numerical solutions of ordinary differential equations

... more 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 ... 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 ... of the basic Euler’s Method, resulting in a whole family of algorithms for the numerical solution of ODEs – – – Improved Euler’s Method Modified Euler’s Method Runge-Kutta algorithms School of...

Ngày tải lên: 19/03/2015, 08:49

... 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) ... 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 READING: Gear, C.W 1971, Numerical...

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 ... (Washington: National Bureau of Standards; reprinted 1968 by Dover Publications, New York), §25.5 [1] Gear, C.W 1971, Numerical Initial Value Problems in Ordinary Differential Equations (Englewood Cliffs,...

Ngày tải lên: 15/12/2013, 04:15

... 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 ... controls are merely indicative of what you might need The routine odeint should be customized to the problem at hand 722 Chapter 16 Integration of Ordinary Differential Equations } nrerror("Too many...

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 ... high-accuracy solutions to ordinary differential equations with minimal computational effort (A possible exception, infrequently encountered in practice, is discussed in §16.7.) Sample page from NUMERICAL ... 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...

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 ... q,f2,f1,delta,*c; 732 Chapter 16 Integration of Ordinary Differential Equations } CITED REFERENCES AND FURTHER READING: Stoer, J., and Bulirsch, R 1980, Introduction to Numerical Analysis (New York: Springer-Verlag),...

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 ... from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software...

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 ... 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 ... #define C2X C3X C4X A2X A3X Integration of Ordinary Differential Equations -0.396296677520e-01 0.550778939579 -0.553509845700e-01 0.462 0.880208333333 As an example of how stiff is used, one can solve...

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 ... 1971, Numerical Initial 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 ... 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

... 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 ... Bruti-Liberati Numerical Solution of Stochastic Differential Equations with Jumps in Finance Eckhard Platen Nicola Bruti-Liberati (1975–2007) School of Finance and Economics Department of Mathematical ... about the numerical solution of stochastic differential equations on the webpage of the first author under Numerical Methods”: http://www.business.uts.edu.au/ finance/staff/Eckhard /Numerical Methods.html...

Ngày tải lên: 19/02/2014, 22:20

... Journal of Inequalities and Applications For the special case that n and p 1, various problems on the solutions of 1.1 , such as the existence of periodic solutions, bifurcations of periodic solutions, ... 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 solutions ... “Existence of periodic solutions of one-dimensional differential-delay equations, ” Tohoku Mathematical Journal, vol 30, no 1, pp 13–35, 1978 S Chapin, “Periodic solutions of differential-delay equations...

Ngày tải lên: 22/06/2014, 02:20

... Numerical Methods for Ordinary Differential Equations Numerical Methods for Ordinary Differential Equations, Second Edition J C Butcher © 2008 John Wiley & Sons, Ltd ISBN: 978-0-470-72335-7 Numerical ... subject of numerical methods for ordinary differential equations Chapter contains a detailed analysis of Runge–Kutta methods It includes studies of the order, stability and convergence of Runge–Kutta ... range of the function f which Numerical Methods for Ordinary Differential Equations, Second Edition J C Butcher © 2008 John Wiley & Sons, Ltd ISBN: 978-0-470-72335-7 NUMERICAL METHODS FOR ORDINARY...

Ngày tải lên: 27/06/2014, 08:20

... 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 ... Bruti-Liberati Numerical Solution of Stochastic Differential Equations with Jumps in Finance Eckhard Platen Nicola Bruti-Liberati (1975–2007) School of Finance and Economics Department of Mathematical ... about the numerical solution of stochastic differential equations on the webpage of the first author under Numerical Methods”: http://www.business.uts.edu.au/ finance/staff/Eckhard /Numerical Methods.html...

Ngày tải lên: 28/06/2014, 17:20

... Delay -differential equations In this thesis we are concerned with the numerical solution of delay -differential equations (DDE's) Delay -differential equations may best be regarded as extensions of ordinary ... we consider the development of numerical software for the solution of such problems Our discussion opens with a brief introduction to the theory of delay -differential equations Attention is paid ... solution of ordinary differential equations (ODE's), we then consider the application of ODE software to evolutionary DDE's Special attention is paid to the occurrence, effect and accommodation of...

Ngày tải lên: 11/10/2014, 04:02

... of the root location algorithm discussed in (2.7) Bibliography 415 418 291 appendix A a formal proof for (1.1.3:14) 292 A formal proof for (1.1.3:14) In this appendix we give a formal proof of ... transitive closure of E Heuristically, by an extension of the above arguments, we may think of Ei as defining the set of points whose continuity bounds can affect the continuity bound, ci , of yi El ... the notation of (5), s q Es q+ whereas t q ,Et 1+ Thus k' < k cannot be true and the result then follows [] We can now use this result to prove part (ii) of Proposition 1: PROOF of Proposition...

Ngày tải lên: 11/10/2014, 04:03

... are often investigated in various fields of science and technology The question of the existence and uniqueness of almost periodic solutions of differential equations is an age-old problem of great ... http://www.springer.com/series/304 2047 a Gani T Stamov Almost Periodic Solutions of Impulsive Differential Equations 123 Gani T Stamov Technical University of Sofia Department of Mathematics Sliven Bulgaria ISBN 978-3-642-27545-6 ... examples considered the systems of impulsive differential equations are given by means of a system of differential equations and conditions of jumps A brief description of impulsive systems is given...

Ngày tải lên: 29/10/2015, 12:30

... 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 ... 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

Ngày tải lên: 14/03/2014, 13:20

... meromorphic 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 ... order of entire solutions of some algebraic differential equations and improve the related results of Bergweiler, Barsegian, and others We also estimate the growth order of entire solutions of a ... 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...

Ngày tải lên: 20/06/2014, 20:20

... existence and uniqueness of solutions for initial value problem of fractional differential equations J Univ Jinan 2010, 24:312-315 Page 12 of 13 Zhao et al Advances in Difference Equations 2011, 2011:10 ... D: Existence of positive solutions for boundary value problems of fractional differential equations J Univ Jinan 2010, 24:205-208 17 Zhao Y, Sun S: On the existence of positive solutions for ... uniqueness of solutions for coupled systems of higher-order nonlinear fractional differential equations Fixed Point Theory Appl 2010, 2010:1-17 30 Babakhani A: Positive solutions for system of nonlinear...

Ngày tải lên: 21/06/2014, 03:20