Obrechkoff方法在求解常微分方程振荡、刚性问题中的应用研究
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摘要
由于常微分方程本身的重要性以及在不同领域的广泛应用，贯穿整个20世纪，常微分方程的数值求解研究得到了巨大的发展。特别是，随着计算机性能的快速提高，一些著名数学软件的不断深化发展，更多的新思想得以实现，更多的复杂方法涌现出来，常微分方程数值求解以及数值方法发展研究的领域有不断深化扩大的趋势。
计算机的数值计算功能对物理学中常微分方程研究的用途不仅仅是可以得到数值结果，更为重要的是，它为物理学家提供了“计算机模拟实验”这个新的研究手段。有了计算机数值计算这个强有力的工具，我们将目光投向物理领域中一些较为复杂的常微分方程(非线性Duffing方程，周期性振荡方程以及刚性方程)的数值求解与相应数值方法的研究。
在物理领域中，常常可以遇到一些应用很是广泛的常微分方程，例如薛定锷方程、非线性Duffing方程、天体轨道方程以及刚性方程等。这些方程多为一阶或二阶的常微分方程，形式简单，却很少能得到解析解。即使数值求解也往往存在着求解精度不高或因方程本身性质特殊造成数值方法求解结果不尽理想。在这些问题中具有代表性的有两类问题：周期性振荡问题与刚性问题。
在本论文中，我们主要集中于这两类问题相应的数值方法研究做出探讨。
对于周期性振荡问题，我们主要关注二阶常微分方程
y″(x)+ω~2y(x)=f(x，y)，y′(0)=y′_0，y(0)=y_0
这类方程的近似解析解中常包含cos(ωx)、sin(ωx)或e~(iωx)等周期性函数。鉴于其周期振荡性质，往往造成数值方法求解困难，结果出现不稳定，甚至发散。
我们研究发现对于具有周期振荡性质的问题必须有匹配的数值方法，即数值方法也需具有周期振荡性质。否则即使原本精度很高的方法，如果与所求解问题的性质不匹配，数值求解的结果也往往是不理想，甚至得到发散的结果。反之，如果数值方法与问题匹配但精度不够，同样也不能得到满意的结果。
为此，我们从两方面出发研究针对周期性振荡问题的数值方法。
Because of the importance of ordinary differential equations and their wide application in diffirent fields, through the whole 20th century, the research on numerical solutions of ODEs made enormous progress. Particularly, with the fast development of electronic computer and some famous mathematics softwares, more and more new ideas and approaches which surpass classic methods could be realized. At the same time, the field of numerical methods has been expanded.
The computational function of computer is not merely to get numerical results for physical research, the more important thing is, and it has offered this new research means of "computer simulation experiment" to physicist. With the powerful tool of electronic computation, we cast sight on numerical solutions to some comparatively complicated differential equations in the physics field (nonlinear Duffing equation, periodic oscillatory equation and stiff equation) and the research on the corresponding numerical methods.
In the physics field, we often encounter some differential equation with wide application, for example, Schr6dinger equation, nonlinear Duffing equation, orbital equation and stiff equation. Most of these equations are first-order or second-order equations with simple form, but seldom of could be analytically solved. Because of low accurate numerical methods and the specific property of the equations, it is hard to get ideal results even in numerical way. There are two representative problems: periodic oscillatory problem and stiff problem.
In this desertation, we focus mainly on numerical methods for oscillatory and stiff problems.
For the oscillatory problem, we focus on second-order differential equation
Commonly, there exist some periodic functions like cos(ωx)、 sin(ωx) or e~(tωx) in the approximately analytic solution to periodic oscillatory problem. Due to its oscillatory attribute, it is difficult to solve the problem numerically, and even if it is done, the numerical results often become unstable or even diffused.
计算机的数值计算功能对物理学中常微分方程研究的用途不仅仅是可以得到数值结果，更为重要的是，它为物理学家提供了“计算机模拟实验”这个新的研究手段。有了计算机数值计算这个强有力的工具，我们将目光投向物理领域中一些较为复杂的常微分方程(非线性Duffing方程，周期性振荡方程以及刚性方程)的数值求解与相应数值方法的研究。
在物理领域中，常常可以遇到一些应用很是广泛的常微分方程，例如薛定锷方程、非线性Duffing方程、天体轨道方程以及刚性方程等。这些方程多为一阶或二阶的常微分方程，形式简单，却很少能得到解析解。即使数值求解也往往存在着求解精度不高或因方程本身性质特殊造成数值方法求解结果不尽理想。在这些问题中具有代表性的有两类问题：周期性振荡问题与刚性问题。
在本论文中，我们主要集中于这两类问题相应的数值方法研究做出探讨。
对于周期性振荡问题，我们主要关注二阶常微分方程
y″(x)+ω~2y(x)=f(x，y)，y′(0)=y′_0，y(0)=y_0
这类方程的近似解析解中常包含cos(ωx)、sin(ωx)或e~(iωx)等周期性函数。鉴于其周期振荡性质，往往造成数值方法求解困难，结果出现不稳定，甚至发散。
我们研究发现对于具有周期振荡性质的问题必须有匹配的数值方法，即数值方法也需具有周期振荡性质。否则即使原本精度很高的方法，如果与所求解问题的性质不匹配，数值求解的结果也往往是不理想，甚至得到发散的结果。反之，如果数值方法与问题匹配但精度不够，同样也不能得到满意的结果。
为此，我们从两方面出发研究针对周期性振荡问题的数值方法。
Because of the importance of ordinary differential equations and their wide application in diffirent fields, through the whole 20th century, the research on numerical solutions of ODEs made enormous progress. Particularly, with the fast development of electronic computer and some famous mathematics softwares, more and more new ideas and approaches which surpass classic methods could be realized. At the same time, the field of numerical methods has been expanded.
