Recursos de colección
Project Euclid (Hosted at Cornell University Library) (198.174 recursos)
Taiwanese Journal of Mathematics
Taiwanese Journal of Mathematics
Guan, Hong-Bo; Shi, Dong-Yang
In this paper, a nonconforming finite element method (NFEM) is proposed for the
constrained optimal control problems (OCPs) governed by parabolic equations. The time
discretization is based on the finite difference methods. The state and co-state
variables are approximated by the nonconforming $EQ_1^{\operatorname{rot}}$ elements,
and the control variable is approximated by the piecewise constant element,
respectively. Some superclose properties are obtained for the above three variables.
Moreover, for the state and co-state, the convergence and superconvergence results
are achieved in $L^2$-norm and the broken energy norm, respectively.
Guan, Hong-Bo; Shi, Dong-Yang
In this paper, a nonconforming finite element method (NFEM) is proposed for the constrained optimal control problems (OCPs) governed by parabolic equations. The time discretization is based on the finite difference methods. The state and co-state variables are approximated by the nonconforming $EQ_1^{\operatorname{rot}}$ elements, and the control variable is approximated by the piecewise constant element, respectively. Some superclose properties are obtained for the above three variables. Moreover, for the state and co-state, the convergence and superconvergence results are achieved in $L^2$-norm and the broken energy norm, respectively.
Saeed, Umer
In this paper, we propose a method for solving some well-known classes of fractional
Lane-Emden type equations which are nonlinear ordinary differential equations on the
semi-infinite domain. The method is proposed by utilizing Haar wavelets in
conjunction with Adomian's decomposition method. The operational matrices for the
Haar wavelets are derived and constructed. Procedure of implementation and
convergence analysis of the method are presented. The method is tested on the
fractional standard Lane-Emden equation and the fractional isothermal gas spheres
equation. We compare the results produce by present method with some well-known
results to show the accuracy and applicability of the method.
Saeed, Umer
In this paper, we propose a method for solving some well-known classes of fractional Lane-Emden type equations which are nonlinear ordinary differential equations on the semi-infinite domain. The method is proposed by utilizing Haar wavelets in conjunction with Adomian's decomposition method. The operational matrices for the Haar wavelets are derived and constructed. Procedure of implementation and convergence analysis of the method are presented. The method is tested on the fractional standard Lane-Emden equation and the fractional isothermal gas spheres equation. We compare the results produce by present method with some well-known results to show the accuracy and applicability of the...
Pastor, Karel
The aim of this paper is to show the equivalence of two classes of nonsmooth
functions. We also compare optimality conditions which have been stated for these
classes.
Pastor, Karel
The aim of this paper is to show the equivalence of two classes of nonsmooth functions. We also compare optimality conditions which have been stated for these classes.
Chen, Jie; Lin, Haibo
Let $(\mathcal{X},d,\mu)$ be a metric measure space satisfying the so-called upper
doubling condition and the geometrically doubling condition. Let $T_*$ be the maximal
Calderón-Zygmund operator and $\vec{b} := (b_1,\ldots,b_m)$ be a finite family of
$\widetilde{\operatorname{RBMO}}(\mu)$ functions. In this paper, the authors
establish the boundedness of the maximal multilinear commutator $T_{*,\vec{b}}$
generated by $T_*$ and $\vec{b}$ on the Lebesgue space $L^p(\mu)$ with $p \in (1,
\infty)$. For $\vec{b} = (b_1,\ldots,b_m)$ being a finite family of Orlicz type
functions, the weak type endpoint estimate for the maximal multilinear commutator
$T_{*,\vec{b}}$ generated by $T_*$ and $\vec{b}$ is also presented. The main tool to
deal with these estimates is the smoothing technique.
