COMPUTATIOAL METHODS AD ALGORITHMS Vol. I umerical Methods for Ordinar Differential Equations and Dnamic Sstems E.A. oviov UMERICAL METHODS FOR ORDIARY DIFFERETIAL EQUATIOS AD DYAMIC SYSTEMS E.A. oviov Institute of Computational Modeling, Russian Academ of Sciences, Krasnoars, Russia Kewords: Analtical methods, Talor series, successive approximations, ewton Kantorovich method, perturbation method, error estimation, Euler method, trapezoid method, RungeKutta method, Adams formulae, stiff sstems, dela differential equations Contents 1. Introduction. Dnamic Sstems 3. Analtical Methods 4. Onestep Methods 5. Stiff Sstems 6. Linear Multistep Methods 7. Error Estimation 8. Dela Differential Equations Glossar Bibliograph Biographical Setch Summar Approximate methods of solution of Cauch problem for sstems of ordinar differential equations, including dela differential equations, are described. Principal methods of local and global error estimation as well as the inequalities for accurac control of computations and stabilit control of numerical schemes are considered. Realization of variable step algorithms and algorithms for nonuniform schemes (explicit and Lstable) with automatic selection of numerical scheme b inequalit for stabilit control is briefl discussed. Examples of particular numerical methods are given. 1. Introduction UESCO EOLSS Earl analsis of ordinar differential Eqs. (ODE) can be traced towards the end of the 17 th centur in connection with the stud of problems of mechanics and certain geometric problems. The unnown in these equations is a function of one independent variable, and the equations include not onl the unnown function but its derivatives of diverse orders as well. ODEs are widel used in mechanics, astronom, phsics, biolog, chemistr etc. It is explained b the fact that such equations often quantitativel are based on phsical laws, which describe some phenomena. Schematic design of electronic schemes,
COMPUTATIOAL METHODS AD ALGORITHMS Vol. I umerical Methods for Ordinar Differential Equations and Dnamic Sstems E.A. oviov modeling of inetics of chemical reactions and computation of dnamics of mechanical sstems is a far from complete list of the problems described b ODE. A large class of problems arises from the approximation of timedependent problems b finite elements or finite differences. In turn, applied problems serve as a source for new problem statement in the theor of differential equations. For instance, in this wa the optimal control mathematical theor had been developed. Elementar ODEs occur when finding for the integral of a given continuous function f (t ), since this is a problem of finding unnown function (t ), which satisfies the equation = f (t ). For proof of solvabilit of this equation the Riemann integral theor had been developed. A natural generalization of this equation is a sstem of ODEs of the first order solved for derivative = f ( t,, (1a) where f : R R R is nown continuous vectorfunction, and R is dimensional linear real space. The relation (1a) is an abridged form of writing the sstem of equations i = f i ( t, 1,, ), 1 i. The solution of (1a) on an interval [ a, b ] is a function ϕ : [ a, b] R, which turns (1a) into identit on [ a, b]: ϕ ( t ) f ( t, ϕ( t )), t [ a, b]. In general situation Eq. (1a) has infinite set of solutions ( parametric famil of solutions). To choose one of them, a supplementar condition is required, for example, requirement for the solution to tae on a given value at the given point t : ) ( t =. (1b) The problem of finding the solution of sstem of ODEs (1a), which satisfies the initial condition (1b), is called Cauch problem or initialvalue problem. In what follows, as problem (1) the problem (1a), (1b) is understood. In order that the problem (1) would have unique solution, it is necessar to impose supplementar requirements on the function f. It is said that function f ( t, satisfies Lipshitz condition in the second argument, if there exists such constant L that for all t R and x, R the inequalit f ( t, x ) f ( t, L x is valid, where is certain norm in R. The number L is called Lipshitz constant. UESCO EOLSS If function f is continuous in the first argument and satisfies Lipshitz condition in the second, then problem (1) has unique solution on an interval of the form [ t, t ], t t. A simple sufficient condition for Lipshitz condition is the following one. If function f is differentiable in the second argument and its derivative is uniforml bounded b some constant L : f ( t, / L for all ( t, R R, then the function satisfies Lipshitz condition with constant L.
