solving an ordinary differential equation

... intentionally left blank AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS This refreshing, introductory textbook covers standard techniques for solving ordinary differential equations, as well ... undergraduate curriculum, and an important tool in the armoury of practising mathematicians, scientists and engineers Although the emphasis in the text is on pencil and paper analysis, and the book in ... obtaining explicit solutions of ordinary differential equations, and then to introduce the ideas of qualitative analysis using phase plane techniques Simple difference equations are also included,...

Ngày tải lên: 12/05/2014, 03:24

... 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) ... 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 IN C: ... The underlying idea of any routine for solving the initial value problem is always this: Rewrite the dy’s and dx’s in (16.0.3) as finite steps ∆y and ∆x, and multiply the equations by ∆x This gives...

Ngày tải lên: 15/12/2013, 04:15

... 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 equation can serve as an initial ... 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 ... values for the variables y[1 n] and their derivatives dydx[1 n] known at x, use the fourth-order Runge-Kutta method to advance the solution over an interval h and return the incremented variables...

Ngày tải lên: 15/12/2013, 04:15

... interesting and worthy of Gilbert and Sullivan’s Mikado: The routine can always achieve an apparent zero error by making the stepsize so small that quantities of order hy add to quantities of ... customized to the problem at hand 722 Chapter 16 Integration of Ordinary Differential Equations } nrerror("Too many steps in routine odeint"); } CITED REFERENCES AND FURTHER READING: Gear, C.W ... 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 [2] Shampine, L.F., and...

Ngày tải lên: 24/12/2013, 12:16

... 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, ... for differential equations containing nonsmooth functions For example, you might have a differential equation whose right-hand side involves a function that is evaluated by table look-up and ... constant, but h changes by varying n in (16.3.1) The importance of this even power series is that, if we play our usual tricks of combining steps to knock out higher-order error terms, we can gain...

Ngày tải lên: 24/12/2013, 12:16

... consider the final answer of a numerical calculation as itself being an analytic function (if a complicated one) of an adjustable parameter like the stepsize h That analytic function can be probed ... 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), ... North America) The quantities Wk can be calculated during the integration The optimal column index q is then defined by 728 Chapter 16 Integration of Ordinary Differential Equations During the...

Ngày tải lên: 24/12/2013, 12:16

... Chapter 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 ... (16.5.6) and set KMAXX = 12 in bsstep CITED REFERENCES AND FURTHER READING: Deuflhard, P 1985, SIAM Review, vol 27, pp 505–535 16.6 Stiff Sets of Equations As soon as one deals with more than one ... for a 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...

Ngày tải lên: 24/12/2013, 12:16

... this replacement for you, so the routines can handle right-hand sides of the form f(y, x) without any special effort on your part We now mention an important point: It is absolutely crucial to scale ... maximum value of n; GROW and SHRNK are the largest and smallest factors by which stepsize can change in one step; ERRCON equals (GROW/SAFETY) raised to the power (1/PGROW) and handles the case when ... routine derivs is a dummy argument and so can have any name However, to maintain “plug-compatibility” with rkqs, bsstep and stiff, the routine jacobn is not an argument and must have exactly this name...

Ngày tải lên: 24/12/2013, 12:16

... (16.7.6) Set x = xn+1 in equation (16.7.6) to get an approximation to yn+1 Differentiate equation (16.7.6) and set x = xn+1 to get an approximation to yn+1 , and similarly for yn+1 and yn+1 Call the ... 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 ... “integrand” is the right-hand side, which depends both on x and on the dependent variables y Thus to advance the solution of y = f(x, y) from xn to x, we have 16.7 Multistep, Multivalue, and Predictor-Corrector...

Ngày tải lên: 24/12/2013, 12:16

... dynamical system and discuss stability including the stable manifold and the Hartman–Grobman theorem for both continuous and discrete systems We prove the Poincar´–Bendixson theorem and investigate ... important equations which can be explicitly solved using some clever transformation In fact, there are reference books like the one by Kamke [17], where you can look up a given equation and find ... Finding an integrating factor is in general as hard as solving the original equation However, in some cases making an ansatz for the form of µ works Consider xy + 3x − 2y = and look for an integrating...

Ngày tải lên: 31/03/2014, 15:56

Ngày tải lên: 12/06/2014, 16:08

... 139–164, 1978 J L Kaplan and J A Yorke, Ordinary differential equations which yield periodic solutions of differential delay equations,” Journal of Mathematical Analysis and Applications, vol ... delay equations via Hamiltonian systems—II,” Nonlinear Analysis: Theory, Methods & Applications, vol 65, no 1, pp 40–58, 2006 14 J Mawhin and M Willem, Critical Point Theory and Hamiltonian Systems, ... studied by many authors since 1970s of the last century, and a lot of remarkable results have been achieved We refer to 1–6 for reference The delay equation 1.1 with more than one delay and p is...