The computational function of computer is not merely to get numerical results for physical research, the more important thing is, and it has offered this new research means of "computer simulation experiment" to physicist. With the powerful tool of electronic computation, we cast sight on numerical solutions to some comparatively complicated differential equations in the physics field (nonlinear Duffing equation, periodic oscillatory equation and stiff equation) and the research on the corresponding numerical methods.
In the physics field, we often encounter some differential equation with wide application, for example, Schr6dinger equation, nonlinear Duffing equation, orbital equation and stiff equation. Most of these equations are first-order or second-order equations with simple form, but seldom of could be analytically solved. Because of low accurate numerical methods and the specific property of the equations, it is hard to get ideal results even in numerical way. There are two representative problems: periodic oscillatory problem and stiff problem.
In this desertation, we focus mainly on numerical methods for oscillatory and stiff problems.
For the oscillatory problem, we focus on second-order differential equation
Commonly, there exist some periodic functions like cos(ωx)、 sin(ωx) or e~(tωx) in the approximately analytic solution to periodic oscillatory problem. Due to its oscillatory attribute, it is difficult to solve the problem numerically, and even if it is done, the numerical results often become unstable or even diffused.
引文
[1] F. Bashforth, J.C. Adams, An attempt to test the theories of capillary action by comparing the theoretical and measured forms of drops of uid, with an explanation of the method of integration employed in constructing the tables which give the theoretical forms of such drops, Cambridge University Press, Cambridge, 1883.
[2] C. Runge, Uber die numerische Aufosung von Differentialgleichungen, Math. Ann., 46 (1895) 167-178.
[3] K. Heun, Neue Methoden zur approximativen Integration der Differential gleichungen einer unabhangigen Veranderlichen, Z. Math. Phys., 45 (1900) 23-38.
[4] W. Kutta, Beitrag zur naherungsweisen Integration totaler Differential gleichungen, Z. Math. Phys., 46 (1901) 435-453.
[5] E. J. Nystrom, Uber die numerische Integration von Differentialgleichungen, Acta Soc. Sci. Fennicae, 50 (13) (1925).
[6] F.R. Moulton, New Methods in Exterior Balistics, University of Chicago, Chicago 1926.
[7] G. Dahlquist, Convergence and stability in the numerical integration of ordinary differential equations, Math. Scand., 4 (1956) 33-53.
[8] G. Dahlquist, Stability and error bounds in the numerical integration of rdinary differential equations, Kungl. Tekn. Hogsk. Handl. Stockholm 130 (1959)1-87.
[9] P. Albrecht, A new theoretical approach to Runge-Kutta methods, SIAM J. Numer. Anal., 24 (1987) 391-406.
[10] R. Alt, Deux theorems sur la A-stabilite des schemas de Runge-Kutta simplement implicites, Rev. Francais d'Automat. Recherche Operationelle Ser. R-36 (1972) 99-104.
[11] J.C. Butcher, Coefficients for the study of Runge-Kutta integration processes, J. Austral. Math. Soc., 3 (1963) 185-201.
[12] J.C. Butcher, An algebraic theory of integration methods, Math. Comp., 26 (1972) 79-106.
[13] R. Frank, J. Schneid, C.W. Ueberhuber, The concept of B-convergence, SIAM J. Numer. Anal., 18 (1981) 753-780.
[14] R. Alexander, Diagonally implicit Runge-Kutta methods for stiff ODEs, SIAM J. Numer. Anal., 14 (1977) 1006-1021.
[15] K. Burrage, A special family of Runge-Kutta methods for solving stiff differential equations, BIT 18 (1978)22-41.
[16] K. Burrage, J.C. Butcher, Stability criteria for implicit Runge-Kutta methods, SIAM J. Numer. Anal., 16 (1979) 46-57.
[17] J.C. Butcher, Implicit Runge-Kutta processes, Math. Comp., 18 (1964) 50-64.
[18] J.C. Butcher, On the attainable order of Runge-Kutta methods, Math. Comput., 19 (1965) 408-417.
[19] J.C. Butcher, The non-existence of ten stage eighth order explicit Runge-Kutta methods, BIT 25 (1985) 521:540.
[20] J.C. Butcher, A stability property of implicit Runge-Kutta methods, BIT 15 (1975) 358-361.
[21] M. Crouzeix, Sur les methodes de Runge-Kutta pour l'approximation des problemes d'evolution, Lecture Notes in Economics and Mathematical Systems, 134, Springer, Berlin, 1974, 206-223.
[22] E. Hairer, C. Lubich, M. Roche, in the Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods, Lecture Notes in Mathematics, 1409, Springer, Berlin, 1989.
[23] Houwen, P. J. van der& Sommeijer, B. P., Explicit Runge-Kutta (-Nystrom) methods with reduced phase errors for computing oscillating solutions. SIAM J. Numer. Anal., 24 1987 595-617
[24] C.W. Gear, Hybrid methods for initial value problems in ordinary differential equations, SIAM J. Numer. Anal., 2 (1965) 69-86.
[25] S. Gill, A process for the step-by-step integration of differential equations in an automatic digital computing machine, Proc. Cambridge Philos. Soc, 47 (1951) 96-108.
[26] R.H. Merson, An operational method for the study of integration processes, Proceedings of the Symposium on Data Processing, Weapons Research Establishment, Salisbury, South Australia, 1957.
[27] A. Prothero, A. Robinson, On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations, Math Comp., 28 (1974) 145-162.
[28] D. Barton, I.M. Willers, R.V.M. Zahar, The automatic solution of ordinary differential equations by the method of Taylor series, Comput. J., 14 (1971) 243-248.
