value problems in ordinary differential equations

... store the initial condition as intervals containing the corresponding decimal values The function string to interval converts a decimal string to a machine interval that contains the decimal value ... is used in chunk 27 The initial condition and endpoint are represented as intervals in VNODE-LP In this example, they are all point values stored as intervals The components of iVector (interval ... all this into 70 orderstudy.cc 70 ≡ #include #include #include #include #include "vnode.h" using namespace std; using namespace vnodelp; Lorenz 19 main program...

Ngày tải lên: 12/01/2014, 22:07

... boundary value problems for higher-order linear differential equations with strong singularities, Boundary Value Problems 2006 (2006), Article ID 83910, 32 pages [2] J V Baxley, Boundary value problems ... defined on the interval [α,τ], then the cross section ᐄ τ; E0 = y(τ) : y ∈ ᐄ E0 (2.25) is a continuum in Rn Reminding that a set-valued mapping Ᏻ, which maps a topological space X into compact ... BVP [13] W Walter, Ordinary Differential Equations, Graduate Texts in Mathematics, vol 182, Springer, New York, 1998 [14] W Wasov, Asymptotic Expressions for Ordinary Differential Equations, John...

Ngày tải lên: 22/06/2014, 22:20

... 710 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 ... stiff equations, relevant both to ordinary differential equations and also to partial differential equations (Chapter 19) Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING ... solution over an interval by combining the information from several Euler-style steps (each involving one evaluation of the right-hand f’s), and then using the information obtained to match a...

Ngày tải lên: 15/12/2013, 04:15

... constructing a good Runge-Kutta code is given in [3] Here is the routine for carrying out one classical Runge-Kutta step on a set of n differential equations You input the values of the independent ... 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 Cliffs, NJ: ... Figure 16.1.2 Midpoint method Second-order accuracy is obtained by using the initial derivative at each step to find a point halfway across the interval, then using the midpoint derivative across...

Ngày tải lên: 15/12/2013, 04:15

... Ordinary Differential Equations } nrerror("Too many steps in routine odeint"); } CITED REFERENCES AND FURTHER READING: Gear, C.W 1971, Numerical Initial Value Problems in Ordinary Differential Equations ... free_vector(ak2,1,n); } Noting that the above routines are all in single precision, don’t be too greedy in specifying eps The punishment for excessive greediness is interesting and worthy of Gilbert ... ending values are always stored Again, these controls are merely indicative of what you might need The routine odeint should be customized to the problem at hand 722 Chapter 16 Integration of Ordinary...

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, ... to ordinary differential equations with minimal computational effort (A possible exception, infrequently encountered in practice, is discussed in §16.7.) Sample page from NUMERICAL RECIPES IN ... not particularly good for differential equations that have singular points inside the interval of integration A regular solution must tiptoe very carefully across such points Runge-Kutta with adaptive...

Ngày tải lên: 24/12/2013, 12:16

... "nrutil.h" #define KMAXX #define IMAXX (KMAXX+1) #define SAFE1 0.25 #define SAFE2 0.7 #define REDMAX 1.0e-5 #define REDMIN 0.7 #define TINY 1.0e-30 #define SCALMX 0.1 Maximum row number used in the extrapolation ... 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 scaling “trick” suggested in ... calculated during the integration The optimal column index q is then defined by 728 Chapter 16 Integration of Ordinary Differential Equations During the first step, when we have no information about...

Ngày tải lên: 24/12/2013, 12:16

... rule for integrating y = f (x, y) 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 ... 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 ... Software Permission is granted for internet users to make one paper copy for their own personal use Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer,...

Ngày tải lên: 24/12/2013, 12:16

... #define #define #define #define #define #define #define #define #define #define #define #define #define #define #define #define #define #define #define GAM 0.231 A21 2.0 A31 4.52470820736 A32 4.16352878860 ... 16 #define #define #define #define #define #define C1X C2X C3X C4X A2X A3X 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); ... re-try step nrerror("exceeded MAXTRY in stiff"); 742 Chapter 16 #define #define #define #define #define C2X C3X C4X A2X A3X Integration of Ordinary Differential Equations -0.396296677520e-01 0.550778939579...

Ngày tải lên: 24/12/2013, 12:16

... We are willing, however, to be corrected Gear, C.W 1971, Numerical Initial Value Problems in Ordinary Differential Equations (Englewood Cliffs, NJ: Prentice-Hall), Chapter [1] Shampine, L.F., ... some initial guess for yn+1 , insert it into the right-hand side of (16.7.2) to get an updated value of yn+1 , insert this updated value back into the right-hand side, and continue iterating But ... 16 Integration of Ordinary Differential Equations x y(x) = yn + f(x , y) dx (16.7.1) xn In a single-step method like Runge-Kutta or Bulirsch-Stoer, the value yn+1 at xn+1 depends only on yn In...