COMPUTATIOAL METHODS AD ALGORITHMS Vol. I umerical Methods for Ordinar Differential Equations and Dnamic Sstems E.A. oviov. Dnamic Sstems Initiall, as dnamic sstem a mechanical sstem with finite number of degrees of freedom was understood. State of such sstem is characterized b its disposition (configuration), and law of motion determines the rate of change of the state of the sstem. In the simplest case the state is characterized b values 1,,, which can tae on arbitrar real values. Two different collections, 1, and ~ ~ 1,, correspond to different states, and proximit of i and ~ i, 1 i means closeness of corresponding states of the sstem. In this case the law of motion is written down in the form of autonomous ODE sstem. Considering the values,, as coordinates of point in dimensional space, one can geometricall represent the states of dnamic sstem b means of this point. Such space is called phase space or state space, and the change of state in time is called phase trajector or state trajector. Later on, the notion of dnamic sstem got wider interpretation and meant arbitrar phsical sstem describable b autonomous sstem of ODE. Dnamic sstem is taled about when qualitative behavior of all trajectories is considered in the whole phase space (global theor or in some its part (local theor. In the theor of dnamic sstems a special attention is paid to behavior of phase trajectories under infinitel increase of time. From particular trajectories usuall those are of interest, whose properties can influence greatl the qualitative pattern, even if local one. Since nonautonomous sstem can be reduced to autonomous b introducing supplementar variable, these two problems are not discriminated below, if the difference is not of principal case. 3. Analtical methods A large part of theor of ODE is devoted to the stud of solutions, which are not nown exactl. This is socalled qualitative theor of differential equations. It includes stabilit theor, which enables us to indicate stabilit properties of the solutions from properties of the equation without nowing the solution. For example, in the control theor there exists a need to ascertain stabilit of solution with regard to small perturbation of initial values in the whole infinite interval t t. Solution, which varies little in infinite interval [ t, ] under small perturbation of initial values, is called stable according to Liapunov. However, in man particular problems it is often necessar to find solution at ever point. Initiall the efforts were concentrated on integration of equations b quadrature methods, i.e. on obtaining formulae expressing (explicitl, implicitl or in parametric form) the dependence of solution on t via elementar functions or integrals of the latter. But in the middle of 19 th centur the first examples of ODE nonintegrable b quadrature methods had been indicated. It turned out that solution in form of formulae can be found for small number of classes of equations (Bernoulli equation, differential equation in total differentials, linear differential equation with constant coefficients). Even the simplest nonlinear equation of the first order = + t cannot be solved b quadrature methods. Therefore methods were required for computation of approximate solutions of differential equations. UESCO EOLSS 1 Historicall, the first method of solution of ODEs used b its author ewton was the method of series expansion. The desired solution is expanded into a series (for instance,
COMPUTATIOAL METHODS AD ALGORITHMS Vol. I umerical Methods for Ordinar Differential Equations and Dnamic Sstems E.A. oviov Talor series) with unnown coefficients. This series is substituted in (1a), and from the obtained equation the coefficients are determined. If function f is analtic, that is, can be expanded into power series in t and : f ( t, = f + f + f t + f t + f t + f +, then solution (t ) of problem (1) 1 1 is analtic as well and can be represented in the form 11 ( t ) = + t + t + with unnown coefficients, 1,,. After substitution of the obtained expressions in (1) and equalization of coefficients at equal powers of t an infinite sstem of equations = f + f + f + = f + f + f + f, is obtained. 1 1, 1 1 1 11 1 + From the first equation 1 is found, from the second and so on. These methods require a large amount of tedious wor. For finding coefficients of Talor series it is necessar to compute derivatives of high orders of the function f ( t,. To find approximate solution, the method of successive approximations can be used. Problem (1) is equivalent to integral equation t ( t t ) = + f ( s, ( s)) ds, t [ t, t ]. () Appling to the obtained equation the method of simple iteration, starting, for instance, () from function ( t ) =, one obtains recurrentl determined sequence of functions ( n ) ( t ) = + f ( s, t t ( n 1) ( s)) ds, n = 1,,. (3) The sequence of functions n (t ) uniforml converges to the solution of Eq. (1), estimation of the rate of convergence being nown. Since this method requires large amount of computational wor, it plas mainl theoretical role for example, it is useful when proving CauchPicard theorem or when proving statements on differentiabilit of solutions with respect to parameter or initial data. UESCO EOLSS In applications, asmptotic methods of approximate solution of ODE are often used. Instead of (1) a simpler integrable problem is solved b quadrature methods, whose solutions approximate solutions of the original one. Asmptotic methods are based on distinguishing in the equation principal terms and the terms relativel small. 1 As an example the method of small parameter for equation = f ( t, ; μ) can be adduced. Asmptotic methods are used both for obtaining analtic expressions which approximate solution and for stud of qualitative behavior of the solution.