[29] J.C. Butcher, A modified multistep method for the numerical integration of ordinary differential equations, J. Assoc. Comput. Mach., 12 (1965) 124-135.
[30] Kesan, Cenk, Taylor polynomial solutions of linear differential equations, Appl. Math. and Compt., 142 (2003) 155-165.
[31] Tian, T.; Burrage, K., Implicit Taylor methods for stiff stochastic differential equations, Appl. Numer. Math., 38 (2001) 167-185.
[32] Nedialkov, N.S.; Jackson, K.R.; Corliss, G.F. .Validated solutions of initial value problems for ordinary differential equations, Appl. Math. and Compt., 105 (1999)21-68.
[33] Ma, Hang; Qin, Qing-Hua, A second-order scheme for integration of one-dimensional dynamic analysis, Compt. and Math. Appl., 49 (2005) 239-252.
[34] Bulos, B.; Khuri, Suheil A., On the modified Taylor's approximation for the solution of linear and nonlinear equations, Appl. Math. and Compt., 160 (2005) 939-953.
[35] N. Obrechkoff, On mechanical quadrature (Bulgarian, French summary), Spisanie Bulgar. Akad. Nauk., 65 (1942) 191-289.
[36] Jackson, Kenneth R.; Nedialkov, Nedialko S. ,Some recent advances in validated methods for IVPs for ODEs, Appl. Num. Math., 42 (2002) 269-284.
[37] M. Crouzeix, Sur la B-stabilite des methodes de Runge-Kutta, Numer. Math., 32 (1979) 75-82.
[38] C.F. Curtiss, J.O. Hirschfelder, Integration of stiff equations, Proc. Nat. Acad. Sci., 38 (1952) 235-243.
[39] G. Dahlquist, A special stability property for linear multistep methods, BIT 3 (1963)27-43.
[40] G. Dahlquist, G-stability is equivalent to A-stability, BIT 18 (1978) 384-401.
[41] C.W. Gear, Hybrid methods for initial value problems in ordinary differential equations, SIAM J. Numer. Anal., 2 (1965) 69-86.
[42] C.W. Gear, Algorithm 407, DIFSUB for solution of ordinary differential equations, Comm. ACM., 14 (1971) 447-451.
[43] W.E. Milne, A note on the numerical integration of dffierential equations, J. Res. Nat. Bur. Standards, 43 (1949) 537-542.
[44] MAURY, J. L. and SEGAL, G. P. "Cowell Type Numerical Integration as Applied to Satellite Orbit Computation," Technical Report X-553-69-46, NASA, 1969,NTIS #N6926703.
[45] Lambert J. D., and Mitchell, A. R., On the solution of y'= f(x,y)by a class of high accuracy difference formulae of low order, Z. Angew. Math. Phys., 13, 1962,223-232.
[46] J. D. Lambert & I. A. Watson, Symmetric multistep methods for periodic initial value problems, J. Inst. Math. Applic., 18 (1976), 189-202.
[47] Henrici, P., Discrete variable methods for ordinary differential equations, Wiley, New York, 1962.
[48] Coleman, J. P. Numerical methods for y" = f(x,y)via rational approximation for the cosin, IMA J. Numer. Anal., 9 (1989) 145-165
[49] U. Ananthakrishnaiah, Adaptive methods for periodic initial value problems of second order differential equations, J. Comput. Appl. Math., 8 1982 101-104
[50] U. Ananthakrishnaiah, P-Stable Obrechkoff methods with minimal phase-lag for periodic initial-value problems, Math. Comput., 49 (1987), 553-559.
[51] Rai, A. Sesappa; U. Ananthakrishnaiah, Obrechkoff method having additional parameters for general second-order differential equations, J Comput. Appl. Math., 79(1997), 167-182.
[52] Neta, B.; Fukushima, T., Obrechkoff versus super-implicit methods for the solution of first-and second-order initial value problems, Compt. and Math. Appl., 45 (2003), 383-390.
[53] Dimovski, Ivan; Kiryakova, Virginia, Obrechkoff integral transform:Properties and relation to a generalized fractional calculus, Numerical Functional Analysis and Optimization, 21 (2000), 121-144.
[54] Stapleton, David P., Numerical simulation of a rigid rotating body by Obrechkoff integration, Compt. Phys. Comm., 98 (1996), 153-166.
[55] Iseries, Arieh, Composite methods for numerical solution of stiff systems of ODEs, SIAM J. Numer. Anal., 21 (1984), 340-351.
[56] Iseries, A, Two-step numerical methos for parabolic differential equations, BIT 21, (1981), 80-96.
[57] Delsarte, Ph.; Genin, Y.; Kamp, Y., Proof of the Daniel-Moore conjectures for A-stable multistep two derivative integration formulas, Philips Journal of Research 36 (1981), 77-86.
[58] Zhongcheng Wang and Yongming Dai, "A twelfth order two-step method for the numerical integration of one-dimensional Schr(?)dinger equation.", Int. J. Mod. Phys. C., 14 (2003), 1087-1105.
[59] Zhongcheng Wang and Yongming Dai, An Eighth-Order Two-Step Formula for the Numerical Integration of the One-Dimensional Schr(?)dinger Equation, N. Math. J. Chin. Univ., 12 (2003), 146-150.
[60] Zhongcheng Wang, Yonghua Ge, Yongming Dai and Deyin Zhao, A Mathematica program for the two-step twelfth-order method with multi-derivative for the numerical solution of a one-dimensional Schrodinger equation, Comput. Phys. Commun., 160 (2004), 23-45.
[61] Dai, Yongming; Wang, Zhongcheng; Zhao, Deying; Song, Xiaolong, A new high efficient and high accurate Obrechkoff four-step method for the periodic nonlinear undamped Duffing's equation, Comput. Phys. Commun., 165 (2005), 110-126.
