Boundary Value Problems最新文献

英文中文

A new weighted fractional operator with respect to another function via a new modified generalized Mittag–Leffler law 利用一个新的改进的广义Mittag-Leffler定律，得到一个关于另一个函数的新的加权分数算子

4区数学 Q1 MATHEMATICS

Boundary Value Problems

Pub Date : 2023-10-06 DOI: 10.1186/s13661-023-01790-7

Sabri T. M. Thabet, Thabet Abdeljawad, Imed Kedim, M. Iadh Ayari

Abstract In this paper, new generalized weighted fractional derivatives with respect to another function are derived in the sense of Caputo and Riemann–Liouville involving a new modified version of a generalized Mittag–Leffler function with three parameters, as well as their corresponding fractional integrals. In addition, several new and existing operators of nonsingular kernels are obtained as special cases of our operator. Many important properties related to our new operator are introduced, such as a series version involving Riemann–Liouville fractional integrals, weighted Laplace transforms with respect to another function, etc. Finally, an example is given to illustrate the effectiveness of the new results.

摘要本文在Caputo和Riemann-Liouville意义下，导出了关于另一函数的新的广义加权分数阶导数，涉及到具有三个参数的广义Mittag-Leffler函数的一个新的修正版本，以及相应的分数阶积分。此外，作为该算子的特例，还得到了一些新的和已有的非奇异核算子。介绍了与我们的新算子相关的许多重要性质，如涉及Riemann-Liouville分数积分的级数版本，关于另一个函数的加权拉普拉斯变换等。最后，通过一个算例说明了新结果的有效性。

引用次数: 0

On nonlinear fractional Choquard equation with indefinite potential and general nonlinearity 具有不定势的非线性分数阶Choquard方程及一般非线性

4区数学 Q1 MATHEMATICS

Boundary Value Problems

Pub Date : 2023-10-05 DOI: 10.1186/s13661-023-01786-3

Fangfang Liao, Fulai Chen, Shifeng Geng, Dong Liu

Abstract In this paper, we consider a class of fractional Choquard equations with indefinite potential $$ (-Delta )^{alpha}u+V(x)u= biggl[ int _{{mathbb{R}}^{N}} frac{M(epsilon y)G(u)}{ vert x-y vert ^{mu}},mathrm{d}y biggr]M( epsilon x)g(u), quad xin {mathbb{R}}^{N}, $$ ( − Δ ) α u + V ( x ) u = [ ∫ R N M ( ϵ y ) G ( u ) | x − y | μ d y ] M ( ϵ x ) g ( u ) , x ∈ R N , where $alpha in (0,1)$ α ∈ ( 0 , 1 ) , $N> 2alpha $ N > 2 α , $0 0 < μ < 2 α , ϵ is a positive parameter. Here $(-Delta )^{alpha}$ ( − Δ ) α stands for the fractional Laplacian, V is a linear potential with periodicity condition, and M is a nonlinear reaction potential with a global condition. We establish the existence and concentration of ground state solutions under general nonlinearity by using variational methods.

摘要本文考虑一类具有不定势$$ (-Delta )^{alpha}u+V(x)u= biggl[ int _{{mathbb{R}}^{N}} frac{M(epsilon y)G(u)}{ vert x-y vert ^{mu}},mathrm{d}y biggr]M( epsilon x)g(u), quad xin {mathbb{R}}^{N}, $$(−Δ) α u + V (x) u =[∫R N M (λ y) G (u) | x−y | μ d y] M (λ x) G (u)， x∈R N，其中$alpha in (0,1)$ α∈(0,1)，$N> 2alpha $ N ＆gt;2 α， $0<mu <2alpha $ 0 ＆lt;μ ＆lt;2 α， ε是一个正参数。其中$(-Delta )^{alpha}$(−Δ) α表示分数阶拉普拉斯势，V是具有周期性条件的线性势，M是具有全局条件的非线性反应势。用变分方法建立了一般非线性条件下基态解的存在性和集中性。

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引用次数: 0

Computing Dirichlet eigenvalues of the Schrödinger operator with a PT-symmetric optical potential 计算具有pt对称光势的Schrödinger算子的狄利克雷特征值

