analytical solutions to ordinary differential equations

... INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS This refreshing, introductory textbook covers standard techniques for solving ordinary differential equations, as well as introducing students to qualitative ... to make the transition from theory to practice Exercises are also provided to test and extend understanding; full solutions for these are available for teachers AN INTRODUCTION TO ORDINARY DIFFERENTIAL ... systematic way 3.1 Ordinary and partial differential equations The most significant distinction is between ordinary and partial differential equations, and this depends on whether ordinary or partial...

Ngày tải lên: 12/05/2014, 03:24

... the ‘order’ of the Taylor Series expansion Need to have the higher order differentials of x(t) to maintain accuracy – • • may not be available due to the complexity of the original equation Method ... applied to a wide range of ODEs However, except for the simplest of problems, the integration interval has to be small for accuracy Time to solution becomes longer when integration intervals are too ... integration intervals to generate the solutions to the underlying ODE • ˆ ˆ h  x(t ) + x(t + h) ˆ (t + h) = x(t ) + hf  ˆ x ,t + ÷ 2  Problem: again, the formula requires which has yet to be calculated...

Ngày tải lên: 19/03/2015, 08:49

... 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 with ... differential equations, ” Transactions of the American Mathematical Society, vol 238, pp 139–164, 1978 J L Kaplan and J A Yorke, Ordinary differential equations which yield periodic solutions of ... that n and p 1, various problems on the solutions of 1.1 , such as the existence of periodic solutions, bifurcations of periodic solutions, and stability of solutions, have been studied by many...

Ngày tải lên: 22/06/2014, 02:20

... 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 ... than Runge-Kutta In recent years Bulirsch-Stoer has been replacing predictor-corrector in many applications, but it is too soon to say that predictor-corrector is dominated in all cases However, ... sophisticated predictor-corrector routines are competitive Accordingly, we have chosen not to give an implementation of predictor-corrector in this book We discuss predictor-corrector further in §16.7,...

Ngày tải lên: 15/12/2013, 04:15

... yout[i]=y[i]+h6*(dydx[i]+dyt[i]+2.0*dym[i]); weights free_vector(yt,1,n); free_vector(dyt,1,n); free_vector(dym,1,n); 714 Chapter 16 Integration of Ordinary Differential Equations } CITED REFERENCES AND FURTHER ... along the trajectory of an ordinary differential equation can serve as an initial point The fact that all steps are treated identically also makes it easy to incorporate Runge-Kutta into relatively ... in mind, however, that the old workhorse’s last trip may well be to take you to the poorhouse: Bulirsch-Stoer or predictor-corrector methods can be very much more efficient for problems where very...

Ngày tải lên: 15/12/2013, 04:15

... errmax,h,htemp,xnew,*yerr,*ytemp; 720 Chapter 16 Integration of Ordinary Differential Equations ak2=vector(1,n); ak3=vector(1,n); ak4=vector(1,n); ak5=vector(1,n); ak6=vector(1,n); ytemp=vector(1,n); for (i=1;i

Ngày tải lên: 24/12/2013, 12:16

... caveats, we believe that the Bulirsch-Stoer method, discussed in this section, is the best known way to obtain high-accuracy solutions to ordinary differential equations with minimal computational ... 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, ... The usefulness of the modified midpoint method to the Bulirsch-Stoer technique (§16.4) derives from a “deep” result about equations (16.3.2), due to Gragg It turns out that the error of (16.3.2),...

Ngày tải lên: 24/12/2013, 12:16

... the Bulirsch-Stoer Method 729 d=matrix(1,nv,1,KMAXX); err=vector(1,KMAXX); x=vector(1,KMAXX); yerr=vector(1,nv); ysav=vector(1,nv); yseq=vector(1,nv); if (eps != epsold) { A new tolerance, so ... 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 ... communication theory to determine the “average expected convergence behavior” of the extrapolation His model produces certain correction factors α(k, q) by which Hk is to be multiplied to try to get convergence...

Ngày tải lên: 24/12/2013, 12:16

... not referenced Also input is htot, the total step to be taken, and nstep, the number of substeps to be used The output is returned as yout[1 nv], with the same storage arrangement as y derivs ... With this storage arrangement you can use bsstep simply by replacing the call to mmid with one to stoerm using the same arguments; just be sure that the argument nv of bsstep is set to 2n You ... (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 compatibility...

Ngày tải lên: 24/12/2013, 12:16

... free_vector(ysav,1,n); free_vector(g4,1,n); free_vector(g3,1,n); free_vector(g2,1,n); free_vector(g1,1,n); free_vector(err,1,n); free_vector(dysav,1,n); free_matrix(dfdy,1,n,1,n); free_vector(dfdx,1,n); free_matrix(a,1,n,1,n); ... Multivalue, and Predictor-Corrector Methods 747 free_vector(ysav,1,nv); free_vector(yerr,1,nv); free_vector(x,1,KMAXX); free_vector(err,1,KMAXX); free_matrix(dfdy,1,nv,1,nv); free_vector(dfdx,1,nv); ... Integration of Ordinary Differential Equations (1.0/2.0) (-3.0/2.0) (121.0/50.0) (29.0/250.0) 1.0 (3.0/5.0) indx=ivector(1,n); a=matrix(1,n,1,n); dfdx=vector(1,n); dfdy=matrix(1,n,1,n); dysav=vector(1,n);...