[62] Yongming Dai, Zhongcheng Wang and Dongmei Wu, A four-step trigonometric fitted P-stable Obrechkoff method for periodic initial-value problems, J. Comput & Appl. Math., 187, 2, March, 2006, 192-201.
[63] Wang, Zhongcheng, Wang, Yuan, A new kind of high-efficient and high-accurate P-stable Obrechkoff three-step method for periodic initial-value problems, Compt. Phys. Commun., 171, n 2, Sep 15, 2005, 79-92.
[64] Zhao, Deyin; Wang, Zhongcheng; Dai, Yongming, Importance of the first-order derivative formula in the Obrechkoff method, Compt. Phys. Commun. 167 (2005), 65-75.
[65] Deyin Zhao, Zhongcheng Wang, Yongming Dai, Yuan Wang, A high-accurate and efficient obrechkoff five-step method for undamped Duffing's equation, Int. J. Mod. Phys. C. 16 (2005), 1027-1041.
[66] Zhongcheng Wang and Yongming Dai, A highly accurate and efficient trigonometrically-fitted P-stable three-step method for periodic intial-value problems, J. Mod. Phys. C., 14 (2005), accepted.
[67] Wang, Zhongcheng; Chen, Qimang, A trigonometrically-fitted one-step method with multi-deriative for the numerical solution to the one-dimensional Schrodinger equation, Comput. Phys. Commun., 170 (2005), 49-64.
[68] Zhongcheng Wang, P-stable linear symmetric multistep methods for periodic initial-value problems, Comput. Phys. Commun., 171 (2005), 162-174.
[69] Kuznetsov, Dmitriy F.,The three-step strong numerical methods of the orders of accuracy 1.0 and 1.5 for Ito stochastic differential equations, Journal of Automation and Information Sciences, 34, n 9-12, 2002, 22-35.
[70] Xie, Jun; Zhou, Yishu;Han, Dajian, Three steps method integrated with statistical techniques for structural damage diagnosis, Zhendong yu Chongji/Journal of Vibration and Shock, 23, n 4, December, 2004, 119-122.
[71] Ford, J.A.; Tharmlikit, S., Three-step fixed-point quasi-Newton methods for unconstrained optimization, Computers and Mathematics with Applications, 50, n 7, October, 2005, Numer. Meth. Comput. Mech., 1051-1060.
[72] Kim, Jong-Nam; Choi, Tae-Sun, Fast three step search algorithm with minimum checking points, Digest of Technical Papers - IEEE International Conference on Consumer Electronics, 1998, TUPM 9.4, 132-133.
[73] Brusa, L. & Nigro, L., A one-step method for direct integration of structural dynamic equations, Int. J. Numer. Meth. Engng., 15 (1980), 553-559.
[74] Cash, J. R.,Higher Order P-stable formulae for the numerical integration of periodic initial value problems, Numer. Math., 37 (1981), 355-370.
[75] Chawla, M. M., Two-step fourth-order P-stable method for second order differential equations, BIT 21 (1981), 190-193
[76] Chawla, M.M & Neta, Families of two-step fourth-order P-stable methods for second order differential equations, J. Comput. Appl. Math., 15 (1986), 213-223.
[77] Chawla, M.M, Rao, P. S & Neta, B. Two-step fourth-order P-stable methods with phase-lag of order six for y"=f(t,y), J. Comput. Appl. Math., 16 (1986), 233-236.
[78] Simos T. E, "A two-step method with phase-lag of order infinity for the numerical integration of second order periodic initial-value problem." Int. J. Comput. Math., 39 1991 135-140.
[79] Simos T. E., Explicit two-step methods with minimal phase-lag for the numerical integration of special second-order initial-value problems and their application to the one-dimensional schroedinger equation, J. Comput. Appl. Math., 39 (1992), 89-94.
[80] Hairer, E., Unconditionally stable methods for second order differential equations, Numer. Math., 32 (1979), 373-379.
[81] Chawla, M.M & Rao, P. S., A Numerov-type method with minimal phase-lag for the integration of second order periodic initial-value problems. Ⅱ: Explicit method, J. Comput. Appl. Math., 15 (1986), 329-337.
[82] Chawla, M.M & Rao, P. S., Explicit sixth-order method with phase-lag of order eight for y" = f(t,y), J. Comput. Appl. Math., 17 (1987), 365-368.
[83] Raptis, A. D & Simos T. E. "A four-step phase fitted method for the numerical integration of second order initial-value problem." BIT 31 (1991) 160-168.
[84] Simos T. E. "Accurate exponentially-fitted four-step method for the numerical solution of the radial Schrodinger equation." Molecular Simulation, 20, n 5, (1998), 285-301.
[85] Ixaru, L.Gr.; Vanden Berghe, G.; De Meyer, H and Van Daele, M., Four-step exponential-fitted methods for nonlinear physical problems, Compt. Phys. Commun., 100 (1997), 56-70.
[86] Simos T. E., Four-step method for the numerical solution of the Schroedinger equation, J of Compt and Appl. Math., 30 (1990), 251.
[87] Simos T. E., Explicit four-step phase-fitted method for the numerical integration of second-order initial-value problems, J of Compt and Appl. Math., 55 (1994), 125.
[88] Van Daele, M.; Berghe, G. V.; De Meyer, H and Ixaru, L.G., Exponential-fitted four-step methods for y' = f(x, y), Int. J. of Compt. Math., 66 (1998), 299-309.
[89] Simos T. E., Numerov-type method for the numerical solution of the radial Schroedinger equation, Appl. Numer. Math., 7 (1991) 201-206.
