共查询到20条相似文献,搜索用时 390 毫秒
1.
本文主要研究了一类多项Caputo分数阶随机微分方程的Euler-Maruyama (EM)方法,并证明了其强收敛性.具体地,我们首先构造了求解多项Caputo分数阶随机微分方程初值问题的EM方法,然后证明分数阶导数的指标满足$\frac{1}{2}<\alpha_{1}<\alpha_{2}<\cdots<\alpha_{m}<1$时,该方法是$\alpha_{m}-\alpha_{m-1}$阶强收敛的.文末的数值试验验证了理论结果的正确性. 相似文献
2.
Chun-fengRen Yi-chenMa 《计算数学(英文版)》2004,22(1):101-112
A two-grid method for the steady penalized incompressible Navier-Stokes equations is presented. Convergence results are proved. If h = O(H^3-s) and ε = O(H^5-2s) (s = 0(n=2);s=1/2(n=3)are chosen, the convergence order of this two-grid method is the same as that of the usual finite element method. Numerical results show that this method is efficient and can save a lot of computation time. 相似文献
3.
Sufficient conditions are derived for the existence of a globally attractive almost periodic solution of a competition system modelled by the nonautonomous Lotka–Volterra delay differential equations $$\begin{gathered} \frac{{{\text{d}}N_1 (t)}}{{{\text{d}}t}} = N_1 (t)\left[ {r_1 (t) - a_{11} (t)N_1 (t - \tau (t)) - a_{12} (t)N_2 (t - \tau (t))} \right], \hfill \\ \frac{{{\text{d}}N_2 (t)}}{{{\text{d}}t}} = N_2 (t)\left[ {r_2 (t) - a_{21} (t)N_1 (t - \tau (t)) - a_{22} (t)N_2 (t - \tau (t))} \right], \hfill \\ \end{gathered} $$ in which $ \tau ,r_i ,a_{ij} (i,j = 1,2) $ are continuous positive almost periodic functions; conditions are also obtained for all positive solutions of the above system to 'oscillate' about the unique almost periodic solution. Some ecobiological consequences of the convergence to almost periodicity and delay induced oscillations are briefly discussed. 相似文献
4.
This paper presents a strong predictor-corrector method for the numerical solution of stochastic delay differential equations (SDDEs) of Itô-type. The method is proved to be mean-square convergent of order min{$1/2, \hat{p}$} under the Lipschitz condition and the linear growth condition, where $\hat{p}$ is the exponent of Hölder condition of the initial function. Stability criteria for this type of method are derived. It is shown that for certain choices of the flexible parameter $p$ the derived method can have a better stability property than more commonly used numerical methods. That is, for some $p$, the asymptotic MS-stability bound of the method will be much larger than that of the Euler-Maruyama method. Numerical results are reported confirming convergence properties and comparing stability properties of methods with different parameters $p$. Finally, the vectorised simulation is discussed and it is shown that this implementation is much more efficient. 相似文献
5.
针对一类变延迟微分方程,应用全隐式方法—平衡方法,研究了其收敛性和稳定性.结果表明平衡方法以$\frac{1}{2}\gamma,\gamma\in(0,1]$阶收敛到精确解;并且强平衡方法和弱平衡方法都能保持解析解的均方稳定性;进一步数值实验验证了算法理论分析的正确性,并且表明全隐式的平衡方法比显式方法—Euler方法具有更好的稳定性. 相似文献
6.
Gao Ruxi 《数学年刊B辑(英文版)》1983,4(3):293-298
This paper deals with the following mixed problem for Quasilinear hyperbolic equationsThe M order uniformly valid asymptotic solutions are obtained and there errors areestimated. 相似文献
7.
