solving systems of differential equations examples

Ngày tải lên: 20/06/2014, 22:20

... equal to I. The proof of uniqueness is complete. To prove the existence we note, first of all, that it suffices to provide a proof if one of the κ i ’s is equal to ±1 and one of the δ j ’s is equal ... solutions of isomonodromy problems for such systems of difference equations. In the case of one-interval gap probability this has been done (in a different language) in [Bor], [BB]. One example of the ... ALEXEI BORODIN Most of the results of the present paper can be carried over to the case of q-difference equations of the form Y (qz)=A(z)Y (z). The q-difference Schlesinger equations are, cf. (3)–(6), (3q) (4q) (5q) (6q) B i (...

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Ngày tải lên: 14/03/2014, 13:20

... Integration of Ordinary Differential Equations 16.0 Introduction Problems involving ordinary differential equations (ODEs) can always be reduced to the study of sets of first-order differential equations. ... auxiliary variables. The generic problem in ordinary differential equations is thus reduced to the study of a set of N coupled first-order differential equations for the functions y i ,i=1,2, ,N, having ... 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). 16.1...

Ngày tải lên: 15/12/2013, 04:15

... 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). 16.1 ... 710 Chapter 16. Integration of Ordinary Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright ... that derive from this basic 712 Chapter 16. Integration of Ordinary Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright...

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 ... ,n−1 y(x+H)≈y n ≡ 1 2 [z n +z n−1 +hf(x + H, z n )] (16.3.2) 714 Chapter 16. Integration of Ordinary Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright ... informationcan beobtained. Obviously, 720 Chapter 16. Integration of Ordinary Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright...

Ngày tải lên: 24/12/2013, 12:16

... 722 Chapter 16. Integration of Ordinary Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright ... Initial Value Problems in Ordinary Differential Equations (Englewood Cliffs, NJ: Prentice-Hall). [1] Cash, J.R., and Karp, A.H. 1990, ACM Transactions on Mathematical Software , vol. 16, pp. 201– 222. ... modified midpoint method, which advances a vector of dependent variables y(x) from a point x to a point x + H by a sequence of n substeps each of size h, h = H/n (16.3.1) In principle, one could...

Ngày tải lên: 24/12/2013, 12:16

... extrapolate each component of a vector of quantities. 728 Chapter 16. Integration of Ordinary Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN ... 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 ... ordinary differential equations with minimal computational effort. (A possible exception, infrequently encountered in practice, is discussed in §16.7.) 726 Chapter 16. Integration of Ordinary Differential...

Ngày tải lên: 24/12/2013, 12:16

... 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 ... 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

... 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 ... nice feature of implicit methods holds only for linear systems, but even in the general case implicit methods give better stability. 742 Chapter 16. Integration of Ordinary Differential Equations Sample ... form of the midpoint rule: y n+1 − y n−1 =2hf  y n+1 + y n−1 2  (16.6.29) 738 Chapter 16. Integration of Ordinary Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC...

Ngày tải lên: 24/12/2013, 12:16

... Problems in Ordinary Differential Equations (Englewood Cliffs, NJ: Prentice-Hall), Chapter 9. [1] Shampine, L.F., and Gordon, M.K. 1975, Computer Solution of Ordinary Differential Equations. The Initial ... adjusting the stepsize is difficult. 748 Chapter 16. Integration of Ordinary Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright ... been, we think, squeezed 752 Chapter 16. Integration of Ordinary Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright...

Ngày tải lên: 24/12/2013, 12:16

... present some examples of applications. In the second part we present a new method for solving systems involving ordinary differential equations which does not need this initial box. The aim of the method ... part. Mots-clefs : syst`emes hybrides, equations differentielles ordinaires, anal- yse par intervalles. References 1. Hansen, E. and Sengupta, S.: Bounding solutions of systems of equations using interval analysis. ... of Hybrid Systems Using Interval Arith- metic Constraints. Hybrid Systems: Computation and Control, 139–142 (2004) 9. Ishii D., Ueda K., Hosobe H., and Goldsztejn A.: Interval-based Solving of...

Ngày tải lên: 12/01/2014, 22:20

... 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 ∈ ... context of derivative pricing. The book does not claim to be a complete account of the state of the art of the subject. Rather it attempts to provide a systematic framework for an understanding of ... order of strong convergence from γ =0.5toγ =1.0. Nonethe- less, the order of weak convergence of the Milstein scheme equals β =1.0, which is not an improvement over the order of weak convergence of...

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