[90] Simos T. E. & Raptis, A. D., Numerov-type methods with minimal phase-lag for the numerical integration of the one-dimensional Schrodinger equation, Computing, 45 (1990), 175-181.
[91] Simos T. E. "A P-stable complete in phase Obrechkoff trigonometric fitted method for periodic initial-value problems", Proc. R. Soc. London A, 6 (1993) 283.
[92] R. van Dooren, Stabilization of Cowell's classical finite difference method for numerical integration, J. Comput. Phys., 16 (1974), 186-192.
[93] Lambert, J. D., Computational methods in ordinary differential equations, John wiley&Sons, London (1973).
[94] Widlund, O. B. "A note on unconditionally stable linear multistep methods", BIT 7 (1967), I, 65-70.
[95] Gear, C. W., The automatic integration of stiff ordiary differential equations, Information Processing 68, ed., A, J. H. Morrell, North Holland Publishing Co., 1969 187-193.
[96] Cryer, C. W., On the instability of high order backward-difference multistep methods, BIT, 12 (1972), I, 17-25.
[97] Cash, J.R., Modified extended backward differentiation formulae for the numerical solution of stiff initial value problems in ODEs and DAEs, J. of Compt. and Appl. Math., 125 (2000), 117-130.
[98] Jackson, Kenneth R., Numerical solution of large systems of stiff IVPs for ODEs, Appl. Numer. Math., 20(1996), 5-20.
[99] Guillard, Herve , Peyret, Roger, On the use of spectral methods for the numerical solution of stiff problems, Comput. Meth. Appl. Mech. Engng., 66 (1988), 17-43.
[100] Novati, P., An Explicit One-step Method for Stiff Problems, Computing (Vienna/New York), 71 (2003), 133-151.
[101] Bond, J. E.; Cash, J. R., block method for the numerical integration of stiff systems of ordinary differential equations, BIT (Copenhagen), 19 (1979), 429-447.
[102] Huang, Chengming , Chang, Qianshun; Xiao, Aiguo, B-convergence of general linear methods for stiff problems, Appl. Numer. Math., 47 (2003), 31-44.
[103] J.C.P. Miller, The Airy integral, In Brit. Assoc. Adv. Sci. Math. Tables, Volume B, Cambridge Press, (1946).
[104] D. Barton, I.M. Willem and R.V.M. Zahar, The automatic solution of ordinary differential equations by the method of Taylor series, Comput J., 14, (1971)243-248.
[105] H.J. Han, The applicability of Taylor series methods in simulation, In Proc. 1983 Summer Computer Simulation Conference, July 10-13, (1983), Vancouver, B.C.
[106] R.E. Moore, Interval Analysis, Prentice Hall, Englewood Cliffs, NJ, (1966).
[107] R.J. Lohner, Enclosing the solutions of ordinary initial and boundary value problems, In Computer Arith metic, Scientific Computation, and Programming Languages, (Edited by E. Kaucher, U. Kulisch and Ch. U M ~*, 255-286, Teubner, Stuttgart, (1987).
[108] L.B. Rail, Automatic differentiation: Techniques and applications, In Springer Lecture Notes in Computer Science, 120, Springer-Verlag, Berlin, (1981).
[109] Y.F. Chang, Automatic solution of differential equations, In Springer Lecture Notes in Math., Constructive and Computational Methods for Differential and Integral Equations, (Edited by D.L. Colton and R.P. Gilbert) 430, 61-94, Springer-Verlag, New York, (1974).
[110] Xin Yuan, Wu, A sixth-order A-stable explicit one-step method for stiff problem systems, Comput. Math. Appl., 35 (1998) 59-64.
[2] C. Runge, Uber die numerische Aufosung von Differentialgleichungen, Math. Ann., 46 (1895) 167-178.
[3] K. Heun, Neue Methoden zur approximativen Integration der Differential gleichungen einer unabhangigen Veranderlichen, Z. Math. Phys., 45 (1900) 23-38.
[4] W. Kutta, Beitrag zur naherungsweisen Integration totaler Differential gleichungen, Z. Math. Phys., 46 (1901) 435-453.
[5] E. J. Nystrom, Uber die numerische Integration von Differentialgleichungen, Acta Soc. Sci. Fennicae, 50 (13) (1925).
[6] F.R. Moulton, New Methods in Exterior Balistics, University of Chicago, Chicago 1926.
[7] G. Dahlquist, Convergence and stability in the numerical integration of ordinary differential equations, Math. Scand., 4 (1956) 33-53.
[8] G. Dahlquist, Stability and error bounds in the numerical integration of rdinary differential equations, Kungl. Tekn. Hogsk. Handl. Stockholm 130 (1959)1-87.
[9] P. Albrecht, A new theoretical approach to Runge-Kutta methods, SIAM J. Numer. Anal., 24 (1987) 391-406.
[10] R. Alt, Deux theorems sur la A-stabilite des schemas de Runge-Kutta simplement implicites, Rev. Francais d'Automat. Recherche Operationelle Ser. R-36 (1972) 99-104.
[11] J.C. Butcher, Coefficients for the study of Runge-Kutta integration processes, J. Austral. Math. Soc., 3 (1963) 185-201.
[12] J.C. Butcher, An algebraic theory of integration methods, Math. Comp., 26 (1972) 79-106.
[13] R. Frank, J. Schneid, C.W. Ueberhuber, The concept of B-convergence, SIAM J. Numer. Anal., 18 (1981) 753-780.
[14] R. Alexander, Diagonally implicit Runge-Kutta methods for stiff ODEs, SIAM J. Numer. Anal., 14 (1977) 1006-1021.
[15] K. Burrage, A special family of Runge-Kutta methods for solving stiff differential equations, BIT 18 (1978)22-41.