4区数学 Q1 MATHEMATICS

Boundary Value Problems

Pub Date : 2023-10-04 DOI: 10.1186/s13661-023-01787-2

Cemile Nur

Abstract We provide estimates for the eigenvalues of non-self-adjoint Sturm–Liouville operators with Dirichlet boundary conditions for a shift of the special potential $4cos ^{2}x+4iVsin 2x$ 4 cos 2 x + 4 i V sin 2 x that is a PT-symmetric optical potential, especially when $|c|=|sqrt{1-4V^{2}}|<2$ | c | = | 1 − 4 V 2 | < 2 or correspondingly $0leq V 0 ≤ V < 5 / 2 . We obtain some useful equations for calculating Dirichlet eigenvalues also for $|c|geq 2$ | c | ≥ 2 or equally $Vgeq sqrt{5}/2$ V ≥ 5 / 2 . We discuss our results by comparing them with the periodic and antiperiodic eigenvalues of the Schrödinger operator. We even approximate complex eigenvalues by the roots of some polynomials derived from some iteration formulas. Moreover, we give a numerical example with error analysis.

摘要本文给出了具有Dirichlet边界条件的非自伴随Sturm-Liouville算子在pt对称光势$4cos ^{2}x+4iVsin 2x$ 4 cos 2 x + 4 i V sin 2 x移位时的特征值估计，特别是当$|c|=|sqrt{1-4V^{2}}|<2$ | c | = | 1−4 V 2 | ＆lt;2或对应$0leq V<sqrt {5}/2$ 0≤V ＆lt;5 / 2。对于$|c|geq 2$ | c |≥2或同样的$Vgeq sqrt{5}/2$ V≥5 / 2，我们也得到了一些计算Dirichlet特征值的有用方程。我们通过与Schrödinger算子的周期特征值和反周期特征值的比较来讨论我们的结果。我们甚至用一些由迭代公式导出的多项式的根来近似复特征值。并给出了数值算例，进行了误差分析。

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引用次数: 0

Analytical mechanics methods in finite element analysis of multibody elastic system 多体弹性系统有限元分析中的分析力学方法

4区数学 Q1 MATHEMATICS

Boundary Value Problems

Pub Date : 2023-10-04 DOI: 10.1186/s13661-023-01784-5

Maria Luminita Scutaru, Sorin Vlase, Marin Marin

Abstract The study of multibody systems with elastic elements involves at the moment the reevaluation of the classical methods of analysis offered by analytical mechanics. Modeling this system with the finite element method requires obtaining the motion equation for an element in the circumstances imposed by a multibody system. The paper aims to present the main analysis methods used by researchers, to make a comparative analysis, and to show the advantages or disadvantages offered by different methods. For the presentation of the main methods (namely Lagrange’s equations, Gibbs–Appell’s equations, Maggi’s formalism, Kane’s equations, and Hamilton’s equations) a unified notation is used. The paper provides a critical evaluation of the studied applications that involved some of these methods, highlighting the reason why it was decided to use them. Also, the paper identifies potential research areas to explore.

具有弹性单元的多体系统的研究目前涉及到对分析力学经典分析方法的重新评价。用有限元法对该系统进行建模，需要得到在多体系统作用下单元的运动方程。本文旨在介绍研究人员使用的主要分析方法，并进行比较分析，显示不同方法的优缺点。对于主要方法(即拉格朗日方程、吉布斯-阿佩尔方程、马吉形式主义、凯恩方程和汉密尔顿方程)的表示，使用统一的符号。本文对涉及其中一些方法的研究应用进行了批判性评估，强调了决定使用这些方法的原因。此外，本文还确定了潜在的研究领域。

引用次数: 0

Remarks on a fractional nonlinear partial integro-differential equation via the new generalized multivariate Mittag-Leffler function 用新的广义多元Mittag-Leffler函数讨论分数阶非线性偏积分微分方程

4区数学 Q1 MATHEMATICS

Boundary Value Problems

Pub Date : 2023-10-03 DOI: 10.1186/s13661-023-01783-6

Chenkuan Li, Reza Saadati, Joshua Beaudin, Andrii Hrytsenko

Abstract Introducing a new generalized multivariate Mittag-Leffler function which is a generalization of the multivariate Mittag-Leffler function, we derive a sufficient condition for the uniqueness of solutions to a brand new boundary value problem of the fractional nonlinear partial integro-differential equation using Banach’s fixed point theorem and Babenko’s technique. This has many potential applications since uniqueness is an important topic in many scientific areas, and the method used clearly opens directions for studying other types of equations and corresponding initial or boundary value problems. In addition, we use Python which is a high-level programming language efficiently dealing with the summation of multi-indices to compute approximate values of the generalized Mittag-Leffler function (it seems impossible to do so by any existing integral representation of the Mittag-Leffler function), and provide an example showing applications of key results derived.

