systems of ordinary differential equations

... subject of stiff equations, relevant both to ordinary differential equations and also to partial differential equations (Chapter 19) Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC ... 1973, Computational Methods in Ordinary Differential Equations (New York: Wiley) Lapidus, L., and Seinfeld, J 1971, Numerical Solution of Ordinary Differential Equations (New York: Academic Press) ... value problem Each comes with its own set of debits and credits that must be understood before it is used 710 Chapter 16 Integration of Ordinary Differential Equations CITED REFERENCES AND FURTHER...

Ngày tải lên: 15/12/2013, 04:15

... sequence of steps in identical manner Prior behavior of a solution is not used in its propagation This is mathematically proper, since any point along the trajectory of an ordinary differential ... free_vector(dym,1,n); 714 Chapter 16 Integration of Ordinary Differential Equations } CITED REFERENCES AND FURTHER READING: Abramowitz, M., and Stegun, I.A 1964, Handbook of Mathematical Functions, Applied ... email to trade@cup.cam.ac.uk (outside North America) x2 x1 712 Chapter 16 Integration of Ordinary Differential Equations yn yn + Figure 16.1.3 Fourth-order Runge-Kutta method In each step the derivative...

Ngày tải lên: 15/12/2013, 04:15

... steps each of size h) Since the basic method is fourth order, the true solution and the two numerical approximations are related by 716 Chapter 16 Integration of Ordinary Differential Equations ... generally useful stepper routine is this: One of the arguments of the routine will of course be the vector of dependent variables at the beginning of a proposed step Call that y[1 n] Let us require ... Integration of Ordinary Differential Equations for the ith equation will be taken to be ∆0 = eps × yscal[i] (16.2.8) ∆0 = h × dydx[i] (16.2.9) This enforces fractional accuracy not on the values of y...

Ngày tải lên: 24/12/2013, 12:16

... powers of h, 724 Chapter 16 Integration of Ordinary Differential Equations } CITED REFERENCES AND FURTHER READING: Gear, C.W 1971, Numerical Initial Value Problems in Ordinary Differential Equations ... method does an excellent job of feeling its way through rocky or discontinuous terrain It is also an excellent choice for quick-and-dirty, low-accuracy solution of a set of equations A second warning ... The techniques described in this section are not for differential equations containing nonsmooth functions For example, you might have a differential equation whose right-hand side involves a...

Ngày tải lên: 24/12/2013, 12:16

... method a degree of robustness for problems with discontinuities Let us remind you once again that scaling of the variables is often crucial for successful integration of differential equations The ... Second-Order Conservative Equations Usually when you have a system of high-order differential equations to solve it is best to reformulate them as a system of first-order equations, as discussed ... class of equations that occurs quite frequently in practice where you can gain about a factor of two in efficiency by differencing the equations directly The equations are second-order systems...

Ngày tải lên: 24/12/2013, 12:16

... 16 Integration of Ordinary Differential Equations Note that for compatibility with bsstep the arrays y and d2y are of length 2n for a system of n second-order equations The values of y are stored ... vol 27, pp 505–535 16.6 Stiff Sets of Equations As soon as one deals with more than one first-order differential equation, the possibility of a stiff set of equations arises Stiffness occurs in ... 16.5 Second-Order Conservative Equations 733 Here zm is y (x0 + H) Henrici showed how to rewrite equations (16.5.2) to reduce roundoff error by using the quantities ∆k ≡ yk+1 −...

Ngày tải lên: 24/12/2013, 12:16

... email to trade@cup.cam.ac.uk (outside North America) x 736 Chapter 16 Integration of Ordinary Differential Equations Of course, we give up accuracy in following the evolution towards equilibrium ... classes of higher-order methods for stiff systems: • Generalizations of the Runge-Kutta method, of which the most useful are the Rosenbrock methods The first practical implementation of these ... Chapter 16 Integration of Ordinary Differential Equations Rosenbrock Methods s y(x0 + h) = y0 + c i ki (16.6.21) i=1 where the corrections ki are found by solving s linear equations that generalize...

Ngày tải lên: 24/12/2013, 12:16

... The second of the equations in (16.7.9) is 752 Chapter 16 Integration of Ordinary Differential Equations you suspect that your problem is suitable for this treatment, we recommend use of a canned ... Value Problems in Ordinary Differential Equations (Englewood Cliffs, NJ: Prentice-Hall), Chapter [1] Shampine, L.F., and Gordon, M.K 1975, Computer Solution of Ordinary Differential Equations The ... Chapter 16 Integration of Ordinary Differential Equations For multivalue methods the basic data available to the integrator are the first few terms of the Taylor series expansion of the solution at...

Ngày tải lên: 24/12/2013, 12:16

Ngày tải lên: 14/03/2014, 13:20

... by a lot of researchers see 7–13 Most of the work contained in literature on 1.1 is the existence and multiplicity of periodic solutions However, except the questions of the existence of periodic ... inequality For other study of Wirtinger’s inequality, one may see 15 and the references therein Now, we are ready to prove our main results We first give the proof of Theorem 1.3 Proof of Theorem 1.3 From ... and Technology 20070049 References M Han, “Bifurcations of periodic solutions of delay differential equations, ” Journal of Differential Equations, vol 189, no 2, pp 396–411, 2003 R D Nussbaum,...

