非線性拋物型方程時(shí)間周期解的一種差分方法

芮 杰

(中國(guó)石油大學(xué)數(shù)學(xué)與計(jì)算科學(xué)學(xué)院,山東東營(yíng) 257061)

非線性拋物型方程時(shí)間周期解的一種差分方法

芮 杰

(中國(guó)石油大學(xué)數(shù)學(xué)與計(jì)算科學(xué)學(xué)院,山東東營(yíng) 257061)

建立了一個(gè)具有時(shí)間周期的非線性拋物型方程的隱式差分格式,差分格式的精度為O(k2+h4),并用離散泛函分析的方法證明了格式的收斂性和穩(wěn)定性.

非線性拋物型方程時(shí)間周期解;差分格式;收斂性;穩(wěn)定性

0 引言

時(shí)滯非線性拋物型方程時(shí)間周期解是生態(tài)學(xué)以及控制論等領(lǐng)域中非常重要的模型,并有很多關(guān)于周期解的存在性與穩(wěn)定性的研究成果[1-5].我們主要考慮一些數(shù)值研究成果,本文研究下列時(shí)滯非線性拋物型問(wèn)題的一種隱式差分格式的構(gòu)造:

其中,D>0是常數(shù);uτ=u(x,t-τ),τ>0表示時(shí)滯;T是正的常數(shù),它代表解的時(shí)間周期,并假設(shè)時(shí)間周期 T是給定的,函數(shù) u0(t),u1(t),f(x,t,u,uτ)在 R+×[0,1]上連續(xù),且關(guān)于 t均為 T周期函數(shù),f(x,t,u,uτ) 關(guān)于 u、uτ是非線性的.

1 差分格式

2 差分格式的收斂性和穩(wěn)定性

3 數(shù)值試驗(yàn)

表1 格式A與解析解的誤差比較

[1] Pao CV.Periodic solutionsof parabolic systemswith time delays[J].JMath Anal Appl,2000,251:251-263.

[2] Pao C V.Numericalmethods for time-periodic solutions of non-linear parabolic boundary value problems[J].Slam J Number Anal,2001,39:647-667.

[3] Pao CV.Satiability and attractively of periodic solutionsof para-bolic systemswith time delays[J].JMath Anal Appl,2005,304:423-450.

[4] Wei F,Xin L.Asymptotic periodicity in diffusive logistic equationswith discrete delays[J].Nonlinear Anal,1996,26:171-178.

[5] Chen W,Jiong R,Wei L.On the existence and stability of the periodic solution in the Cohen-Grossberg neural network with time delay and high-order terms[J].Applied Mathematics and Com-putation,2006,177:194-210.

[6] 舒阿秀.時(shí)滯非線性拋物型方程時(shí)間周期解的有限差分方法 [J].合肥工業(yè)大學(xué)學(xué)報(bào):自然科學(xué)版,2008,31(2):296-300.

[7] Zhou YL.Applications of discrete Functional Analysis to the Finite Difference Method[M].Beijing:International Academic Publishers,1990.

Abstract:In this paper,an implicit scheme for the time-periodic solution of nonlinear parabolic equation.The truncation errorof the three schemes is O(k2+h4).The convergence and stability are proved by the discrete functions analysismethod.

Key words:time-periodic solution of nonlinear parabolic equation;difference schemes;convergence;stability

[責(zé)任編輯:李春紅]

One Difference Schemes for Nonlinear Parabolic Equation with Time Delay

RUIJie

(College of Mathematics,Computational Science in China University of Petroleum,Dongying Shandong 257061,China)

O241.82

A

1671-6876(2010)06-0471-05

2010-08-03

芮杰(1979-),女,漢族,山東新泰人,講師,碩士,研究方向?yàn)槠⒎址匠淘趫D像處理中的應(yīng)用.