Chen, Jie; Lin, Haibo
Let $(\mathcal{X},d,\mu)$ be a metric measure space satisfying the so-called upper doubling condition and the geometrically doubling condition. Let $T_*$ be the maximal Calderón-Zygmund operator and $\vec{b} := (b_1,\ldots,b_m)$ be a finite family of $\widetilde{\operatorname{RBMO}}(\mu)$ functions. In this paper, the authors establish the boundedness of the maximal multilinear commutator $T_{*,\vec{b}}$ generated by $T_*$ and $\vec{b}$ on the Lebesgue space $L^p(\mu)$ with $p \in (1, \infty)$. For $\vec{b} = (b_1,\ldots,b_m)$ being a finite family of Orlicz type functions, the weak type endpoint estimate for the maximal multilinear commutator $T_{*,\vec{b}}$ generated by $T_*$ and $\vec{b}$ is also presented. The main tool to...
Zhou, Shidi; Geng, Jiansheng
In this paper, we prove a Nekhoroshev type theorem for high dimensional NLS
(nonlinear Schrödinger equations):\[ \mathrm{i} \partial_{t} u - \Delta u + V * u +
\partial_{\overline{u}} g(x,u,\overline{u}) = 0, \quad x \in \mathbb{T}^d, \; t
\in \mathbb{R} \] where real-valued function $V$ is sufficiently smooth and $g$ is an
analytic function. We prove that, for any given $M \in \mathbb{N}$, there exists an
$\varepsilon_0 \gt 0$, such that for any solution $u = u(t,x)$ with initial data $u_0
= u_0(x)$ whose Sobolev norm $\|u_{0}\|_{s} = \varepsilon \lt \varepsilon_0$, during
the time $|t| \leq \varepsilon^{-M}$, its Sobolev norm $\|u(t)\|_s$ remains bounded
by $C_s \varepsilon$.
Zhou, Shidi; Geng, Jiansheng
In this paper, we prove a Nekhoroshev type theorem for high dimensional NLS (nonlinear Schrödinger equations):\[ \mathrm{i} \partial_{t} u - \Delta u + V * u + \partial_{\overline{u}} g(x,u,\overline{u}) = 0, \quad x \in \mathbb{T}^d, \; t \in \mathbb{R} \] where real-valued function $V$ is sufficiently smooth and $g$ is an analytic function. We prove that, for any given $M \in \mathbb{N}$, there exists an $\varepsilon_0 \gt 0$, such that for any solution $u = u(t,x)$ with initial data $u_0 = u_0(x)$ whose Sobolev norm $\|u_{0}\|_{s} = \varepsilon \lt \varepsilon_0$, during the time $|t| \leq \varepsilon^{-M}$, its Sobolev norm...
Shao, Xiang; Yin, Zheng
In this paper, we obtain a conditional variational principle for the topological
entropy of level sets of Birkhoff averages for maps with the gluing orbit property.
Our result can be easily extended to flows.
Shao, Xiang; Yin, Zheng
In this paper, we obtain a conditional variational principle for the topological entropy of level sets of Birkhoff averages for maps with the gluing orbit property. Our result can be easily extended to flows.
Chen, Bochao; Gao, Yixian; Li, Yong
This paper is concerned with the existence of families of time-periodic solutions for
the nonlinear wave equations with Hamiltonian perturbations on one-dimensional tori.
We obtain the result by a new method: a para-differential conjugation together with a
classical iteration scheme, which have been used for the nonlinear Schrödinger
equation in [22]. Avoiding the use of KAM theorem and Nash-Moser iteration method,
though a para-differential conjugation, an equivalent form of the investigated
nonlinear wave equations can be obtained, while the frequencies are fixed in a
Cantor-like set whose complement has small measure. Applying the non-resonant
conditions on each finite-dimensional subspaces, solutions can be constructed to the
block diagonal equation on...
Chen, Bochao; Gao, Yixian; Li, Yong
This paper is concerned with the existence of families of time-periodic solutions for the nonlinear wave equations with Hamiltonian perturbations on one-dimensional tori. We obtain the result by a new method: a para-differential conjugation together with a classical iteration scheme, which have been used for the nonlinear Schrödinger equation in [22]. Avoiding the use of KAM theorem and Nash-Moser iteration method, though a para-differential conjugation, an equivalent form of the investigated nonlinear wave equations can be obtained, while the frequencies are fixed in a Cantor-like set whose complement has small measure. Applying the non-resonant conditions on each finite-dimensional subspaces, solutions...
