求解带非线性边界条件的反应扩散方程数值解的组合迭代方法

1　引　　言我们首先考虑如下抛物型方程ｕｔ-ＤΔｕ =ｆ(ｘ ,ｔ ,ｕ) (ｔ∈ ( 0 ,Ｔ],ｘ∈Ω ) ｕ/ ν+ βｕ =ｇ(ｘ ,ｔ ,ｕ) (ｔ∈ ( 0 ,Ｔ],ｘ∈ Ω )ｕ(ｘ ,0 ) =ψ(ｘ) (ｘ∈Ω )( 1 .1 )其中Ｔ为正常数 ,Ω 是ＲＰ 空间的有界区域 记ＱＴ=Ω × ( 0 ,Ｔ],ＳＴ= Ω × ( 0 ,Ｔ],假设在ＱＴ上Ｄ≡ｄ(ｘ ,ｔ) >0 ,在ＳＴ 上β≡β(ｘ ,ｔ)≥ 0 又设 ｆ(ｘ ,ｔ,ｕ) ,ｇ(ｘ ,ｔ,ｕ)为关于ｕ的非线性函数 ,且对ｘ ,ｔ各参数满足Ｈ¨ｏｌｄｅｒ连续条件 将 ( 1 .1 )离散化之后我们得到相应的有限差分系统 ,当 ｇ(ｘ ,ｔ,ｕ)为ｕ的线性…     

Young Measure Solutions of a Class of Singular Diffusion Equations

Ｃｏｎｓｉｄｅｒｔｈｅｆｉｒｓｔｉｎｉｔｉａｌ ｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍ ｕ ｔ=ｄｉｖ ｑ( ｕ) ,　 (ｘ,ｔ) ∈ＱＴ,(1 )ｕ(ｘ,ｔ) =0 ,　 (ｘ,ｔ) ∈ Ω× (0 ,Ｔ) ,(2 )ｕ(ｘ,0 ) =ｕ0 (ｘ) ,　ｘ∈Ω ,(3 )ｗｈｅｒｅΩｉｓａｂｏｕｎｄｅｄｄｏｍａｉｎｉｎＲＮｗｉｔｈｓｍｏｏｔｈｂｏｕｎｄａｒｙ Ω ,ＱＴ=Ω× (0 ,Ｔ) , ｑ = φ ,φ∈Ｃ1(ＲＮ) ,ａｎｄ φ , ｑｓａｔｉｓｆｙｔｈｅｓｔｒｕｃｔｕｒｅｃｏｎｄｉｔｉｏｎｓ(λ｜ξ｜1+δ-1 ) +≤ φ(ξ) ≤Λ｜ξ｜1+δ+ 1 ,　 ξ∈ＲＮ,(4 )｜ ｑ(ξ) …     

具吸收项的退化抛物问题的有限元误差估计

1　引　　言本文将考虑下列退化抛物方程的Ｇａｌｅｒｋｉｎ逼近ｕｔ =Δβ(ｕ) - ｆ(ｕ)　　在Ω× ( 0 ,Ｔ]内 ( 1 .1 )ｕ(ｘ ,ｔ) =0　　　　　　　在 Ω× ( 0 ,Ｔ]上 ( 1 .2 )ｕ(ｘ ,0 ) =ｕ0 (ｘ)　　在Ω内(1 .3)其中Ω Ｒｎ 是有界凸域 ,0 <Ｔ <∞ .β(ｖ) (ｖ∈Ｒ)是满足 β( 0 ) =β′( 0 ) =0且 β′≥ 0的函数 因此 ,( 1 1 )是退化的非线性抛物方程 方程 ( 1 1 )具有深刻的物理背景[1] ,文献 [2 - 3]讨论了方程 ( 1 1 )的特殊形式—多孔介质方程 (ＰＭＥ)的数值方法 关于ＰＭＥ解的存在性、唯一性和正则性已有许多结果 …     

关于不用计算导数的大范围收敛迭代法的注记

1　引　　言在文 [1 ]中我们借助于动力系统方法导出了求连续函数 ｆ(ｘ)在区间 [ａ ,ｂ]上单零点ｘ 的一个大范围收敛的连续性方法 .此处 ｆ(ｘ)满足李氏条件 ,且 ｆ(ａ) <0 ,ｆ(ｂ) >0 .这个连续性方法由动力系统ｄｘｄｔ =- ｆ(ｘ)ｘ( 0 ) =ｘ0 ∈ [ａ ,ｂ]( 1 )确定 ,其解析解ｘ(ｔ ,ｘ0 )具有性质ｌｉｍｔ→ +∞ｘ(ｔ,ｘ0 ) =ｘ ,　　ｘ0 ∈ [ａ ,ｂ].　　为了数值地求出ｘ ,我们利用显式欧拉法ｘｎ+ 1=ｘｎ -ｈｎｆ(ｘｎ)ｘ0 =ｂ　ｏｒａ ( 2 )来求 ( 1 )式的解 .其中ｈｎ>0 ,为步长 .它的选择满足文 [1 ]中的不等式ａ<ｘｎ+ 1<ｘｎ,…     