We study limit behavior for sums of the form $\frac{1}{|\Lambda_{L|}}\sum_{x\in \Lambda_{L}}u(t,x),$ where the field $\Lambda_L=\left\{x\in {\bf{Z^d}}:|x|\le L\right\}$ is composed of solutions of the parabolic Anderson equation $$u(t,x) = 1 + \kappa \mathop{\int}_{0}^{t} \Delta u(s,x){\rm d}s + \mathop{\int}_{0}^{t}u(s,x)\partial B_{x}(s). $$ The index set is a box in Z d , namely $\Lambda_{L} = \left\{x\in {\bf Z}^{\bf d} : |x| \leq L\right\}$ and L = L(t) is a nondecreasing function $L : [0,\infty)\rightarrow {\bf R}^{+}. $ We identify two critical parameters $\eta(1) < \eta(2)$ such that for $\gamma > \eta(1)$ and L(t) = eγ t , the sums $\frac{1}{|\Lambda_L|}\sum_{x\in \Lambda_L}u(t,x)$ satisfy a law of large numbers, or put another way, they exhibit annealed behavior. For $\gamma > \eta(2)$ and L(t) = eγ t , one has $\sum_{x\in \Lambda_L}u(t,x)$ when properly normalized and centered satisfies a central limit theorem. For subexponential scales, that is when $\lim_{t \rightarrow \infty} \frac{1}{t}\ln L(t) = 0,$ quenched asymptotics occur. That means $\lim_{t\rightarrow \infty}\frac{1}{t}\ln\left (\frac{1}{|\Lambda_L|}\sum_{x\in \Lambda_L}u(t,x)\right) = \gamma(\kappa),$ where $\gamma(\kappa)$ is the almost sure Lyapunov exponent, i.e. $\lim_{t\rightarrow \infty}\frac{1}{t}\ln u(t,x)= \gamma(\kappa).$ We also examine the behavior of $\frac{1}{|\Lambda_L|}\sum_{x\in \Lambda_L}u(t,x)$ for L = e γ t with γ in the transition range $(0,\eta(1))$ 相似文献
8.
K. Oskolkov 《Journal of Fourier Analysis and Applications》1998,4(3):341-356
Let $h(t,x): = p.v. \sum\limits_{n \in Z\backslash \left| 0 \right|} {\frac{{e^{\pi i(tn^2 + 2xn)} }}{{2\pi in}}} = \mathop {\lim }\limits_{N \to \infty } \sum\limits_{0< \left| n \right| \leqslant N} {\frac{{e^{\pi i(tn^2 + 2xn)} }}{{2\pi in}}} $ ( $(i = \sqrt { - 1;} t,x$ -real variables). It is proved that in the rectangle $D: = \left\{ {(t,x):0< t< 1,\left| x \right| \leqslant \frac{1}{2}} \right\}$ , the function h satisfies the followingfunctional inequality: $\left| {h(t,x)} \right| \leqslant \sqrt t \left| {h\left( {\frac{1}{t},\frac{x}{t}} \right)} \right| + c,$ where c is an absolute positive constant. Iterations of this relation provide another, more elementary, proof of the known global boundedness result $\left\| {h; L^\infty (E^2 )} \right\| : = ess sup \left| {h(t,x)} \right|< \infty .$ The above functional inequality is derived from a general duality relation, of theta-function type, for solutions of the Cauchy initial value problem for Schrödinger equation of a free particle. Variation and complexity of solutions of Schrödinger equation are discussed. 相似文献
9.
Mirko Lepović 《Journal of Applied Mathematics and Computing》2004,14(1-2):39-49
LetG be a simple graph and let $\bar G$ denotes its complement. We say thatG is integral if its spectrum consists entirely of integers. If $\overline {\alpha K_a \cup \beta K_b } $ is integral we show that it belongs to the class of integral graphs $$\overline {[\frac{{kt}}{\tau }x_o + \frac{{mt}}{\tau }z]K(t + \ell n)k + \ell m \cup [\frac{{kt}}{\tau }y_o + \frac{{(t + \ell n)k + \ell m}}{\tau }z]nK\ell m,} $$ where (i) t, k, l, m, n ∈ ? such that (m, n) =1, (n, t) =1 and (l, t)=1; (ii) τ=((t+ln)k+lm, mt) such that τ| kt; (iii) (x0, y0) is aparticular solution of the linear Diophantine equation ((t+ln)k+lm)x-(mt)y=τ and (iv) z≥z0 where z0 is the least integer such that $(\frac{{kt}}{\tau }x_0 + \frac{{mt}}{\tau }z_0 ) \geqslant 1$ and $(\frac{{kt}}{\tau }y_0 + \frac{{(t + \ell n)k + \ell m}}{\tau }z_0 ) \geqslant 1$ . 相似文献
10.
