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1.
We consider the solution of the Korteweg–de Vries (KdV) equation with periodic initial value where C , A , k , μ, and β are constants. The solution is shown to be uniformly bounded for all small ɛ, and a formal expansion is constructed for the solution via the method of multiple scales. By using the energy method, we show that for any given number T > 0 , the difference between the true solution v ( x , t ; ɛ) and the N th partial sum of the asymptotic series is bounded by ɛ N +1 multiplied by a constant depending on T and N , for all −∞ < x < ∞, 0 ≤ t ≤ T /ɛ , and 0 ≤ɛ≤ɛ0 . 相似文献
2.
On the Nevanlinna Order of Meromorphic Solutions to Linear Analytic Difference Equations 总被引:1,自引:0,他引:1
For various classes of linear ordinary analytic difference equations with meromorphic coefficients, we study Nevanlinna order properties of suitable meromorphic solutions. For a large class of first-order equations with coefficient of order ρ∈[0, ∞), we explicitly construct meromorphic solutions of order ≤ρ+ 1. For higher-order equations with coefficients of order ρ∈[0, ∞), we show that meromorphic solutions with increase of order ≤ρ+ 1 in a certain strip have order ≤ρ+ 1. The assumptions made in the latter setting may seem quite restrictive, but they are satisfied for several classes of second-order difference equations that have been studied in recent years. The latter include Harper-type equations, "reflectionless" equations, Askey–Wilson-type equations, and equations of relativistic Calogero–Moser type. 相似文献
3.
We consider a mixed initial boundary-value problem for the nonlinear parabolic functional-differential equation. The functional part of the equation depends on a generalized superposition of the sought solution and a transformation of the one-dimensional spatial argument. An approximate projection difference scheme is proposed for a wide class of measurable (including non-invertible) transformations. An O( + h
1 + ) bound is obtained for the rate of convergence in the L
2(Q) norm to the generalized solutions of the original problem without prior assumptions about invertibility of the transformation, smoothness of the solution, or compatibility of grid increments. 相似文献
4.
Multiple Positive Solutions for a Nonlinear Elliptic Equation Involving Hardy–Sobolev–Maz'ya Term 下载免费PDF全文
In this paper, we study the existence and nonexistence of multiple positive solutions for the following problem involving Hardy–Sobolev–Maz'ya term:-Δu- λu/|y|2=|u|pt-1u/|y|t+ μf(x), x ∈Ω,where Ω is a bounded domain in RN(N ≥ 2), 0 ∈Ω, x =(y, z) ∈ Rk× RN-kand pt =N +2-2t N-2(0 ≤ t ≤2). For f(x) ∈ C1(Ω)\{0}, we show that there exists a constant μ* 0 such that the problem possessesat least two positive solutions if μ∈(0, μ*) and at least one positive solution if μ = μ*. Furthermore,there are no positive solutions if μ∈(μ*, +∞). 相似文献
5.
陈娇 《数学物理学报(B辑英文版)》2012,32(3):1167-1180
In this article,the author studies the Cauchy problem of the damped wave equation with a nonlinear convection term in multi-dimensions.The author shows that a classical solution to the Cauchy problem e... 相似文献
6.
J. E. Macías-Díaz J. Villa-Morales 《Journal of Difference Equations and Applications》2017,23(4):799-820
Departing from a general stochastic differential equation with Brownian diffusion, we establish that the distribution of probability of the stopping time is governed by a parabolic partial differential equation. A particular form of the problem under investigation may be associated to a stochastic generalization of the well-known Paris’ law from structural mechanics, in which case, the solution of the boundary-value problem represents the probability distribution of the hitting time. An implicit, convergent and probability-based discretization to approximate the solution of the boundary-value problem is proposed in this work. Using a convenient vector representation of our scheme, we prove that the method preserves the most relevant properties of a probability distribution function, namely, the non-negativity, the boundedness from above by 1, and the monotonicity. In addition, we establish that our method is a convergent technique, and provide some illustrative comparisons against known exact solutions. 相似文献
7.