Monotone Iterative Method for Set valued Quasilinear Elliptic Boundary Value Problems

§ 1.ＰｒｏｂｌｅｍａｎｄＡｓｓｕｍｐｔｉｏｎｓ　Ｔｈｉｓｐａｐｅｒｄｅａｌｓｗｉｔｈｔｈｅｓｏｌｕｔｉｏｎｓｏｆｔｈｅｆｏｌｌｏｗｉｎｇｄｉｆｆｅｒｅｎｔｉａｌｉｎｃｌｕｓｉｏｎｐｒｏｂｌｅｍ :Ａｕ∈ｆ(ｘ,ｕ) ,ｘ∈Ω ;ｕ=0 ,ｘ∈ Ω ,(1 )ｗｈｅｒｅＡｕ(ｘ) =-∑Ｎｉ=1Ｄｉ[ａｉ(ｘ ,Ｄｕ(ｘ) ) ] ,Ω ＲＮｉｓａｂｏｕｎｄｅｄｄｏｍａｉｎｗｉｔｈｐｉｅｃｅｗｉｓｅＬｉｐｓｃｈｉｔｚｂｏｕｎｄａｒｙ Ω ,Ｄｕ =(Ｄ1ｕ ,Ｄ2 ｕ ,… ,ＤＮｕ) ,Ｄｉｕ = ｕ ｘｉ,ｉ=1 ,2 ,… ,Ｎ ,ａｎｄｆ:Ω×Ｒ→ 2 Ｒｉｓａ…     

EXISTENCE TIME FOR THE SEMILINEAR SCHR?DINGER EQUATION

§ 1　ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｉｓｐａｐｅｒ ,ｗｅｗｉｌｌｃｏｎｓｉｄｅｒｔｈｅｓｅｍｉｌｉｎｅａｒＳｃｈｒ ｄｉｎｇｅｒｅｑｕａｔｉｏｎｉｎｏｎｅｓｐａｃｅｄｉｍｅｎｓｉｏｎｏｆｔｈｅｔｙｐｅｕｔ-ｉｕｘｘ =Ｆ(ｕ) .　 (ｘ ,ｔ)∈Ｒ×Ｒ+,( 1 )ｕ(ｘ ,0 ) =ｕ0 ,　ｘ ∈Ｒ ,( 2 )ｗｈｅｒｅｕ =ｕ(ｘ ,ｔ)ｉｓｃｏｍｐｌｅｘ ｖａｌｕｅｄｆｕｎｃｔｉｏｎ ,ａｎｄＦｉｓａｓｍｏｏｔｈｆｕｎｃｔｉｏｎｏｆｕｓｕｃｈｔｈａｔ｜Ｆ(ｕ) ｜ =Ｏ( ｜ｕ｜α+1)ｆｏｒ |ｕ|ｓｕｆｆｉｃｉｅｎｔｌｙｓｍａｌｌａｎｄ…     

Periodic Solutions of Porous Medium Equations with Weakly Nonlinear Sources

§ 1.Ｉｎｔｒｏｄｕｃｔｉｏｎ　ＴｈｉｓｐａｐｅｒｉｓｃｏｎｃｅｒｎｅｄｗｉｔｈｔｈｅｔｉｍｅｐｅｒｉｏｄｉｃｓｏｌｕｔｉｏｎｓｏｆｔｈｅｐｏｒｏｕｓｍｅｄｉｕｍｅｑｕａｔｉｏｎｓｗｉｔｈｗｅａｋｌｙｎｏｎｌｉｎｅａｒｓｏｕｒｃｅｓａｎｄｗｉｔｈｔｈｅＤｉｒｉｃｈｌｅｔｂｏｕｎｄａｒｙｖａｌｕｅｃｏｎｄｉｔｉｏｎ ,ｎａｍｅｌｙ ,ｔｈｅｐｒｏｂｌｅｍ ｕ ｔ =Δ(｜ｕ｜ｍ- 1ｕ) +Ｂ(ｘ ,ｔ,ｕ) +ｆ(ｘ ,ｔ)　ｉｎΩ×Ｒ ,(1 .1 )ｕ(ｘ ,ｔ) =0　ｏｎ Ω×Ｒ , (1 .2 )ｕ(ｘ ,ｔ+ω) =ｕ(ｘ ,ｔ)　ｉｎ Ω×Ｒ ,…     