More work is done to study the explicit, weak and strong implicit difference solution for the first boundary problem of quasilinear parabolic system: $$\begin{gathered} u_t = ( - 1)^{M + 1} A(x,t,u, \cdots ,u_x M - 1)u_x 2M + f(x,t,u, \cdots u_x 2M - 1), \hfill \\ (x,t) \in Q_T = \left| {0< x< l,0< t \leqslant T} \right|, \hfill \\ u_x ^k (0,t) = u_x ^k (l,t) = 0 (k = 0,1, \cdots ,M - 1),0< t \leqslant T, \hfill \\ u(x,0) = \varphi (x),0 \leqslant x \leqslant l, \hfill \\ \end{gathered} $$ whereu, ?, andf arem-dimensional vector valued functions, A is anm×m positively definite matrix, and $u_t = \frac{{\partial u}}{{\partial t}},u_x ^k = \frac{{\partial ^k u}}{{\partial x^k }}$ . For this problem, the convergence of iteration for the general difference schemes is proved. 相似文献
11.
P(t,n)和C(t,n)分别表示在阶为n的路和圈中添加t条边后得到的图的最小直径;f(t,k)表示从直径为k的图中删去t条边后得到的连通图的最大直径.这篇文章证明了t≥4且n≥5时,P(t,n)≤(n-8)/(t 1) 3;若t为奇数,则C(t,n)≤(n-8)/(t 1) 3;若t为偶数,则C(t,n)≤(n-7)/(t 2) 3.特别地,「(n-1)/5」≤P(4,n)≤「(n 3)/5」,「n/4」-1≤C(3,n)≤「n/4」.最后,证明了:若k≥3且为奇数,则f(t,k)≥(t 1)k-2t 4.这些改进了某些已知结果. 相似文献
12.
Bang-He Li 《数学研究》2016,49(4):319-324
Let $ζ(s)$ be the Riemann zeta function, $s=\sigma+it$. For $0 < \sigma < 1$, we expand $ζ(s)$ as the following series convergent in the space of slowly increasing distributions
with variable $t$ : $$ζ(\sigma+it)=\sum\limits^∞_{n=0}a_n(\sigma)ψ_n(t),$$ where $$ψ_n(t)=(2^nn!\sqrt{\pi})^{-1 ⁄ 2}e^{\frac{-t^2}{2}}H_n(t),$$ $H_n(t)$ is the Hermite polynomial, and $$a_n(σ)=2\pi(-1)^{n+1}ψ_n(i(1-σ))+(-i)^n\sqrt{2\pi}\sum\limits^∞_{m=1}\frac{1}{m^σ}ψ_n(1nm).$$ This paper is concerned with the convergence of the above series for $σ > 0.$ In the deduction,
it is crucial to regard the zeta function as Fourier transfomations of Schwartz'
distributions. 相似文献
13.
M. M. Kabardov 《Vestnik St. Petersburg University: Mathematics》2009,42(2):90-94
In this paper, we consider a method for inverting the Laplace transform F(s) = \(\int\limits_0^\infty {e^{ - st} f(t)dt} \), which consists in representing the original function by the Laguerre series
$f(t) = \sum\limits_{k = 0}^\infty {a_k L_k (bt).} $
(1)
First, we perform a conformal mapping of the plane (s), which depends on parameter ξ. The value of the parameter is determined by the location of the singular points of the given representation. Under this mapping, series (1) takes the form
$f(t) = \frac{{b - \xi }}{b}\exp (\xi t)\sum\limits_{k = 0}^\infty {c_k L_k ((b - \xi )t).} $
It is demonstrated that such inverting scheme is equivalent to applying the Picone-Tricomi method with further acceleration of the rate of convergence of series (1) using the Euler-Knopp nonlinear procedure
$\sum\limits_{k = 0}^\infty {a_k z^k } = \sum\limits_{k = 0}^\infty {A_k (p)\frac{{z^k }}{{(1 - pz)^{k + 1} }},} A_k (p) = \sum\limits_{j = 0}^k {\left( \begin{gathered} k \hfill \\ j \hfill \\ \end{gathered} \right)( - p)^{k - j} a_j } .$
Under this approach, the original function is represented by the series where parameters ξ and p are related by the formula p = x/(ξ ? b). Unlike many other methods for summation of series, in the scheme suggested, there is no need to investigate the regularity conditions.
相似文献
$f(t) = \exp \left( {\frac{{bpt}}{{p - 1}}} \right)\sum\limits_{k = 0}^\infty {\frac{{A_k (p)}}{{(1 - p)^{k + 1} }}L_k } \left( {\frac{{bpt}}{{1 - p}}} \right),$
14.