COMPUTATIOAL METHODS AD ALGORITHMS Vol. I umerical Methods for Ordinar Differential Equations and Dnamic Sstems E.A. oviov TO ACCESS ALL THE 4 PAGES OF THIS CHAPTER, Visit: http://www.eolss.net/eolsssampleallchapter.aspx Bibliograph Deer K., Verwer J.G. (1984). Stabilit of RungeKutta methods for stiff nonlinear differential equations, 33 pp. orth Holland Amsterdam ew Yor Oxford: Elsevier Science Publishers B. V. [Stabilit theor of RungeKutta methods for stiff nonlinear sstems of differential equations is studied] Hall G., Watt J.M. (ed.) (1976). Modern numerical methods for ordinar differential equations, 31 pp. Oxford: Clarendon Press. [Collective monograph, contains a detailed presentation of theor and practical applications of numerical methods of solution of differential equations] Hairer E., orsett S.P., Waner G. (1987). Solving ordinar differential equations I. onstiff Problems, 51 pp. Berlin Heidelberg: SpringerVerlag. [Theor and practice of numerical solution of nonstiff sstems of differential equations is presented, a large number of examples is included] Hairer E., Waner G. (1996). Solving ordinar differential equations II. Stiff and differentialalgebraic problems, 685 pp. Berlin Heidelberg ew Yor Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Too: SpringerVerlag. [Theor and practice of modern numerical methods of solution of stiff and algebrodifferential sstems of differential equations is studied] oviov E.A. (1997). Explicit methods for stiff sstems, 19 pp. ovosibirs: aua. (In Russian) [Theor and issues of practical realization of explicit methods as applied to solution of sstems of differential equations of intermediate stiffness are described] Stetter H.J. (1973). Analsis of discretization methods for ordinar differential equations, 461 pp. Berlin Heidelberg ew Yor: SpringerVerlag. [The boo is devoted to theoretic aspects of numerical methods of solution of Cauch problem for sstems of differential equations] Vinogradov I.M. (ed.) (1979). Mathematical encclopedia. Moscow: Sovetsaa entsilopedija, V., 898. [Qualitative theor and approximate methods of solution of differential equations are discussed] Biographical Setch UESCO EOLSS Evgenii A. oviov was born in Voronezh, Russia. He completed his Diploma in Applied mathematics at the Voronezh State Universit, Voronezh, Russia in 1978. In 198 he too the Russian degree of Candidate in Phsics and Mathematics at the Computer Center of the Russian Academ of Sciences, ovosibirs. In 199 E.A.oviov was awarded the Russian degree of Doctor in Phsics and Mathematics from Computer Center of the Russian Academ of Sciences, ovosibirs with the thesis Onestep noniterative methods of solution of stiff sstems. In 1993 he too Professor Diploma at the Chair Mathematical support of discrete devices and sstems. From 1978 to 1985 he is as a research worer at the Computing center SB RAS in ovosibirs. Since 1985 E.A. oviov is HeadScientist at the Computer Center of the Russian Academ of Sciences in Krasnoars, Russia (now renamed the Institute of Computational Modeling of the Russian Academ of Sciences). E.A. oviov is a nown specialist in the field of computational mathematics. He is author of one monograph and over 1 scientific papers, devoted to numeric methods of solution of ordinar differential equations, and their applications.