A. V. Domrin 《Proceedings of the Steklov Institute of Mathematics》2006,253(1):37-50
It is very likely that all local holomorphic solutions of integrable (1+1)-dimensional parabolic-type evolution equations can be obtained from the zero solution by formal gauge transformations that belong (as formal power series) to appropriate Gevrey classes. We describe in detail the construction of solutions by means of convergent gauge transformations and prove an assertion converse to the above conjecture; namely, we suggest a simple necessary condition for the existence of a local holomorphic solution to the Cauchy problem for the evolution equations under consideration in terms of scattering data of initial conditions. 相似文献
8.
赵增勤 《数学物理学报(A辑)》2005,25(3):393-403
设(i) f(t,u): (0,1)×(0,+∞)→[0,+∞)连续,关于u 单调增加; (ii) 存在函数g:[1,+∞)→(0,+∞),g(b)0,G(t,s)是相应问题的Green函数。 相似文献
9.
魏利 《数学物理学报(A辑)》2010,30(4):1111-1116
该文利用非线性增生映射值域的扰动理论研究了与广义p-Laplace算子相关的Neumann边值问题在L~s(Ω)空间中解的存在性,其中2≤p≤s+∞.文中采用了一些新的证明技巧,推广和补充了笔者以往的一些工作. 相似文献
10.
曾六川 《数学物理学报(A辑)》2005,25(2):281-288
该文研究集值映象方程0∈T(z)的解的迭代逼近,其中T是极大强单调算子.设{x^k}与{e^k}是由不精确邻近点算法x^{k+1}+c_kT(x^{k+1})> x^k+e^{k+1}生成的序列,满足‖e^{k+1}‖≤η_k‖x^{k+1}_x^k‖, ∑^∞_{k=0}(η_k-1)<+∞且inf_(k≥0) η_k=μ≥1.在适当的限制下证明了,{x^k}收敛到T的一个根当且仅当
lim inf_{k→+∞} d(x^k,Z)=0,其中Z是方程0∈T(z)的解集 相似文献
11.
In the parameter variation method, a scalar parameterk, k[0, 1], is introduced into the differential equations. The parameterk is inserted in such a way that, whenk=0, the solution of the boundary-value problem is known or readily calculated and, whenk=1, the problem is identical with the original problem. Thus, bydeforming the solution step-by-step throughk-space fromk=0 tok=1, the original problem may be solved. These solutions then provide good starting values for any convergent, iterative scheme such as the Newton-Raphson method.The method is applied to the solution of problems with various types of boundary-value specifications and is further extended to take account of situations arising in the solution of problems from variational calculus (e.g., total elapsed time not specified, optimum control not a simple function of the variables). 相似文献
12.
13.
冉启康 《数学物理学报(A辑)》2004,24(3):354-361
该文讨论了二阶拟线性椭圆型问题u|\-\{Ω=0: -div[(d+|u|\+2)\+\{〖SX(〗p〖〗2〖SX)〗-1u]
=λ\-1u\+\{p-1+g(x,u),〓 x∈Ω正解的存在性和唯一性,其中 Ω是 R\+N 中的有界区域, λ\-1 是-△\-p 在 Ω上对应于零Dirichlet边界条件的第一特征根,
g(x, t) 满足增长条件lim[DD(X]t→+∞[DD)]〖SX(〗g(x,t)〖〗t\+\{p-1〖SX)〗=0, p>1, 0≤d<+∞〖HT5”H〗关键词:〖HT5”SS〗拟线性椭圆问题; 鞍点; 正解. 相似文献
14.
研究了工件满足一致性,批容量无界的两台同类机在线分批排序问题,目标为极小化工件的最大完工时间和极小化工件的最大流程时间,三元素法分别表示为Q_2|r_ir_j?p_i≤p_j,B=∞, on-line|C_(max),Q_2|r_ir_j?p_i≥p_j,B=∞, on-line|F_(max).不失一般性,假设第一台机器速度为1,第二台机器速度为s,s≥1.对于上述两类问题设计了一个在线算法,并分析了算法竞争比的上界.对第一类问题该在线算法的竞争比不超过s+α,这里α为α~2+sα-1=0的正根,特别地,当s=1时,该算法的竞争比不超过1.618.对第二类排序问题,该在线算法的竞争比不超过1+1/α. 相似文献
15.