关于复合函数的三个问题

复合函数是形如 ｙ =ｆ[ｇ(ｘ) ]的函数 ,如 ｙ =ｌｏｇ3(ｘ2 -2ｘ 3 )由 ｙ =ｌｏｇ3ｕ ,ｕ =ｘ2-2ｘ 3复合而成 ;ｙ =( 3ｘ 1) - 13是由 ｙ =ｕ- 13,ｕ =3ｘ 1复合而成 ,ｙ =ａｓｉｎｘ(ａ >0且ａ≠ 1)由ｙ =ａｕ,ｕ =ｓｉｎｘ复合而成 ,其中ｇ(ｘ) 称为内层函数 ,ｙ =ｆ(ｕ)称为外层函数 ,且均为基本函数 .关于复合函数一般有三个问题要研究 .1　已知 ｙ =ｆ[ｇ(ｘ) ]的表达式 ,求 ｆ(ｘ)的表达式 .例 1　已知 ｆ( 2ｘ -1) =ｘ2 (ｘ∈Ｒ) ,求ｆ(ｘ) 的表达式 .解法 1　 (换元法 )令 2ｘ -1=ｔ ,则ｘ =ｔ 12 .∴ ｆ(ｔ) =14 (ｔ 1) …     

Pointwise Estimate for Baskakov Operators

§ 1.Ｉｎｔｒｏｄｕｃｔｉｏｎ　Ｆｏｒｔｈｅｗｅｌｌ ｋｎｏｗｎＢｅｒｎｓｔｅｉｎｐｏｌｙｎｏｍｉａｌＢｎ(ｆ;ｘ) = ｎｋ=0ｆ ｋｎ ｐｎ,ｋ(ｘ) ,　ｐｎ ,ｋ(ｘ) =ｎｋ ｘｋ( 1 -ｘ) ｎ-ｋ,ＢｅｒｅｎｓａｎｄＬｏｒｅｎｔｚ[1]ｐｒｏｖｅｄｔｈａｔｆｏｒｆ∈Ｃ[0 ,1 ] ,0 <α<2 ,ｏｎｅｈａｓ｜ (Ｂｎｆ-ｆ) (ｘ) ｜=Ｏ ｘ( 1 -ｘ)ｎα/ 2 ω2 (ｆ;ｔ) =Ｏ(ｔα) . ( 1 .1 )Ｏｎｔｈｅｏｔｈｅｒｈａｎｄ ,ＤｉｔｚｉａｎａｎｄＴｏｔｉｋ[2 ]ｏｂｔａｉｎｅｄｔｈａｔｆｏｒｆ∈Ｃ[0 ,1 ] ,0 <α<2 ,ｏｎｅｈａｓ｜ (Ｂｎｆ -ｆ)…     

一类抛物型偏泛函微分方程解的强迫振动性

本文研究抛物型偏泛函微分方程γ／γｔ［ｕ－ｍΣｔ－１Ｃｔ（ｔ）ｕ（ｘ，ｔ－τｔ）］＝ａ（ｔ）Δｕ－Ｐ（ｘ，ｔ）ｕ－Ｑ（ｘ，ｔ）Ｇ［ｕ（ｘ，ｐ（ｔ）］＋Ｆ（ｘ，ｔ），（ｘ，ｔ）包含Ｄ×［０，＋∞］解的强近振动性，其中Ｄ为Ｒ＾ｎ中具有逐片光滑边办γＤ的有界区域，ｕ＝ｕ（ｘ，ｔ），Δ是Ｒ＾ｎ中的Ｌａｐｌａｃｅ算子。     

A Vasicek-Type Short Rate Model With Memory Effect

We introduce a Vasicek-type short rate model which has two additional parameters representing memory effect. This model presents better results in yield curve fitting than the classical Vasicek model. We derive closed-form expressions for the prices of bonds and bond options. Although the model is non-Markov, there exists an associated Markov process that allows one to apply usual numerical methods to the model. We derive analogs of an affine term structure and term structure equations for the model, and, using them, we present a numerical method to evaluate contingent claims.     

Entropy Unilateral Solution for Some Noncoercive Nonlinear Parabolic Problems Via a Sequence of Penalized Equations

We give an existence result of the obstacle parabolic equations(b(x,u))/(t)-div(a(x,t,u,▽u))+div(φ(x,t,u))=f in Q_T,where b(x,u) is bounded function of u,the term-div(a(x,t,u,▽u)) is a Leray-Lions type operator and the function φ is a nonlinear lower order and satisfy only the growth condition.The second term f belongs to L~1(Q_T).The proof of an existence solution is based on the penalization methods.     

一类奇异二阶边值问题的正周期解

考察了一类奇异二阶周期边值问题,其中非线性项f(t,u)是局部Caratheódory函数.主要工具是高度函数,它描述了非线性项f(t,u)在有界集合上的增长特性.通过考察高度函数的积分获得了单个或多重正解存在的几个充分条件.我们的工作表明这种存在性与非线性项f(t,u)在u=0附近的性质无关.     