Tyagi, Jagmohan; Verma, Ram Baran
In this article, we study the existence and multiplicity of nontrivial solutions to
the problem \[\begin{cases} -\epsilon^{2} F(x,D^{2}u) = f(x,u) + \psi(Du)
&\textrm{in $\Omega$}, \\ u = 0 &\textrm{on $\partial
\Omega$},\end{cases}\]where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{n}$,
$n \gt 2$. We show that the problem possesses nontrivial solutions for small value of
$\epsilon$ provided $f$ and $\psi$ are continuous and $f$ has a positive zero. We
employ degree theory arguments and Liouville type theorem for the multiplicity of the
solutions.
Tyagi, Jagmohan; Verma, Ram Baran
In this article, we study the existence and multiplicity of nontrivial solutions to the problem \[\begin{cases} -\epsilon^{2} F(x,D^{2}u) = f(x,u) + \psi(Du) &\textrm{in $\Omega$}, \\ u = 0 &\textrm{on $\partial \Omega$},\end{cases}\]where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{n}$, $n \gt 2$. We show that the problem possesses nontrivial solutions for small value of $\epsilon$ provided $f$ and $\psi$ are continuous and $f$ has a positive zero. We employ degree theory arguments and Liouville type theorem for the multiplicity of the solutions.
Luo, Huxiao; Tang, Xianhua; Li, Shengjun
We use two variant fountain theorems to prove the existence of infinitely many weak
solutions for the following fractional $p$-Laplace equation\[ (-\Delta)^\alpha_p u +
V(x) |u|^{p-2}u = f(x,u), \quad x \in \mathbb{R}^N,\]where $N \geq 2$, $p \geq 2$,
$\alpha \in (0,1)$, $(-\Delta)^\alpha_p$ is the fractional $p$-Laplacian and $f$ is
either asymptotically linear or subcritical $p$-superlinear growth. Under appropriate
assumptions on $V$ and $f$, we prove the existence of infinitely many nontrivial high
or small energy solutions. Our results generalize and extend some existing
results.
Luo, Huxiao; Tang, Xianhua; Li, Shengjun
We use two variant fountain theorems to prove the existence of infinitely many weak solutions for the following fractional $p$-Laplace equation\[ (-\Delta)^\alpha_p u + V(x) |u|^{p-2}u = f(x,u), \quad x \in \mathbb{R}^N,\]where $N \geq 2$, $p \geq 2$, $\alpha \in (0,1)$, $(-\Delta)^\alpha_p$ is the fractional $p$-Laplacian and $f$ is either asymptotically linear or subcritical $p$-superlinear growth. Under appropriate assumptions on $V$ and $f$, we prove the existence of infinitely many nontrivial high or small energy solutions. Our results generalize and extend some existing results.
Pei, Jingwen; Yu, Zhixian; Zhou, Huiling
In this paper, we mainly investigate exponential stability of traveling wavefronts
for delayed $2D$ lattice differential equation with nonlocal interaction. For all
non-critical traveling wavefronts with the wave speed $c \gt c_*(\theta)$, where
$c_*(\theta) \gt 0$ is the critical wave speed and $\theta$ is the direction of
propagation, we prove that these traveling waves are asymptotically stable, when the
initial perturbation around the traveling waves decay exponentially at far fields,
but can be allowed arbitrarily large in other locations. Our approach adopted in this
paper is the weighted energy method and the squeezing technique with the help of
Gronwall's inequality. Furthermore, from stability result, we prove the uniqueness
(up...
Pei, Jingwen; Yu, Zhixian; Zhou, Huiling
In this paper, we mainly investigate exponential stability of traveling wavefronts for delayed $2D$ lattice differential equation with nonlocal interaction. For all non-critical traveling wavefronts with the wave speed $c \gt c_*(\theta)$, where $c_*(\theta) \gt 0$ is the critical wave speed and $\theta$ is the direction of propagation, we prove that these traveling waves are asymptotically stable, when the initial perturbation around the traveling waves decay exponentially at far fields, but can be allowed arbitrarily large in other locations. Our approach adopted in this paper is the weighted energy method and the squeezing technique with the help of Gronwall's inequality....