Ngày tải lên: 24/12/2013, 12:16

... that can be used to control accuracy and to adjust stepsize If one corrector step is good, aren’t many better? Why not use each corrector as an improved predictor and iterate to convergence on ... predictor-corrector approach using an explicit method for the predictor, or by Newton iteration for stiff systems Why go to all the trouble of introducing a whole new method that turns out to be ... 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...

Ngày tải lên: 24/12/2013, 12:16

... blank AN INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS A complete introduction to partial differential equations, this textbook provides a rigorous yet accessible guide to students in mathematics, ... graphs to provide insight into the numerical examples Solutions and hints to selected exercises are included for students whilst extended solution sets are available to lecturers from solutions@ cambridge.org ... Classification 1.3 Differential operators and the superposition principle 1.4 Differential equations as mathematical models 1.5 Associated conditions 1.6 Simple examples 1.7 Exercises First-order equations...

Ngày tải lên: 16/02/2014, 15:20

... differential equations are given all over the world in various forms The traditional approach to the subject is to introduce a number of analytical techniques, enabling the student to derive exact solutions ... introduction to this topic must focus on methods suitable for computers But these methods often rely on deep analytical insight into the equations We must therefore take great care not to throw away ... adopt to such problems, even if these equations may be hard to handle by an analytical approach In Chapter 12 we give a brief introduction to the Fourier transform and its application to partial...

Ngày tải lên: 31/03/2014, 15:56

... familiarity with probability theory, measure theory, ordinary differential equations, and perhaps partial differential equations as well This is all too much to expect of undergrads But white noise, Brownian ... trajectories of systems modeled by (ODE) not in fact behave as predicted: X(t) x0 Sample path of the stochastic differential equation Hence it seems reasonable to modify (ODE), somehow to include ... Brownian motion and the random calculus are wonderful topics, too good for undergraduates to miss out on Therefore as an experiment I tried to design these lectures so that strong students could...

Ngày tải lên: 31/03/2014, 15:56

... 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 ... solution to the posed problem! Mathematica expects you to know that there are other solutions and how to get them Out[4]= {{x[t] → Problem 1.14 Try to find solutions of the following differential equations: ... Transform both equations into a first order system (ii) Compute the solution to the approximate system corresponding to the given initial condition Compute the time it takes for the stone to hit the...

Ngày tải lên: 31/03/2014, 15:56

... familiarity with probability theory, measure theory, ordinary differential equations, and perhaps partial differential equations as well This is all too much to expect of undergrads But white noise, Brownian ... trajectories of systems modeled by (ODE) not in fact behave as predicted: X(t) x0 Sample path of the stochastic differential equation Hence it seems reasonable to modify (ODE), somehow to include ... Brownian motion and the random calculus are wonderful topics, too good for undergraduates to miss out on Therefore as an experiment I tried to design these lectures so that strong students could...

Ngày tải lên: 08/04/2014, 12:24

Ngày tải lên: 12/06/2014, 16:08

... defined by x t θ , for θ ∈ −∞, , φ belongs to P and φ 0 The fractional derivative is xt θ understood here in the Caputo sense The aim of our paper is to study the solvability of 1.1 and present ... follows the continuity of S t in the uniform operator topology for t > that I1 tends to 0, as t2 → t1 The continuity of h ensures that I2 tends to 0, as t2 → t1 For I3 , we have t2 ∞ I3 ≤ q t1 ... Difference Equations, vol 2010, Article ID 691721, 10 pages, 2010 M M El-Borai, “Some probability densities and fundamental solutions of fractional evolution equations, ” Chaos, Solitons and Fractals,...

Ngày tải lên: 21/06/2014, 05:20

... of solutions of degenerate nonlinear equations on manifolds,” Differential and Integral Equations, vol 13, no 10–12, pp 1445–1458, 2000 M M Cavalcanti and V N D Cavalcanti, “Existence and asymptotic ... semigroups in Banach spaces to give the existence and uniqueness of the solution for problem 1.5 – 1.7 Cavalcanti et al 7–11 studied the existence and asymptotic behavior of solutions evolution problem ... 1.8 – 1.10 , we refer the reader to 3, 6, 19–31 and their references In these papers, existence, boundedness, asymptotic behavior, and nonexistence of global solutions for problem 1.8 – 1.10...

Ngày tải lên: 21/06/2014, 11:20

... this paper, our aim is to study the existence of n distinct pairs of nontrivial solutions to the Dirichlet boundary problem for the second-order impulsive differential equations u t λh t, u t ... periodic solutions of first-order singular differential equations, ” Bulletin of the London Mathematical Society, vol 40, no 1, pp 143–150, 2008 D Qian and X Li, “Periodic solutions for ordinary ... differential equations, ” Journal of Mathematical Analysis and Applications, vol 318, no 2, pp 726–741, 2006 10 J J Nieto and D O’Regan, “Variational approach to impulsive differential equations, ”...

Ngày tải lên: 21/06/2014, 11:20