摘要引入了一种新的广义多元Mittag-Leffler函数，它是对多元Mittag-Leffler函数的推广，利用Banach不动点定理和Babenko技术，给出了一类全新的分数阶非线性偏积分微分方程边值问题解的唯一性的充分条件。这有许多潜在的应用，因为唯一性在许多科学领域都是一个重要的话题，所使用的方法为研究其他类型的方程和相应的初值或边值问题开辟了明确的方向。此外，我们使用Python，这是一种高级编程语言，有效地处理多索引的求和，以计算广义Mittag-Leffler函数的近似值(似乎不可能通过任何现有的Mittag-Leffler函数的积分表示来做到这一点)，并提供了一个示例，展示了推导出的关键结果的应用。

引用次数: 0

New generalized Halanay inequalities and relative applications to neural networks with variable delays 新的广义Halanay不等式及其在变延迟神经网络中的相关应用

4区数学 Q1 MATHEMATICS

Boundary Value Problems

Pub Date : 2023-09-28 DOI: 10.1186/s13661-023-01773-8

Chunsheng Wang, Han Chen, Runpeng Lin, Ying Sheng, Feng Jiao

Abstract The asymptotic behavior of solutions for a new class of generalized Halanay inequalities is studied via the fixed point method. This research provides a new approach to the study of the stability of Halanay inequality. To make the application of fixed point method in stability research more flexible and feasible, we introduce corresponding functions to construct an operator according to different characteristics of coefficients. The results obtained in this paper are applied to the stability study of a neural network system, which has high value in application. Moreover, three examples and simulations are given to illustrate the results. The conclusions in this paper greatly improve and generalize the relative results in the current literature.

摘要利用不动点法研究了一类新的广义Halanay不等式解的渐近性。本研究为研究Halanay不等式的稳定性提供了一种新的途径。为了使不动点法在稳定性研究中的应用更加灵活和可行，我们根据系数的不同特征引入相应的函数来构造算子。所得结果可应用于神经网络系统的稳定性研究，具有较高的应用价值。最后给出了三个算例和仿真结果。本文的结论大大改进和概括了现有文献的相关结果。

引用次数: 0

The well posedness of solutions for the 2D magnetomicropolar boundary layer equations in an analytic framework 二维磁层微极边界层方程解的适定性

4区数学 Q1 MATHEMATICS

Boundary Value Problems

Pub Date : 2023-09-25 DOI: 10.1186/s13661-023-01782-7

Xiaolei Dong

Abstract In this paper, we prove the existence and uniqueness of solutions to the 2D magnetomicropolar boundary layer equations on the half-plane by using the classical bootstrap argument in an analytic framework.

摘要本文利用解析框架下的经典自举论证，证明了二维磁微极边界层方程在半平面上解的存在唯一性。

引用次数: 0

Initial boundary value problem for a viscoelastic wave equation with Balakrishnan–Taylor damping and a delay term: decay estimates and blow-up result 具有Balakrishnan-Taylor阻尼和时滞项的粘弹性波动方程的初边值问题:衰减估计和爆破结果

4区数学 Q1 MATHEMATICS

Boundary Value Problems

Pub Date : 2023-09-19 DOI: 10.1186/s13661-023-01781-8

Billel Gheraibia, Nouri Boumaza

Abstract In this paper, we study the initial boundary value problem for the following viscoelastic wave equation with Balakrishnan–Taylor damping and a delay term where the relaxation function satisfies $g'(t)leq -xi (t)g^{r}(t)$ g ′ ( t ) ≤ − ξ ( t ) g r ( t ) , $tgeq 0$ t ≥ 0 , $1leq r< frac{3}{2}$ 1 ≤ r < 3 2 . The main goal of this work is to study the global existence, general decay, and blow-up result. The global existence has been obtained by potential-well theory, the decay of solutions of energy has been established by introducing suitable energy and Lyapunov functionals, and a blow-up result has been obtained with negative initial energy.

