... Integration of Ordinary Differential Equations 16.0 IntroductionProblems involving ordinarydifferentialequations (ODEs) can always bereduced to the study of sets of first-order differential equations. ... auxiliary variables.The generic problem in ordinarydifferentialequations is thus reduced to thestudy of a set of N coupled first-order differentialequations for the functionsyi,i=1,2, ,N, ... 1973,Computational Methods in OrdinaryDifferential Equations (New York: Wiley).Lapidus, L., and Seinfeld, J. 1971,Numerical Solution ofOrdinaryDifferential Equations (NewYork: Academic...
... 1973,Computational Methods in OrdinaryDifferential Equations (New York: Wiley).Lapidus, L., and Seinfeld, J. 1971,Numerical Solution ofOrdinaryDifferential Equations (NewYork: Academic ... 710Chapter 16. Integration ofOrdinaryDifferential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright ... that derive from this basic 712Chapter 16. Integration ofOrdinaryDifferential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright...
... 722Chapter 16. Integration ofOrdinaryDifferential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright ... ,n−1y(x+H)≈yn≡12[zn+zn−1+hf(x + H, zn)](16.3.2) 714Chapter 16. Integration ofOrdinaryDifferential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright ... informationcan beobtained. Obviously, 720Chapter 16. Integration ofOrdinaryDifferential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright...
... 722Chapter 16. Integration ofOrdinaryDifferential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright ... Initial Value Problems in OrdinaryDifferential Equations (EnglewoodCliffs, 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 nsubsteps each of size h,h = H/n (16.3.1)In principle, one could...
... extrapolate eachcomponent of a vector of quantities. 728Chapter 16. Integration ofOrdinaryDifferential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN ... methoda degree of robustness for problems with discontinuities.Let us remind you once again that scaling of the variables is often crucial forsuccessful integration ofdifferential equations. The ... differential equations. For nksubdivisions in H, the number of function evaluations can be found from the recurrenceA1= n1+1Ak+1= Ak+ nk+1(16.4.6) 730Chapter 16. Integration ofOrdinary Differential...
... vol. 27, pp. 505–535.16.6 Stiff Sets of Equations As soon as one deals with more than one first-order differential equation, thepossibility of a stiff set ofequations arises. Stiffness occurs ... Second-Order Conservative Equations Usually when you have a system of high-order differentialequations to solve it is bestto reformulate them as a system of rst-order equations, as discussed ... 734Chapter 16. Integration ofOrdinaryDifferential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright...
... form of the midpoint rule:yn+1− yn−1=2hfyn+1+ yn−12(16.6.29) 738Chapter 16. Integration ofOrdinaryDifferential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC ... intermediate values of y and x.y[i]=ysav[i]+A31*g1[i]+A32*g2[i]; 736Chapter 16. Integration ofOrdinaryDifferential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING ... vol. 27, pp. 505–535.16.6 Stiff Sets of Equations As soon as one deals with more than one first-order differential equation, thepossibility of a stiff set ofequations arises. Stiffness occurs...
... Problems in OrdinaryDifferential Equations (EnglewoodCliffs, NJ: Prentice-Hall), Chapter 9. [1]Shampine, L.F., and Gordon, M.K. 1975,Computer Solution ofOrdinaryDifferential Equations. The ... been, we think, squeezed 752Chapter 16. Integration ofOrdinaryDifferential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright ... adjustingthe stepsize is difficult. 748Chapter 16. Integration ofOrdinaryDifferential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright...
... neighborhood of the singular pointz = 0 and we can now try to go the opposite way. Given a solution of thesystem of linear equations (4.38), where α is an eigenvalue of A0we get asolution of our ... block of A0. Or ml>ml−1and (α + ml− A0)ul,ml= 0, that is, α + mlis another eigenvalue of A0.So we have found a quite complete picture of the possible forms of solu-tions of ... manuscript provides an introduction to ordinary differential equations and dynamical systems. We start with some simple examples of explicitly solvable equations. Then we prove the fundamental...
... Classification ofdifferentialequations 113.1 Ordinary and partial differentialequations 113.2 The order of a differential equation 133.3 Linear and nonlinear 133.4 Different types of solution ... PrefaceThe aim of this book is to deal with all of the elementary methods for obtainingexplicit solutions ofordinarydifferential 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...