关于奇异二阶微分方程（Φ（X’））’＋f（t，X）＝0的某些可解准则

本文讨论了一类右端具有奇异性的二阶常微分方程的边值问题，证明了当ｆ（ｔ，ｘ）关于ｔ的奇异程度大于某值时，边值问题无解，而当奇异程度小于某值时，边值问题有解．关于ｕ在ｕ＝０的奇异程度没有限制．     

奇异三阶两点边值问题解的一个存在定理

By constructing suitable Banach space.an existence theorem is established under a condition of linear growth for the third-order boundary value problem u'"(t) f(t,u(t),t'(t),u"(t))=0,0＜t＜1,u(0)=u'(0)=u'(1)=0,where the nonlinear term contains first and second derivatives of unknown function.In this theorem the nonlinear term f(t,u,v,w)may be singular at t=0 and t=1.The main ingredient is Leray-Schauder nonlinear alternative.     

Numerical scheme for a scalar conservation law with memory

We consider approximation of solutions to conservation laws with memory represented by a Volterra term with a smooth decreasing but possibly unbounded kernel. The numerical scheme combines Godunov method with a treatment of the integral term following from product integration rules. We prove stability for both linear and nonlinear flux functions and demonstrate the expected order of convergence using numerical experiments. The problem is motivated by modeling advective transport in heterogeneous media with subscale diffusion.Copyright © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 239–264, 2014     

一类含隅角和弯矩的奇异梁方程三个正解的存在性

利用格林函数方法和Avery-Peterson不动点定理研究了一类非线性四阶两点边值问题u(4)(t) =f(t,u(t),u′(t),u″(t)), 0 ＜ t ＜ 1,u(0) =u′(1) =u″(0) =u′″(1) =0多个正解的存在性,其中允许非线性项f(t,u,v,w)在t=0,t=1,u=0,v=0,w=0处奇异.在力学上该问题模拟了左端简单支撑右端被滑动夹子夹住的弹性梁的挠曲.由于非线性项同时涉及隅角和弯矩,因此主要结论对于梁的稳定性分析是有益的.最后我们给出了一个例子,进一步证实本文理论的严密性和可行性.     

Quenching for a non-local diffusion equation with a singular absorption term and Neumann boundary condition

This paper deals with the quenching phenomenon for a non-local diffusion equation $$u_t(x,t)=\int\limits_\Omega{J(x-y)(u(y,t)-u(x,t))}{\rm d}y-f(u(x,t)),(x,t)\in\Omega\times[0,T),$$ with a general singular absorption term and Neumann boundary condition. The local existence and uniqueness of the solution are proved, and the solution of the equation quenches in finite time is shown. Moreover, under appropriate condition, the only quenching point is x?=?0, and the estimate of the quenching rate is obtained. Finally, some numerical experiments are performed, which illustrate our results.     

一类中立型时滞抛物偏微分方程的强迫振动性

研究了一类中立型时滞抛物偏微分方程:t(u(x,t)-pu(x,t-τ))-∑rk=1ak(t)Δu(x,t-ρk(t))+∑mj=1qj(t)u(x,t-σj(t))=e(x,t),的强迫振动性(其中(x,t)∈Ω×[0,∞)≡G,Ω是n维欧几里得空间Rn中带有逐段光滑边界Ω的有界区域,Δ是Rn中带有三类不同边值条件的拉普拉斯算子,强迫项e(x,t)是定义在G上的一个振荡函数),给出了一些新的振动性判据,这些结果推广了已知的一些结论.     

New Class of Kirchhoff Type Equations with Kelvin-Voigt Damping and General Nonlinearity: Local Existence and Blow-up in Solutions

In this paper, we consider a class of Kirchhoff equation, in the presence of a Kelvin-Voigt type damping and a source term of general nonlinearity forms. Where the studied equation is given as follows\begin{equation*}u_{tt} -\mathcal{K}\left( \mathcal{N}u(t)\right)\left[ \Delta_{p(x)}u +\Delta_{r(x)}u_{t}\right]=\mathcal{F}(x, t, u).\end{equation*}Here, $\mathcal{K}\left( \mathcal{N}u(t)\right)$ is a Kirchhoff function, $\Delta_{r(x)}u_{t}$ represent a Kelvin-Voigt strong damping term, and $\mathcal{F}(x, t, u)$ is a source term. According to an appropriate assumption, we obtain the local existence of the weak solutions by applying the Galerkin's approximation method. Furthermore, we prove a non-global existence result for certain solutions with negative/positive initial energy. More precisely, our aim is to find a sufficient conditions for $p(x), q(x), r(x), \mathcal{F}(x,t,u)$ and the initial data for which the blow-up occurs.