摘要本文研究了具有Balakrishnan-Taylor阻尼且松弛函数满足$g'(t)leq -xi (t)g^{r}(t)$ g ' (t)≤- ξ (t) g r (t)， $tgeq 0$ t≥0,$1leq r< frac{3}{2}$ 1≤r ＆lt的时滞项的粘弹性波动方程的初边值问题;3 .答案:b。本工作的主要目的是研究整体存在性、一般衰变和爆炸结果。通过引入合适的能量和李雅普诺夫泛函，建立了能量解的衰减性，并得到了初始能量为负的爆破结果。

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引用次数: 0

The fundamental solution and blow-up problem of an anisotropic parabolic equation 一类各向异性抛物方程的基本解和爆破问题

4区数学 Q1 MATHEMATICS

Boundary Value Problems

Pub Date : 2023-09-15 DOI: 10.1186/s13661-023-01780-9

Huashui Zhan

Abstract This paper is devoted to the study of anisotropic parabolic equation related to the $p_{i}$ p i -Laplacian with a source term $f(u)$ f ( u ) . If $f(u)=0$ f ( u ) = 0 , then the fundamental solution of the equation is constructed. If there are some restrictions on the growth order of u in the source term, the initial energy $E(0)$ E ( 0 ) is positive and has a super boundedness, which depends on the Sobolev imbedding index, then the local solution may blow up in finite time.

摘要本文研究了源项为$f(u)$ f(u)的与$p_{i}$ pi -拉普拉斯方程有关的各向异性抛物方程。如果$f(u)=0，则构造方程的基本解。如果源项中u的生长阶数存在一定的限制，初始能量$E(0)$ E(0)为正且具有超有界性，且该超有界性依赖于Sobolev嵌入指标，则局部解可能在有限时间内爆炸。

引用次数: 0

Applying periodic and anti-periodic boundary conditions in existence results of fractional differential equations via nonlinear contractive mappings 利用非线性压缩映射在分数阶微分方程存在性结果中应用周期和反周期边界条件

4区数学 Q1 MATHEMATICS

Boundary Value Problems

Pub Date : 2023-09-13 DOI: 10.1186/s13661-023-01778-3

Sumati Kumari Panda, Velusamy Vijayakumar, Kottakkaran Sooppy Nisar

Abstract We introduce a notion of nonlinear cyclic orbital $(xi -mathscr{F})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>(</mml:mo> <mml:mi>ξ</mml:mi> <mml:mo>−</mml:mo> <mml:mi>F</mml:mi> <mml:mo>)</mml:mo> </mml:math> -contraction and prove related results. With these results, we address the existence and uniqueness results with periodic/anti-periodic boundary conditions for: 1. The nonlinear multi-order fractional differential equation $$ mathcal{L}(mathcal{D})theta (varsigma )=sigma bigl(varsigma , theta ( varsigma ) bigr), quad varsigma in mathscr{J}=[0,mathscr{A}], mathscr{A}>0, $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> <mml:mo>(</mml:mo> <mml:mi>D</mml:mi> <mml:mo>)</mml:mo> <mml:mi>θ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>ς</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>σ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ς</mml:mi> <mml:mo>,</mml:mo> <mml:mi>θ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>ς</mml:mi> <mml:mo>)</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mspace /> <mml:mi>ς</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>J</mml:mi> <mml:mo>=</mml:mo> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>A</mml:mi> <mml:mo>]</mml:mo> <mml:mo>,</mml:mo> <mml:mi>A</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> </mml:math> where $$begin{aligned} &mathcal{L}(mathcal{D})=gamma _{w} ,{}^{c} mathcal{D}^{delta _{w}}+ gamma _{w-1} ,{}^{c} mathcal{D}^{delta _{w-1}}+cdots+gamma _{1} ,{}^{c} mathcal{D}^{delta _{1}}+gamma _{0} ,{}^{c} mathcal{D}^{delta _{0}}, &gamma _{flat}in mathbb{R}quad (flat =0,1,2,3,ldots,w), qquad gamma _{w} neq 0, &0leq delta _{0}< delta _{1}< delta _{2}< cdots< delta _{w-1}< delta _{w}< 1; end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtable> <mml:mtr> <mml:mtd /> <mml:mtd> <mml:mi>L</mml:mi> <mml:mo>(</mml:mo> <mml:mi>D</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>γ</mml:mi> <mml:mi>w</mml:mi> </mml:msub> <mml:mspace /> <mml:msup> <mml:mrow /> <mml:mi>c</mml:mi> </mml:msup> <mml:msup> <mml:mi>D</mml:mi> <mml:msub> <mml:mi>δ</mml:mi> <mml:mi>w</mml:mi> </mml:msub> </mml:msup> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>γ</mml:mi> <mml:mrow> <mml:mi>w</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mspace /> <mml:msup> <mml:mrow /> <mml:mi>c</mml:mi> </mml:msup> <mml:msup> <mml:mi>D</mml:mi> <mml:msub> <mml:mi>δ</mml:mi> <mml:mrow> <mml:mi>w</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:msup> <mml:mo>+</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>γ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mspace /> <mml:msup> <mml:mrow /> <mml:mi>c</mml:mi> </mml:msup> <mml:msup> <mml:mi>D</mml:mi> <mml:msub> <mml:mi>δ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:msup> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>γ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mspace /> <mml:msup> <mml:mrow /> <mml:mi>c</mm