[16] K. Burrage, J.C. Butcher, Stability criteria for implicit Runge-Kutta methods, SIAM J. Numer. Anal., 16 (1979) 46-57.
[17] J.C. Butcher, Implicit Runge-Kutta processes, Math. Comp., 18 (1964) 50-64.
[18] J.C. Butcher, On the attainable order of Runge-Kutta methods, Math. Comput., 19 (1965) 408-417.
[19] J.C. Butcher, The non-existence of ten stage eighth order explicit Runge-Kutta methods, BIT 25 (1985) 521:540.
[20] J.C. Butcher, A stability property of implicit Runge-Kutta methods, BIT 15 (1975) 358-361.
[21] M. Crouzeix, Sur les methodes de Runge-Kutta pour l'approximation des problemes d'evolution, Lecture Notes in Economics and Mathematical Systems, 134, Springer, Berlin, 1974, 206-223.
[22] E. Hairer, C. Lubich, M. Roche, in the Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods, Lecture Notes in Mathematics, 1409, Springer, Berlin, 1989.
[23] Houwen, P. J. van der& Sommeijer, B. P., Explicit Runge-Kutta (-Nystrom) methods with reduced phase errors for computing oscillating solutions. SIAM J. Numer. Anal., 24 1987 595-617
[24] C.W. Gear, Hybrid methods for initial value problems in ordinary differential equations, SIAM J. Numer. Anal., 2 (1965) 69-86.
[25] S. Gill, A process for the step-by-step integration of differential equations in an automatic digital computing machine, Proc. Cambridge Philos. Soc, 47 (1951) 96-108.
[26] R.H. Merson, An operational method for the study of integration processes, Proceedings of the Symposium on Data Processing, Weapons Research Establishment, Salisbury, South Australia, 1957.
[27] A. Prothero, A. Robinson, On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations, Math Comp., 28 (1974) 145-162.
[28] D. Barton, I.M. Willers, R.V.M. Zahar, The automatic solution of ordinary differential equations by the method of Taylor series, Comput. J., 14 (1971) 243-248.
[29] J.C. Butcher, A modified multistep method for the numerical integration of ordinary differential equations, J. Assoc. Comput. Mach., 12 (1965) 124-135.
[30] Kesan, Cenk, Taylor polynomial solutions of linear differential equations, Appl. Math. and Compt., 142 (2003) 155-165.
[31] Tian, T.; Burrage, K., Implicit Taylor methods for stiff stochastic differential equations, Appl. Numer. Math., 38 (2001) 167-185.
[32] Nedialkov, N.S.; Jackson, K.R.; Corliss, G.F. .Validated solutions of initial value problems for ordinary differential equations, Appl. Math. and Compt., 105 (1999)21-68.
[33] Ma, Hang; Qin, Qing-Hua, A second-order scheme for integration of one-dimensional dynamic analysis, Compt. and Math. Appl., 49 (2005) 239-252.
[34] Bulos, B.; Khuri, Suheil A., On the modified Taylor's approximation for the solution of linear and nonlinear equations, Appl. Math. and Compt., 160 (2005) 939-953.
[35] N. Obrechkoff, On mechanical quadrature (Bulgarian, French summary), Spisanie Bulgar. Akad. Nauk., 65 (1942) 191-289.
[36] Jackson, Kenneth R.; Nedialkov, Nedialko S. ,Some recent advances in validated methods for IVPs for ODEs, Appl. Num. Math., 42 (2002) 269-284.
[37] M. Crouzeix, Sur la B-stabilite des methodes de Runge-Kutta, Numer. Math., 32 (1979) 75-82.
[38] C.F. Curtiss, J.O. Hirschfelder, Integration of stiff equations, Proc. Nat. Acad. Sci., 38 (1952) 235-243.
[39] G. Dahlquist, A special stability property for linear multistep methods, BIT 3 (1963)27-43.
[40] G. Dahlquist, G-stability is equivalent to A-stability, BIT 18 (1978) 384-401.
[41] C.W. Gear, Hybrid methods for initial value problems in ordinary differential equations, SIAM J. Numer. Anal., 2 (1965) 69-86.
[42] C.W. Gear, Algorithm 407, DIFSUB for solution of ordinary differential equations, Comm. ACM., 14 (1971) 447-451.
[43] W.E. Milne, A note on the numerical integration of dffierential equations, J. Res. Nat. Bur. Standards, 43 (1949) 537-542.
[44] MAURY, J. L. and SEGAL, G. P. "Cowell Type Numerical Integration as Applied to Satellite Orbit Computation," Technical Report X-553-69-46, NASA, 1969,NTIS #N6926703.
[45] Lambert J. D., and Mitchell, A. R., On the solution of y'= f(x,y)by a class of high accuracy difference formulae of low order, Z. Angew. Math. Phys., 13, 1962,223-232.
[46] J. D. Lambert & I. A. Watson, Symmetric multistep methods for periodic initial value problems, J. Inst. Math. Applic., 18 (1976), 189-202.
[47] Henrici, P., Discrete variable methods for ordinary differential equations, Wiley, New York, 1962.
[48] Coleman, J. P. Numerical methods for y" = f(x,y)via rational approximation for the cosin, IMA J. Numer. Anal., 9 (1989) 145-165
[49] U. Ananthakrishnaiah, Adaptive methods for periodic initial value problems of second order differential equations, J. Comput. Appl. Math., 8 1982 101-104
[50] U. Ananthakrishnaiah, P-Stable Obrechkoff methods with minimal phase-lag for periodic initial-value problems, Math. Comput., 49 (1987), 553-559.
