孫立強 田獻珍
摘 要:研究線性多步法求解Rα,β類非線性中立型時滯微分方程的數值穩(wěn)定性.在適當條件下,獲得了G(c,p,0)—代數穩(wěn)定的線性多步法的穩(wěn)定性及漸近穩(wěn)定性的充分條件.
關鍵詞:中立型時滯微分方程;線性多步法;G(c,p,0)—代數穩(wěn)定;漸近穩(wěn)定性
中圖分類號:O241.8 DOI:10.16375/j.cnki.cn45-1395/t.2018.01.003
0 引言
中立型時滯微分方程在自動控制、生物、醫(yī)學、化學、經濟等領域有著廣泛應用.由于其理論解難以獲得,數值方法的研究及穩(wěn)定性分析就顯得尤為重要,此方面的研究取得了許多成果[1-5].余越昕等[6]在研究中立型時滯積分微分方程時,獲得了線性θ-方法的漸近穩(wěn)定性;并研究了單支方法求解一類剛性中立型時滯微分方程時的穩(wěn)定性和漸近穩(wěn)定性[7].而線性多步法關于非線性中立型時滯微分方程的數值穩(wěn)定性研究尚未見于文獻.本文研究復空間中G(c,p,q)—代數穩(wěn)定[8]的線性多步法求解Rα,β類非線性中立型時滯微分方程時的穩(wěn)定性,獲得了線性多步法的穩(wěn)定及漸近穩(wěn)定的充分條件.
1 中立型時滯微分方程的穩(wěn)定性
參考文獻
[1]WANG W S,ZHANG Y, LI S F.Nonlinear stability of one-leg methods for delay differential equations of neutral type [J].Applied Numerical Mathematics, 2008 , 58(2) :122-130
[2]WANG W S,LI S F,SU K.Nonlinear stability of Runge-Kutta methods for neutral delay differential equations[J].Elsevier Science Inc. , 2007 , 214(1) :175-185.
[3]WANG W S,LI S F.Nonlinear stability of general linear methods for neutral delay differential equations [J].Journal of Computational & Applied Mathematics, 2009 , 224(2) :592-601.
[4]ZHANG C J. Nonlinear stability of natural Runge-Kutta methods for neutral delay differential equations [J].計算數學(英文版),2002 , 20(6) :583-590.
[5]余越昕,文立平,李壽佛.非線性中立型延遲微分方程線性方法的漸近穩(wěn)定性[J].高等學校計算數學學報,2006,28(2):103-110.
[6]余越昕,劉忠艷,江春華.中立型延遲積分微分方程線性方法的漸近穩(wěn)定性[J].山東大學學報(理學版),2010,45(1):102-106.
[7]余越昕,房松林,李壽佛.一類剛性中立型延遲微分方程單支方法的穩(wěn)定性分析[J].湘潭大學自然科學學報,2011,33(4): 1-3.
[8]李壽佛.剛性常微分方程及剛性泛函微分方程數值分析[M].湘潭:湘潭大學出版社,2010.
[9]Dahlquist G.Error analysis for a class of methods for stiff non-linear initial value problems[C]. Springer Berlin Heidelberg , 1976,506 :60-72
[10]Dahlquist G.G-stability is equivalent to A-stability [J].Bit Numerical Mathematics, 1978 , 18(4) :384-401.
Abstract: This paper studied the numerical stability of linear multi-step methods for a class Rα,β of nonlinear neutral delay differential equations. Under the suitable conditions, the sufficient conditions of the stability and asymptotic stability of G(c, p, 0)- algebraically stable linear multi-step methods were obtained.
Key words: neutral delay differential equations; linear multi-step methods; G(c, p, 0)-algebraically stable; asymptotic stability
(學科編輯:張玉鳳)