Ngày tải lên: 24/12/2013, 12:16

... called integrating factor if µ(x, y)p(x, y)y + µ(x, y)q(x, y) = is exact Finding an integrating factor is in general as hard as solving the original equation However, in some cases making an ... establish the Frobenius method for linear equations in the complex domain and investigates Sturm–Liouville type boundary value problems including oscillation theory Next we introduce the concept of a ... corresponding initial value problem using In[ 2]:= Out[2]= DSolve[{x [t] == x[t]Sin[t], x[0] == 1}, x[t], t] {{x[t] → e1−Cos[t] }} and plot it using In[ 3]:= Plot[x[t] / %, {t, 0, 2π}]; 14 Introduction...

Ngày tải lên: 31/03/2014, 15:56

... This page intentionally left blank AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS This refreshing, introductory textbook covers standard techniques for solving ordinary differential equations, ... 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, ... differential equations as models of reality One point to bear in mind in this chapter is the distinction to be made between finding the solution of a differential equation, and interpreting this solution...

Ngày tải lên: 12/05/2014, 03:24

Ngày tải lên: 12/06/2014, 16:08

... nonlinear age-structured population model Nonlinear Anal Real World Appl 8, 1096–1112 (2007) [14] Geng, F.-Z.: Solving singular second order three-point boundary value problems using reproducing ... to Boundary Value Problems December 7, 2011 Introduction Initial and boundary value problems of ordinary differential equations play an important role in many fields Various applications of boundary ... machine learning and precessing image Recently, using the reproducing kernel method, Cui and Geng [12, 13, 14, 15, 16] have make much effort to solve some special boundary value problems According...

Ngày tải lên: 20/06/2014, 20:20

... four-point nonlocal boundary value problem of nonlinear integrodifferential equations of fractional order by applying some fixed point theorems On the other hand, impulsive differential equations ... order is considered Advances in Difference Equations In addition, the nonlinear term f t, u t , u t involves u t Evidently, problem 1.5 not only includes boundary value problems mentioned above but ... Substituting the value of b1 into 2.10 and 2.13 , we obtain 2.7 Now, we introduce the fixed point theorem which was established by O’Regan in 22 This theorem will be applied to prove our main results...

Ngày tải lên: 21/06/2014, 07:20

... initial problem 3.1 for systems of ordinary differential equations with constant coefficients and a single delay, satisfying a constant initial condition, has an nparametric family of linearly independent ... boundary value problem for system 3.1 3.8 Advances in Difference Equations 3.1 Fredholm Boundary Value Problem Using the results in 8, , it is easy to derive statements for a general boundary value ... Continuing this process, by assuming that 4.17 holds, it follows by induction that the zi t, cρ of the series 4.18 can be determined, from the relevant coefficients zi t, ci boundary value problems...

Ngày tải lên: 21/06/2014, 16:20

... give a generalized Wirtinger’s inequality Then we turn to consider the problems on 1.1 and 1.2 by using the inequality In order to state our main results, we make the following definitions Definition ... of Lemma 2.1, the inequality inequality 2.1 Proof If x ∈ HT and T x t dt 2.4 0, then 2.1 follows 2.4 on taking z x implies Wirtinger’s y We call 2.4 a generalized Wirtinger’s inequality For other ... generalize Wirtinger’s inequality to a more general form which includes 2.1 as a special case We prove the following lemma Lemma 2.2 Suppose that z ∈ HT and y ∈ L2 0, T ; Rp with z, y Proof Since T y...

Ngày tải lên: 22/06/2014, 02:20

... non-linear discrete systems on in nite intervals, Mathematical Methods in the Applied Sciences 22 (1999), no 1, 91–99 , In nite Interval Problems for Differential, Difference and Integral Equations, ... varying function [11], (E2 ) ∃γ > such that f (k,u)/uγ is nonincreasing in u We start by briefly summarizing some basic properties of solutions of (1.1), which were analyzed in detail in [12] In ... difference equations, or sum difference equations Several approaches can be used in order to treat boundary value problems on in nite intervals; besides the classical ones, such as, for instance,...

Ngày tải lên: 22/06/2014, 22:20

... 4g (3.12) Inserting (3.12) in (3.11), we obtain, in Ω ω, the following inequality: LΨ + | u|−2 Wk Ψ,k ≥ 2G f g2 α− h 2 − h2 |∇h|2 − , (3.13) Maximum principles for a class of elliptic problems ... for terms containing Ψ,k Combining (3.9) with (2.5), we obtain the inequality u,ik u,ik ≥ (α + βu)2 f2 + , G2 in Ω ω, (3.10) where ω := {x ∈ Ω : ∇u(x) = 0} is the set of critical points of u and ... principles for a class of nonlinear second-order elliptic equations, Journal of Differential Equations 37 (1980), no 1, 39–48 M H Protter and H F Weinberger, Maximum Principles in Differential Equations, ...

Ngày tải lên: 22/06/2014, 22:20