[51] Rai, A. Sesappa; U. Ananthakrishnaiah, Obrechkoff method having additional parameters for general second-order differential equations, J Comput. Appl. Math., 79(1997), 167-182.
[52] Neta, B.; Fukushima, T., Obrechkoff versus super-implicit methods for the solution of first-and second-order initial value problems, Compt. and Math. Appl., 45 (2003), 383-390.
[53] Dimovski, Ivan; Kiryakova, Virginia, Obrechkoff integral transform:Properties and relation to a generalized fractional calculus, Numerical Functional Analysis and Optimization, 21 (2000), 121-144.
[54] Stapleton, David P., Numerical simulation of a rigid rotating body by Obrechkoff integration, Compt. Phys. Comm., 98 (1996), 153-166.
[55] Iseries, Arieh, Composite methods for numerical solution of stiff systems of ODEs, SIAM J. Numer. Anal., 21 (1984), 340-351.
[56] Iseries, A, Two-step numerical methos for parabolic differential equations, BIT 21, (1981), 80-96.
[57] Delsarte, Ph.; Genin, Y.; Kamp, Y., Proof of the Daniel-Moore conjectures for A-stable multistep two derivative integration formulas, Philips Journal of Research 36 (1981), 77-86.
[58] Zhongcheng Wang and Yongming Dai, "A twelfth order two-step method for the numerical integration of one-dimensional Schr(?)dinger equation.", Int. J. Mod. Phys. C., 14 (2003), 1087-1105.
[59] Zhongcheng Wang and Yongming Dai, An Eighth-Order Two-Step Formula for the Numerical Integration of the One-Dimensional Schr(?)dinger Equation, N. Math. J. Chin. Univ., 12 (2003), 146-150.
[60] Zhongcheng Wang, Yonghua Ge, Yongming Dai and Deyin Zhao, A Mathematica program for the two-step twelfth-order method with multi-derivative for the numerical solution of a one-dimensional Schrodinger equation, Comput. Phys. Commun., 160 (2004), 23-45.
[61] Dai, Yongming; Wang, Zhongcheng; Zhao, Deying; Song, Xiaolong, A new high efficient and high accurate Obrechkoff four-step method for the periodic nonlinear undamped Duffing's equation, Comput. Phys. Commun., 165 (2005), 110-126.
[62] Yongming Dai, Zhongcheng Wang and Dongmei Wu, A four-step trigonometric fitted P-stable Obrechkoff method for periodic initial-value problems, J. Comput & Appl. Math., 187, 2, March, 2006, 192-201.
[63] Wang, Zhongcheng, Wang, Yuan, A new kind of high-efficient and high-accurate P-stable Obrechkoff three-step method for periodic initial-value problems, Compt. Phys. Commun., 171, n 2, Sep 15, 2005, 79-92.
[64] Zhao, Deyin; Wang, Zhongcheng; Dai, Yongming, Importance of the first-order derivative formula in the Obrechkoff method, Compt. Phys. Commun. 167 (2005), 65-75.
[65] Deyin Zhao, Zhongcheng Wang, Yongming Dai, Yuan Wang, A high-accurate and efficient obrechkoff five-step method for undamped Duffing's equation, Int. J. Mod. Phys. C. 16 (2005), 1027-1041.
[66] Zhongcheng Wang and Yongming Dai, A highly accurate and efficient trigonometrically-fitted P-stable three-step method for periodic intial-value problems, J. Mod. Phys. C., 14 (2005), accepted.
[67] Wang, Zhongcheng; Chen, Qimang, A trigonometrically-fitted one-step method with multi-deriative for the numerical solution to the one-dimensional Schrodinger equation, Comput. Phys. Commun., 170 (2005), 49-64.
[68] Zhongcheng Wang, P-stable linear symmetric multistep methods for periodic initial-value problems, Comput. Phys. Commun., 171 (2005), 162-174.
[69] Kuznetsov, Dmitriy F.,The three-step strong numerical methods of the orders of accuracy 1.0 and 1.5 for Ito stochastic differential equations, Journal of Automation and Information Sciences, 34, n 9-12, 2002, 22-35.
[70] Xie, Jun; Zhou, Yishu;Han, Dajian, Three steps method integrated with statistical techniques for structural damage diagnosis, Zhendong yu Chongji/Journal of Vibration and Shock, 23, n 4, December, 2004, 119-122.
[71] Ford, J.A.; Tharmlikit, S., Three-step fixed-point quasi-Newton methods for unconstrained optimization, Computers and Mathematics with Applications, 50, n 7, October, 2005, Numer. Meth. Comput. Mech., 1051-1060.
[72] Kim, Jong-Nam; Choi, Tae-Sun, Fast three step search algorithm with minimum checking points, Digest of Technical Papers - IEEE International Conference on Consumer Electronics, 1998, TUPM 9.4, 132-133.
[73] Brusa, L. & Nigro, L., A one-step method for direct integration of structural dynamic equations, Int. J. Numer. Meth. Engng., 15 (1980), 553-559.
[74] Cash, J. R.,Higher Order P-stable formulae for the numerical integration of periodic initial value problems, Numer. Math., 37 (1981), 355-370.
[75] Chawla, M. M., Two-step fourth-order P-stable method for second order differential equations, BIT 21 (1981), 190-193
[76] Chawla, M.M & Neta, Families of two-step fourth-order P-stable methods for second order differential equations, J. Comput. Appl. Math., 15 (1986), 213-223.
[77] Chawla, M.M, Rao, P. S & Neta, B. Two-step fourth-order P-stable methods with phase-lag of order six for y"=f(t,y), J. Comput. Appl. Math., 16 (1986), 233-236.
[78] Simos T. E, "A two-step method with phase-lag of order infinity for the numerical integration of second order periodic initial-value problem." Int. J. Comput. Math., 39 1991 135-140.
