the whole graph on the other. Namely, we show that for any \(k\in\mathbb{N}\), the infimal energy among all admissible \(k\)-partitions is bounded from above by \(\Sigma\), and if it is strictly below \(\Sigma\), then a spectral minimal \(k\)-partition exists. We illustrate our results with several examples of existence and non-existence of minimal partitions of unbounded and infinite graphs, with and without potentials.
The nature of the proofs, a key ingredient of which is a version of the characterization of the infimum of the essential spectrum known as Persson's theorem for quantum graphs, strongly suggests that corresponding results should hold for Schrödinger operator-based partitions of unbounded domains in Euclidean space.An exact formula for the number of negative eigenvalues for zigzag carbon nanotubes with \(\delta\) impuritieshttps://zbmath.org/1534.820322024-06-14T15:52:26.737412Z"Niikuni, Hiroaki"https://zbmath.org/authors/?q=ai:niikuni.hiroakiSummary: In this paper, we consider quantum graphs corresponding to zigzag carbon nanotubes with finite number of impurities. Recall that the quantum graph for a zigzag carbon nanotube without impurities is the triplet of the metric graph for the zigzag carbon nanotube \(\Gamma^N\) with \(N\)-zigzags, the Schrödinger operator on \(\Gamma^N\) and the Kirchhoff-Neumann vertex condition. To express impurities, we utilize the \(\delta\)-type vertex condition. In this paper, we study the case where the impurities are located symmetrically with respect to the rotation as a finite number of impurity rings. After we construct a general spectral theory for the setting, we give an exact formulae for the number of negative eigenvalues in the case of a single ring in terms of the strength \(\beta\) of impurities. Throughout this paper, we shall find an interesting difference between zigzag carbon nanotubes and the one-dimensional Schrödinger operator with the \(\delta\) point interactions.An efficient multi-derivative numerical method for chemical boundary value problemshttps://zbmath.org/1534.920032024-06-14T15:52:26.737412Z"Celik, Esra"https://zbmath.org/authors/?q=ai:celik.esra"Tunc, Huseyin"https://zbmath.org/authors/?q=ai:tunc.huseyin"Sari, Murat"https://zbmath.org/authors/?q=ai:sari.muratSummary: The singular and singularly perturbed boundary value problems (SBVPs and SPBVPs) arise in the modeling of various chemical processes such as the isothermal gas sphere, electroactive polymer film, thermal explosion, and chemical reactor theory. Efficient numerical methods are desirable for solving such problems with a wide scope of influence. Here we derive the implicit-explicit local differential transform method (IELDTM) based on the Taylor series to solve chemical SBVPs and SPBVPs. The differential equations are directly utilized to determine the local Taylor coefficients and the entire system of algebraic equations is assembled using explicit/implicit continuity relations regarding the direction parameter. The IELDTM has an effective \(h\)-\(p\) refinement property and increasing the order of the method does not affect the degrees of freedom. We have validated the theoretical convergence results of the IELDTM with various numerical experiments and provided detailed discussions. It has been proven that the IELDTM yields more accurate solutions with fewer CPU times than various recent numerical methods for solving chemical BVPs.The null boundary controllability for a fourth-order parabolic equation with Samarskii-Lonkin-type boundary conditionshttps://zbmath.org/1534.930452024-06-14T15:52:26.737412Z"Oner, Isil"https://zbmath.org/authors/?q=ai:oner.isilIn this work, the author investigates the null boundary controllability for the fourth-order parabolic equations \(u_t +u_{xxxx} -cu=0\) on \([0,1]\times [0,T]\) with Samarkii-Ionkin-type conditions. Due to the absence of self-adjointness under these boundary conditions, the author initially illustrates that the eigenfunctions of the auxiliary spectral problem of the the backward adjoint system form a Riesz basis in \(L^2(0,1)\). In addition, the author establishes the existence and uniqueness of the adjoint problem and, applying the moment method, derives necessary and sufficient conditions for null boundary controllability of the given system for some classes of initial data.
Reviewer: Michela Egidi (Rostock)