Ngày tải lên: 22/06/2014, 02:20

... accurate solutions The size of the integration interval also influences the accuracy of the solutions (should be less than 10% of the time constant) • • st Solutions of sets of order ODEs are done ... following set of differential equations Example: Multiple Reactions Process • Where k1=0.2 hr -1 and k2=0.1 hr -1 and at time t=0, Ca=1mol/L and Cb=Cc=0mol/L Solve the system of equations and ... interval × function of gradient • This is the basis for a family of algorithms used to provide numerical solutions of ODEs • A particular class is the Runge-Kutta algorithms School of Chemical Engineering...

Ngày tải lên: 19/03/2015, 08:49

... §1.1 Newton’s equations §1.2 Classification of differential equations §1.3 First order autonomous equations §1.4 Finding explicit solutions 11 §1.5 Qualitative analysis of first order equations 16 ... integral equations I hope that, after the previous sections, you are by now convinced that integral equations are an important tool in the investigation of differential equations Moreover, the proof of ... 85 90 Part Dynamical systems Chapter §6.1 §6.2 §6.3 §6.4 §6.5 §6.6 Dynamical systems Dynamical systems The flow of an autonomous equation Orbits and invariant sets Stability of fixed points Stability...

Ngày tải lên: 31/03/2014, 15:56

... this paper as “the nature of the vector field.” Therefore, a combination of properties of the associated vector field with the Kneser’s property of the cross sections of the solutions’ funnel is ... the continuation of solutions and the singularity of ᏷ at the point P0 , the set ᏷(P0 ) = ∅ Taking into account the nature of the vector field and the definition of the singularity of the map ᏷, this ... the analogous of (3.121) with respect to ξ00 + ξ10 instead of ξ0 holds true, if the analogous of (3.120) with respect to (3.122) ξ00 + ξ10 instead of ξ1 holds true This definition of ξ01 and ξ11...

Ngày tải lên: 22/06/2014, 22:20

... Solution of Stochastic Differential Equations with Jumps in Finance Eckhard Platen Nicola Bruti-Liberati (1975–2007) School of Finance and Economics Department of Mathematical Sciences University of ... collection of events, sigma-algebra A filtration E(X) expectation of X E(X | A) conditional expectation of X under A P (A) probability of A P (A | B) probability of A conditioned on B ∈ element of ∈ ... approximation of continuous solutions of SDEs The discrete time approximation of SDEs with jumps represents the focus of the monograph The reader learns about powerful numerical methods for the solution of...

Ngày tải lên: 19/02/2014, 22:20

... equivalence of systems of differential equations, Results of mathematic science 40 (1985) 245 (Russian) [4] M Svec, Itegral and asymptotic equivelence of two systems of diffrential equations, ... equivalence of systems of differential equations, IZV.Acad Nauk ASSR (1975) 35 (Russian) [14] C.K Sung, H.G Yoon, J.K Nam, Asymptotic equivalence between to linear differential systems, Ann Differential ... Journal of Science, Mathematics - Physics 23 (2007) 63-69 Main results 2.1 The uniformly stable of null solution of delay differential equations Let us consider the delay differential equations...

Ngày tải lên: 22/03/2014, 11:20

... of the derivations of L/k that commute with the derivation on L (3) The field LG of G-invariant elements of L is equal to k Proof An intuitive proof of (1) and (2) L is the field of fractions of ... isomorphism translates into ZL = (ra,b )GL A proof of Lemma 1.29 finishes the proof of the theorem Proof of lemma 1.29 The proof is rather similar to the one of lemma 1.23 The only thing that we have ... containing a fundamental set of solutions of L(y) = 0, whose field of constants is the same as that of k such that no proper subfield contains a fundamental set of solutions of L(y) = 1.4 The Differential...

Ngày tải lên: 31/03/2014, 15:56

... Preface The aim of this book is to deal with all of the elementary methods for obtaining explicit solutions of ordinary differential equations, and then to introduce the ideas of qualitative analysis ... discussion of the issues of existence and uniqueness of solutions, and treats the standard classes of first order differential equations that can be solved explicitly, as well as covering exact equations ... which covers reduction of order, the method of variation of constants, and series solutions Part IV turns aside from differential equations, motivating the study of difference equations by discussing...

Ngày tải lên: 12/05/2014, 03:24

Ngày tải lên: 12/06/2014, 16:08

... solutions of algebraic differential equations Acta Math Sci (in press, in Chinese) 17 [11] Gu, RM, Ding, JJ, Yuan, WJ: On the estimate of growth order of solutions of a class of systems of algebraic ... Further results of the estimate of growth of entire solutions of some classes of algebraic differential equations Qi Jianming1,3 , Li Yezhou2 and Yuan Wenjun∗3 Department of Mathematics and ... algebraic differential equations of higher order In 2009, Yuan et al [9], improved their results and gave a general estimate of order of w(z), which depends on the degrees of coefficients of differential...

Ngày tải lên: 20/06/2014, 20:20