some basic results on ordinary differential equations

... the differential equations Boundary conditions can be as simple as requiring that certain variables have certain numerical values, or as complicated as a set of nonlinear algebraic equations among ... 16.6 of this chapter treats the subject of stiff equations, relevant both to ordinary differential equations and also to partial differential equations (Chapter 19) Sample page from NUMERICAL RECIPES ... Chapter 16 Integration of Ordinary Differential Equations CITED REFERENCES AND FURTHER READING: Gear, C.W 1971, Numerical Initial Value Problems in Ordinary Differential Equations (Englewood Cliffs,...

Ngày tải lên: 15/12/2013, 04:15

... (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 ... first value? The answer will become clear only in the next section, but in brief is this: This call may not be your only one with these starting conditions You may have taken a previous step with ... 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 equation can serve as an initial point...

Ngày tải lên: 15/12/2013, 04:15

... “worst-offender” equation How is ∆0 , the desired accuracy, related to some looser prescription like “get a solution good to one part in 106 ”? That can be a subtle question, and it depends on exactly what ... 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America) c∗ i 37 378 718 Chapter 16 Integration of Ordinary Differential Equations for the ith equation will be taken ... solutions by y1 (one step 2h) and y2 (2 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...

Ngày tải lên: 24/12/2013, 12:16

... Extrapolation and the Bulirsch-Stoer Method The techniques described in this section are not for differential equations containing nonsmooth functions For example, you might have a differential equation ... stepsize, contains only even 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 ... to compare equation (4.2.4) with equation (16.3.4) above You will see that the transition in Chapter to the idea of Richardson extrapolation, as embodied in Romberg integration of §4.3, is exactly...

Ngày tải lên: 24/12/2013, 12:16

... Integration of Ordinary Differential Equations During the first step, when we have no information about the solution, the stepsize reduction check is made for all k Afterwards, we test for convergence ... [3] 16.5 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 ... readers.) The second idea has to with what kind of fitting function is used Bulirsch and Stoer first recognized the strength of rational function extrapolation in Richardsontype applications That strength...

Ngày tải lên: 24/12/2013, 12:16

... 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 a problem ... system of n = nv/2 equations On input y[1 nv] contains y in its first n elements and y in its second n elements, all evaluated at xs d2y[1 nv] contains the right-hand side function f (also evaluated ... 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America) Then for k = 1, , m − 1, set 734 Chapter 16 Integration of Ordinary Differential Equations Note that for...

Ngày tải lên: 24/12/2013, 12:16

... cost of Jacobian evaluations as well as function evaluations We count one Jacobian evaluation as equivalent to N function evaluations, where N is the number of equations • Once again the user-supplied ... 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 ... also give you one of the Kaps-Rentrop sets as an option Some proposed parameter sets require function evaluations outside the domain of integration; we prefer to avoid that complication The calling...

Ngày tải lên: 24/12/2013, 12:16

... fortunately some good ones such as the Gear formulas are known.) We then appear to have two choices for solving the implicit equations: functional iteration to convergence, or Newton iteration However, ... America only),or send email to trade@cup.cam.ac.uk (outside North America) be satisfied The second of the equations in (16.7.9) is 752 Chapter 16 Integration of Ordinary Differential Equations you ... America only),or send email to trade@cup.cam.ac.uk (outside North America) out in the middle There is possibly only one exceptional case: high-precision solution of very smooth equations with...

Ngày tải lên: 24/12/2013, 12:16

... Introduction §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 ... contained introduction to the field of ordinary differential equations with emphasize on the view point of dynamical systems It only requires some basic knowledge from calculus, complex functions, ... iii iv Contents Chapter §4.1 §4.2 §4.3 §4.4 Differential equations in the complex domain The basic existence and uniqueness result Linear equations The Frobenius method Second order equations 61...

Ngày tải lên: 31/03/2014, 15:56

... ‘tricks’ for nonlinear equations 10.1 Exact equations 10.2 Substitution methods Exercises Part II Second order linear equations with constant coefficients 11 Second order linear equations: general ... Classification of differential equations 11 3.1 Ordinary and partial differential equations 11 3.2 The order of a differential equation 13 3.3 Linear and nonlinear 13 3.4 Different types of solution 14 ... portraits for linear equations 31.1 *Jordan canonical form Exercises Part VI Coupled nonlinear equations 32 Coupled nonlinear equations 32.1 Some comments on phase portraits 32.2 Competition of species...