Wei Zhang 《计算数学(英文版)》2020,38(6):903-932
The key aim of this paper is to show the strong convergence of the truncated Euler-Maruyama method for neutral stochastic differential delay equations (NSDDEs) with
Markovian switching (MS) without the linear growth condition. We present the truncated Euler-Maruyama method of NSDDEs-MS and consider its moment boundedness under
the local Lipschitz condition plus Khasminskii-type condition. We also study its strong
convergence rates at time $T$ and over a finite interval $[0, T]$. Some numerical examples are
given to illustrate the theoretical results. 相似文献
15.
16.
Yan Ziqian 《数学年刊B辑(英文版)》1982,3(1):67-78
In this paper initial value problems and nonlinear mixed boundary value problems for the quasilinear parabolic systems below
$\[\frac{{\partial {u_k}}}{{\partial t}} - \sum\limits_{i,j = 1}^n {a_{ij}^{(k)}} (x,t)\frac{{{\partial ^2}{u_k}}}{{\partial {x_i}\partial {x_j}}} = {f_k}(x,t,u,{u_x}),k = 1, \cdots ,N\]$
are discussed.The boundary value conditions are
$\[{u_k}{|_{\partial \Omega }} = {g_k}(x,t),k = 1, \cdots ,s,\]$
$\[\sum\limits_{i = 1}^n {b_i^{(k)}} (x,t)\frac{{\partial {u_k}}}{{\partial {x_i}}}{|_{\partial \Omega }} = {h_k}(x,t,u),k = s + 1, \cdots N.\]$
Under some "basically natural" assumptions it is shown by means of the Schauder type estimates of the linear parabolic equations and the embedding inequalities in Nikol'skii spaces,these problems have solutions in the spaces $\[{H^{2 + \alpha ,1 + \frac{\alpha }{2}}}(0 < \alpha < 1)\]$.For the boundary value problem with $\[b_i^{(k)}(x,t) = \sum\limits_{j = 1}^n {a_{ij}^{(k)}} (x,t)\cos (n,{x_j})\]$ uniqueness theorem is proved. 相似文献
17.
We consider the effect of perturbations of A on the solution to the following semi-linear parabolic stochastic partial differential equation: $$\left\{\begin{array}{ll}{\rm d}U(t) & = AU(t)\,{\rm d}t + F(t,U(t))\,{\rm d}t + G(t,U(t))\,{\rm d}W_H(t), \quad t > 0;\\U(0)& = x_0. \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad({\rm SDE})\end{array} \right.$$ Here, A is the generator of an analytic C 0-semigroup on a UMD Banach space X, H is a Hilbert space, W H is an H-cylindrical Brownian motion, ${G:[0,T]\times X\rightarrow \mathcal{L}(H, X_{\theta_G}^{A})}$ , and ${F : [0, T]\times X \rightarrow X_{\theta_F}^{A}}$ for some ${\theta_G > -\frac{1}{2}, \theta_F > -\frac{3}{2}+\frac{1}{\tau}}$ , where ${\tau\in [1, 2]}$ denotes the type of the Banach space and ${X_{\theta_F}^{A}}$ denotes the fractional domain space or extrapolation space corresponding to A. We assume F and G to satisfy certain global Lipschitz and linear growth conditions. Let A 0 denote the perturbed operator and U 0 the solution to (SDE) with A substituted by A 0. We provide estimates for ${\|U - U_0\|_{L^p(\Omega;C([0,T];X))}}$ in terms of ${D_{\delta}(A, A_0) := \|R(\lambda : A) - R(\lambda : A_0)\|_{\mathcal{L}(X^{A}_{\delta-1},X)}}$ . Here, ${\delta\in [0, 1]}$ is assumed to satisfy ${0\leq \delta < {\rm min}\{\frac{3}{2} - \frac{1}{\tau} + \theta_F,\, \frac{1}{2} - \frac{1}{p} + \theta_G \}}$ . The work is inspired by the desire to prove convergence of space approximations of (SDE). In this article, we prove convergence rates for the case that A is approximated by its Yosida approximation. 相似文献
18.
In this paper we prove that the Cauchy problem associated with the generalized KdV-BO equation ut + uxxx + λH(uxx) + u^2ux = 0, x ∈ R, t ≥ 0 is locally wellposed in Hr^s(R) for 4/3 〈r≤2, b〉1/r and s≥s(r)= 1/2- 1/2r. In particular, for r = 2, we reobtain the result in [3]. 相似文献
19.