Ngày tải lên: 22/06/2014, 02:20

... m1 and furthermore, for any t ∈ (r1 , r2 ), there is an r ∈ (r0 , r1 ) with r n−1 ρ (r) = t n−1 ρ (t) (3.95) A P Palamides and T G Yannopoulos 21 Since r < t, it follows that (for all such r and ... Palamides and T G Yannopoulos 25 A numerical approach By the previous and especially in view of Theorems 3.8 and 3.11, it is obvious that we cannot find out theoretically the maximal point ξM and (then ... Indiana University Mathematics Journal 30 (1981), no 1, 141– 157 [4] D Bonheure, J M Gomes, and L Sanchez, Positive solutions of a second-order singular ordinary differential equation, Nonlinear Analysis...

Ngày tải lên: 22/06/2014, 22:20

... METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS were worked on and improved and some of which were ultimately discarded Many individual colleagues, both in Auckland and overseas, have read and worked ... Subsection 110 without any explanation Stiff problems typically have large Lipschitz constants, but many have more manageable one-sided Lipschitz constants, and this can be an aid in obtaining realistic ... to me and I express my heartfelt thanks Amongst my many supportive colleagues, I particularly want to name Christian Brouder, Robert Chan, Tina Chan, David Chen, Allison Heard, Shirley Huang,...

Ngày tải lên: 27/06/2014, 08:20

... methods based on “rectangular” and “trapezoidal” rules cannot be used School of Chemical Engineering and Advanced Materials Newcastle University Integration via Taylor Series Expansion • Alternative ... Engineering and Advanced Materials Newcastle University 24 Solving sets of • st order ODEs Chemical engineering systems are often described by more than ODE, e.g dynamic material and energy balances ... constant, k1 and reaction #2 (BC) with rate constant k2 • The following set of differential equations Example: Multiple Reactions Process • Where k1=0.2 hr -1 and k2=0.1 hr -1 and at time t=0,...

Ngày tải lên: 19/03/2015, 08:49

... fourth-order equation We further point out that almost all the equations we have seen here, and also other important equations such as Maxwell’s equations, the Schr¨ dinger equation and Newton’s equation ... by Hamilton led to major advances in optics and in analytical mechanics The Fourier method enabled the solution of heat transfer and wave Introduction propagation, and Green’s method was instrumental ... linear differential equation naturally defines a linear operator: the equation can be expressed as L[u] = f , where L is a linear operator and f is a given function A linear differential equation...

Ngày tải lên: 16/02/2014, 15:20

... properties and refer the reader for more detailed discussions to standard texts on analysis and functional analysis (e.g 8]) 4.1 Completely continuous mappings Let E and X be Banach spaces and let be an ... di erential equations and provide qualitative and quantitative descriptions of solutions At the same time, we develop methods of analysis which may be applied to carry out the above and which have ... 0k g and de ne M = fu : B ( ) ! E such that u is continuous, u( ) = u0 ; ku( )?u0k0 r; and kuk0 = sup 2B ( ) ku( )k < +1g Then M is a closed subset of a Banach space and (35) de nes an equation...

Ngày tải lên: 31/03/2014, 15:56

... Ordinary and Partial Differential Equation Routines in C, C++, Fortran, Java®, Maple®, and MATLAB® H.J Lee and W.E Schiesser CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London ... easily show that yi+1 of Equation 1.25 and yi+1 of Equation 1.26c are the same While Equation 1.25 and Equations 1.26c are mathematically equivalent, Equations 1.26 have an advantage when used in ... single ODE, Equation 1.3, subject to initial condition Equation 1.4, by the Euler and modified Euler methods, Equation 1.28 and Equations 1.30 The analytical solution, Equation 1.5, can be used...

Ngày tải lên: 08/04/2014, 10:27

... NGLES Angles can be classified into the following categories: acute, right, obtuse, and straight ■ An acute angle is an angle that measures less than 90 degrees Acute Angle 179 – THE GRE QUANTITATIVE ... A right angle is an angle that measures exactly 90 degrees A right angle is sumbolized by a square at the vertex Right Angle Symbol ■ An obtuse angle is an angle that measures more than 90 degrees, ... 90-degree angle l2 l1 178 – THE GRE QUANTITATIVE SECTION – Angles y An angle is formed by an endpoint, or vertex, and two rays ray Endpoint, or Vertex N AMING A NGLES There are three ways to name an angle...

Ngày tải lên: 18/06/2014, 17:20

... Wang et al Advances in Difference Equations 2011, 2011:2 http://www.advancesindifferenceequations.com/content/2011/1/2 Page of 11 as an important area of investigation For the general theory and ... nonlinear fractional differential equations J Math Anal Appl 2010, 371:57-68 Wang G, Ahmad B, Zhang L: Impulsive anti-periodic boundary value problem for non-linear differential equations of fractional ... 42:1353-1363 Yang Ch, Zhai Ch, Yan J: Positive solutions of the three-point boundary value problem for second order differential equations with an advanced argument Nonlinear Anal 2006, 65:2013-2023 Jankowski...

Ngày tải lên: 21/06/2014, 03:20