We consider a singularly perturbed convection–diffusion equation, , defined on two domains: a quarter plane, ( x , y ) ∈ (0, ∞) × (0, ∞) , and an infinite strip, ( x , y ) ∈ (−∞, ∞) × (0, 1) . We consider for both problems discontinuous Dirichlet boundary conditions: u ( x , 0) = 0 and u (0, y ) = 1 for the first one and u ( x , 0) =χ[ a , b ] ( x ) and u ( x , 1) = 0 for the second. For each problem, asymptotic expansions of the solution are obtained from an integral representation in two limits: (a) when the singular parameter ε→ 0+ (with fixed distance r to the discontinuity points of the boundary condition) and (b) when that distance r → 0+ (with fixed ε). It is shown that in both problems, the first term of the expansion at ε= 0 is an error function or a combination of error functions. This term characterizes the effect of the discontinuities on the ε-behavior of the solution and its derivatives in the boundary or internal layers. On the other hand, near the discontinuities of the boundary condition, the solution u ( x , y ) of both problems is approximated by a linear function of the polar angle at the discontinuity points. 相似文献
16.
D是C^n空间中具有逐块C(1)边界的有界域,该文建立了D上一个具有离散局部全纯核的(0,q)形式的Koppelman积分公式及其相应的方程解的积分表示和它的内闭一致估计式。 相似文献
17.
J. B. McLeod 《Studies in Applied Mathematics》1997,98(4):311-333
The integral equation:
φμ (s) = (1/3 π)∫π 0 ((sin φμ (t))/(μ −1 + ∫t 0 sin φμ (u) d u )) (log((sin½( s + t ))/ (sin½( s − t )))d t
was derived by Nekrasov to describe waves of permanent form on the surface of a nonviscous, irrotational, infinitely deep flow, the function φμ giving the angle that the wave surface makes with the horizontal. The wave of greatest height is the singular case μ=∞, and it is shown that there exists a solution φ∞ to the equation in this case and that it can be obtained as the limit (in a specified sense) as μ→∞ of solutions for finite μ. Stokes conjectured that φ∞ ( s )→⅙π as s ↓0, so that the wave is sharply crested in the limit case; and Krasovskii conjectured that sup s ∈[0,π] φμ ( s )≤⅙π for all finite μ. Stokes' conjecture was finally proved by Amick, Fraenkel, and Toland, and the present article shows Krasovskii's conjecture to be false for sufficiently large μ, the angle exceeding ⅙π in what is a boundary layer. 相似文献
φ
was derived by Nekrasov to describe waves of permanent form on the surface of a nonviscous, irrotational, infinitely deep flow, the function φ
18.
On the basis of a model of a thermosensitive body, we propose an analytic-numerical method for the construction of the solution
of an axisymmetric quasistationary problem of thermoelasticity for a half-space heated by an instantaneous linear heat source
and exchanging heat through a bounding surface by convective heat transfer with the environment. Using the perturbation method,
we reduce the problem to the solution of a sequence of boundary-value problems for the Poisson equations, whose solutions
are constructed in the form of rapidly convergent series for each approximation by using expansions in multiple probability
integrals. 相似文献
19.
A. P. Oskolkov 《Journal of Mathematical Sciences》1997,85(1):1715-1721
Smooth convergent ε-approximations (11)–(13) for the equations of Oldroyd (8) and Kelvin-Voight (9), (10) fluids are constructed.
It is shown that the first initial boundary-value problem for two-dimensional system (11) and three-dimensional systems (12)
and (13) for every ε>0 has a unique classical solution, and as ε → 0 these solutions converge to classical solutions of the
first initial boundary-value problem for Eqs. (8), (9), and (10) respectively. Bibliography: 10 titles.
Dedicated to L. D. Faddeev on the occasion of his 60th birthday
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 215, 1994, pp. 246–255.
Translated by O. A. Ivanov. 相似文献
20.
Qasem M. Al-Mdallal Muhammed I. Syam 《Communications in Nonlinear Science & Numerical Simulation》2012,17(6):2299-2308
In this paper, we discuss a numerical solution of a class of non-linear fractional singularly perturbed two points boundary-value problem. The method of solution consists of solving reduced problem and boundary layer correction problem. A series method is used to solve the boundary layer correction problem, and then the series solutions is approximated by the Pade’ approximant of order [m, m]. Some theoretical results are established and proved. Two numerical examples are discussed to illustrate the efficiency of the present scheme. 相似文献