Erich Novak Ian H. Sloan Henryk Wozniakowski 《Foundations of Computational Mathematics》2004,4(2):121-156
We study the approximation problem (or problem of optimal recovery in the
$L_2$-norm) for weighted Korobov spaces with smoothness
parameter $\a$. The weights $\gamma_j$ of the Korobov spaces moderate
the behavior of periodic functions with respect to successive variables.
The nonnegative smoothness parameter $\a$ measures the decay
of Fourier coefficients. For $\a=0$, the Korobov space is the
$L_2$ space, whereas for positive $\a$, the Korobov space
is a space of periodic functions with some smoothness
and the approximation problem
corresponds to a compact operator. The periodic functions are defined on
$[0,1]^d$ and our main interest is when the dimension $d$ varies and
may be large. We consider algorithms using two different
classes of information.
The first class $\lall$ consists of arbitrary linear functionals.
The second class $\lstd$ consists of only function values
and this class is more realistic in practical computations.
We want to know when the approximation problem is
tractable. Tractability means that there exists an algorithm whose error
is at most $\e$ and whose information cost is bounded by a polynomial
in the dimension $d$ and in $\e^{-1}$. Strong tractability means that
the bound does not depend on $d$ and is polynomial in $\e^{-1}$.
In this paper we consider the worst case, randomized, and quantum
settings. In each setting, the concepts of error and cost are defined
differently and, therefore, tractability and strong tractability
depend on the setting and on the class of information.
In the worst case setting, we apply known results to prove
that strong tractability and tractability in the class $\lall$
are equivalent. This holds
if and only if $\a>0$ and the sum-exponent $s_{\g}$ of weights is finite, where
$s_{\g}= \inf\{s>0 : \xxsum_{j=1}^\infty\g_j^s\,<\,\infty\}$.
In the worst case setting for the class $\lstd$ we must assume
that $\a>1$ to guarantee that
functionals from $\lstd$ are continuous. The notions of strong
tractability and tractability are not equivalent. In particular,
strong tractability holds if and only if $\a>1$ and
$\xxsum_{j=1}^\infty\g_j<\infty$.
In the randomized setting, it is known that randomization does not
help over the worst case setting in the class $\lall$. For the class
$\lstd$, we prove that strong tractability and tractability
are equivalent and this holds under the same assumption
as for the class $\lall$ in the worst case setting, that is,
if and only if $\a>0$ and $s_{\g} < \infty$.
In the quantum setting, we consider only upper bounds for the class
$\lstd$ with $\a>1$. We prove that $s_{\g}<\infty$ implies strong
tractability.
Hence for $s_{\g}>1$, the randomized and quantum settings
both break worst case intractability of approximation for
the class $\lstd$.
We indicate cost bounds on algorithms with error at
most $\e$. Let $\cc(d)$ denote the cost of computing $L(f)$ for
$L\in \lall$ or $L\in \lstd$, and let the cost of one arithmetic
operation be taken as unity.
The information cost bound in the worst case setting for the
class $\lall$ is of order $\cc (d) \cdot \e^{-p}$
with $p$ being roughly equal to $2\max(s_\g,\a^{-1})$.
Then for the class $\lstd$
in the randomized setting,
we present an algorithm with error at most $\e$ and whose total cost is
of order $\cc(d)\e^{-p-2} + d\e^{-2p-2}$, which for small $\e$ is roughly
$$
d\e^{-2p-2}.
$$
In the quantum setting, we present a quantum algorithm
with error at most $\e$ that
uses about only $d + \log \e^{-1}$ qubits
and whose total cost is of order
$$
(\cc(d) +d) \e^{-1-3p/2}.
$$
The ratio of the costs of the algorithms in the quantum setting and
the randomized setting is of order
$$
\frac{d}{\cc(d)+d}\,\left(\frac1{\e}\right)^{1+p/2}.
$$
Hence, we have a polynomial speedup of order $\e^{-(1+p/2)}$.
We stress that $p$ can be arbitrarily large, and in this case
the speedup is huge. 相似文献
20.
本文探索了一种能多变量综合优化的方法,即对喷管进行参数化设计后,用均匀试验设计(UED)将试验样本均匀散布在设计区间内,求出各性能参数后,利用径向基神经网络(RBF)对试验样本进行拟合,再用粒子群算法(PSO)对训练好的神经网络进行寻优,找出了更好的双喉道气动矢量喷管设计参数组合。数值模拟结果显示,优化后的双喉道气动矢量喷管的矢量角有了明显提高。试验表明这种优化方法具有很好的优化能力,可以用来对喷管几何外形进行参数优化。 相似文献