摘要引入了非线性循环轨道$(xi -mathscr{F})$ (ξ−F) -收缩的概念，并证明了相关结果。利用这些结果，我们讨论了周期/反周期边界条件下的存在唯一性结果:非线性多阶分数阶微分方程$$ mathcal{L}(mathcal{D})theta (varsigma )=sigma bigl(varsigma , theta ( varsigma ) bigr), quad varsigma in mathscr{J}=[0,mathscr{A}], mathscr{A}>0, $$ L (D) θ (ς) = σ (ς， θ (ς))， ς∈J = [0, A]， A ＆gt;0，其中$$begin{aligned} &mathcal{L}(mathcal{D})=gamma _{w} ,{}^{c} mathcal{D}^{delta _{w}}+ gamma _{w-1} ,{}^{c} mathcal{D}^{delta _{w-1}}+cdots+gamma _{1} ,{}^{c} mathcal{D}^{delta _{1}}+gamma _{0} ,{}^{c} mathcal{D}^{delta _{0}}, &gamma _{flat}in mathbb{R}quad (flat =0,1,2,3,ldots,w), qquad gamma _{w} neq 0, &0leq delta _{0}< delta _{1}< delta _{2}< cdots< delta _{w-1}< delta _{w}< 1; end{aligned}$$ L (D) = γ w c D δ w + γ w−1 c D δ w−1 +⋯⋯+ γ 1 c D δ 1 + γ 0 c D δ 0， γ∈R(≈0,1,2,3，…，w)， γ w≠0,0≤δ 0 ＆lt;δ 1 ＆lt;δ 2 ＆lt;⋯＆lt;δ w−1 ＆lt;δ w ＆lt;1;2. 非线性多项分数阶时滞微分方程$$begin{aligned} &mathcal{L}(mathcal{D})theta (varsigma ) =sigma bigl(varsigma , theta ( varsigma ),theta (varsigma -tau ) bigr), quad varsigma in mathscr{J}=[0, mathscr{A}], mathscr{A}>0; &theta (varsigma ) =bar{sigma}(varsigma ),quad varsigma in [-tau ,0], end{aligned}$$ L (D) θ (ς) = σ (ς， θ (ς)， θ (ς−τ))， ς∈J = [0, A]， A ＆gt;0;θ (ς) = σ¯(ς)， ς∈[- τ， 0]，其中$$begin{aligned} &mathcal{L}(mathcal{D})=gamma _{w} ,{}^{c} mathcal{D}^{delta _{w}}+ gamma _{w-1} ,{}^{c} mathcal{D}^{delta _{w-1}}+cdots+gamma _{1} ,{}^{c} mathcal{D}^{delta _{1}}+gamma _{0} ,{}^{c} mathcal{D}^{delta _{0}}, &gamma _{flat}in mathbb{R}quad (flat =0,1,2,3,ldots,w), qquad gamma _{w} neq 0, &0leq delta _{0}< delta _{1}< delta _{2}< cdots< delta _{w-1}< delta _{w}< 1; end{aligned}$$ L (D) = γ w c D δ w + γ w - 1 c D δ w - 1 +⋯⋯+ γ 1 c D δ 1 + γ 0 c D δ 0， γ∈R(≈0,1,2,3，…，w)， γ w≠0,0≤δ 0 ＆lt;δ 1 ＆lt;δ 2 ＆lt;⋯＆lt;δ w−1 ＆lt;δ w ＆lt;1;此外，${}^{c}mathcal{D}^{delta}$ D δ c主要被称为δ阶卡普托分数导数。

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