Ngày tải lên: 12/05/2014, 03:24

Ngày tải lên: 12/06/2014, 16:08

... x be a nontrivial T -periodic solution of the autonomous delay differential equation 1.1 with the second derivative Suppose that the function f : Rp → Rp is κ-Lipschitz continuous Then one has ... Foundation of Nanjing University of Information Science and Technology 20070049 References M Han, “Bifurcations of periodic solutions of delay differential equations, ” Journal of Differential Equations, ... bifurcation theorem for retarded functional differential equations, ” Transactions of the American Mathematical Society, vol 238, pp 139–164, 1978 J L Kaplan and J A Yorke, Ordinary differential equations...

Ngày tải lên: 22/06/2014, 02:20

... unbounded and monotone solutions of the singular second Painlev´ equation on the half-line, Nonlinear Analysis 57 (2004), no 3, 401–419 e [12] Y Rocard, Thermodynamique, chapter 5, Masson, Paris, ... Walter, Ordinary Differential Equations, Graduate Texts in Mathematics, vol 182, Springer, New York, 1998 [14] W Wasov, Asymptotic Expressions for Ordinary Differential Equations, John Wiley & Sons, ... definition of ξ01 and ξ11 is well posed because, since the function r n−1 ρ (r) is decreasing on [r1 ,r2 ] and ρ = ρ(r) is an increasing one on [r1 ,r2 ], we may apply the usual continuation theorem...

Ngày tải lên: 22/06/2014, 22:20

... technique for differential equations with nonlinear boundary conditions, Nonlinear Stud (2001), no 3, 381–388 , Ordinary differential equations with nonlinear boundary conditions, Georgian Math J (2002), ... solutions for first order ordinary differential equations with nonlinear boundary conditions, Appl Math Comput 153 (2004), no 3, 793– 802 T Jankowski, Monotone iterative technique for differential equations ... differential equations of type (1.1) to obtain quite general existence results It is the first paper where the monotone technique is applied for delay differential equations when a boundary condition has...

Ngày tải lên: 23/06/2014, 00:20

... 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 Methods for Ordinary ... PREFACE TO THE SECOND EDITION xix corrections on the first edition or made constructive suggestions on early drafts of this new version In addition to people acknowledged in some other way, I ... differential equations alone does not generally define a unique solution, and it is necessary to add to the formulation of the problem a number of additional conditions These are either ‘boundary conditions’,...

Ngày tải lên: 27/06/2014, 08:20

... of liquid is constant The concentration in the feed is C Ai = 10 mol/L The volume of the tank is V = 10 L and the initial concentration of A is 10 mol/L The differential equation for this problem ... Reactions Process • The reactions are A B C occur within the tank where, reaction #1 (A B) with rate constant, k1 and reaction #2 (BC) with rate constant k2 • The following set of differential ... 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 sequentially...

Ngày tải lên: 19/03/2015, 08:49

... is a basic, simple model from population biology which involves di erential equations Consider an ecological system consisting of one prey species and one predator species The prey population would ... Newton's equations of motion give x1 = x2 = 00 00 ;zx1 ;zx2 ; g and the fact that the rod has a xed length gives the additional constraint x21 + x22 = 1: Chapter 1: Ordinary Di erential Equations ... very long time period) 16 Chapter 1: Ordinary Di erential Equations Throughout this book we will pay attention to such additional considerations, especially when they extend our investigation in...

Ngày tải lên: 21/12/2016, 10:35

... been done for nonoscillation of all solutions of nonlinear functional differential equations In this connection, Parhi 10 established some sufficient conditions for oscillation of all solutions of ... differential equations, ” Memoirs on Differential Equations and Mathematical Physics, vol 31, pp 109–111, 2004 S Padhi, On oscillatory linear third order forced differential equations, ” Differential Equations ... differential equations, ” Acta Mathematica Hungarica, vol 54, no 1-2, pp 79–88, 1989 10 N Parhi, “Sufficient conditions for oscillation and nonoscillation of solutions of a class of second order functional-differential...

Ngày tải lên: 22/06/2014, 02:20

Ngày tải lên: 14/03/2014, 13:20

Ngày tải lên: 14/03/2014, 13:20