[79] Simos T. E., Explicit two-step methods with minimal phase-lag for the numerical integration of special second-order initial-value problems and their application to the one-dimensional schroedinger equation, J. Comput. Appl. Math., 39 (1992), 89-94.
[80] Hairer, E., Unconditionally stable methods for second order differential equations, Numer. Math., 32 (1979), 373-379.
[81] Chawla, M.M & Rao, P. S., A Numerov-type method with minimal phase-lag for the integration of second order periodic initial-value problems. Ⅱ: Explicit method, J. Comput. Appl. Math., 15 (1986), 329-337.
[82] Chawla, M.M & Rao, P. S., Explicit sixth-order method with phase-lag of order eight for y" = f(t,y), J. Comput. Appl. Math., 17 (1987), 365-368.
[83] Raptis, A. D & Simos T. E. "A four-step phase fitted method for the numerical integration of second order initial-value problem." BIT 31 (1991) 160-168.
[84] Simos T. E. "Accurate exponentially-fitted four-step method for the numerical solution of the radial Schrodinger equation." Molecular Simulation, 20, n 5, (1998), 285-301.
[85] Ixaru, L.Gr.; Vanden Berghe, G.; De Meyer, H and Van Daele, M., Four-step exponential-fitted methods for nonlinear physical problems, Compt. Phys. Commun., 100 (1997), 56-70.
[86] Simos T. E., Four-step method for the numerical solution of the Schroedinger equation, J of Compt and Appl. Math., 30 (1990), 251.
[87] Simos T. E., Explicit four-step phase-fitted method for the numerical integration of second-order initial-value problems, J of Compt and Appl. Math., 55 (1994), 125.
[88] Van Daele, M.; Berghe, G. V.; De Meyer, H and Ixaru, L.G., Exponential-fitted four-step methods for y' = f(x, y), Int. J. of Compt. Math., 66 (1998), 299-309.
[89] Simos T. E., Numerov-type method for the numerical solution of the radial Schroedinger equation, Appl. Numer. Math., 7 (1991) 201-206.
[90] Simos T. E. & Raptis, A. D., Numerov-type methods with minimal phase-lag for the numerical integration of the one-dimensional Schrodinger equation, Computing, 45 (1990), 175-181.
[91] Simos T. E. "A P-stable complete in phase Obrechkoff trigonometric fitted method for periodic initial-value problems", Proc. R. Soc. London A, 6 (1993) 283.
[92] R. van Dooren, Stabilization of Cowell's classical finite difference method for numerical integration, J. Comput. Phys., 16 (1974), 186-192.
[93] Lambert, J. D., Computational methods in ordinary differential equations, John wiley&Sons, London (1973).
[94] Widlund, O. B. "A note on unconditionally stable linear multistep methods", BIT 7 (1967), I, 65-70.
[95] Gear, C. W., The automatic integration of stiff ordiary differential equations, Information Processing 68, ed., A, J. H. Morrell, North Holland Publishing Co., 1969 187-193.
[96] Cryer, C. W., On the instability of high order backward-difference multistep methods, BIT, 12 (1972), I, 17-25.
[97] Cash, J.R., Modified extended backward differentiation formulae for the numerical solution of stiff initial value problems in ODEs and DAEs, J. of Compt. and Appl. Math., 125 (2000), 117-130.
[98] Jackson, Kenneth R., Numerical solution of large systems of stiff IVPs for ODEs, Appl. Numer. Math., 20(1996), 5-20.
[99] Guillard, Herve , Peyret, Roger, On the use of spectral methods for the numerical solution of stiff problems, Comput. Meth. Appl. Mech. Engng., 66 (1988), 17-43.
[100] Novati, P., An Explicit One-step Method for Stiff Problems, Computing (Vienna/New York), 71 (2003), 133-151.
[101] Bond, J. E.; Cash, J. R., block method for the numerical integration of stiff systems of ordinary differential equations, BIT (Copenhagen), 19 (1979), 429-447.
[102] Huang, Chengming , Chang, Qianshun; Xiao, Aiguo, B-convergence of general linear methods for stiff problems, Appl. Numer. Math., 47 (2003), 31-44.
[103] J.C.P. Miller, The Airy integral, In Brit. Assoc. Adv. Sci. Math. Tables, Volume B, Cambridge Press, (1946).
[104] D. Barton, I.M. Willem and R.V.M. Zahar, The automatic solution of ordinary differential equations by the method of Taylor series, Comput J., 14, (1971)243-248.
[105] H.J. Han, The applicability of Taylor series methods in simulation, In Proc. 1983 Summer Computer Simulation Conference, July 10-13, (1983), Vancouver, B.C.
[106] R.E. Moore, Interval Analysis, Prentice Hall, Englewood Cliffs, NJ, (1966).
[107] R.J. Lohner, Enclosing the solutions of ordinary initial and boundary value problems, In Computer Arith metic, Scientific Computation, and Programming Languages, (Edited by E. Kaucher, U. Kulisch and Ch. U M ~*, 255-286, Teubner, Stuttgart, (1987).
[108] L.B. Rail, Automatic differentiation: Techniques and applications, In Springer Lecture Notes in Computer Science, 120, Springer-Verlag, Berlin, (1981).
[109] Y.F. Chang, Automatic solution of differential equations, In Springer Lecture Notes in Math., Constructive and Computational Methods for Differential and Integral Equations, (Edited by D.L. Colton and R.P. Gilbert) 430, 61-94, Springer-Verlag, New York, (1974).
[110] Xin Yuan, Wu, A sixth-order A-stable explicit one-step method for stiff problem systems, Comput. Math. Appl., 35 (1998) 59-64.