application to solution of linear ordinary differential equations with constant coefficients

... and monotone iterative technique to investigate the existence of minimal nonnegative solution for a class of second-order nonlinear impulsive differential equations on an infinite interval with an ... some recent developments and applications of the case that impulse effects on a finite interval with a finite number of impulsive times to a variety of problems from Nieto and Rodr´guez-Lopez 14–16 ... & Applications, vol 25, no 4, pp 327–337, 1995 D Guo, “Existence of solutions of boundary value problems for nonlinear second order impulsive differential equations in Banach spaces,” Journal of...

Ngày tải lên: 21/06/2014, 11:20

Ngày tải lên: 22/03/2014, 11:20

... Functional differential equations with state-dependent delays: theory and applications, in: A Canada, P Drabek, A Fonda (Eds.), Handbook of Differential Equations: Ordinary Differential Equations, ... uniqueness of solutions of (12) Now we look for an a priori estimate to prove the continuation of solutions um of (12) on any time interval [0, T ] and then use it for the proof (by the method of compactness; ... O.D.E.s with SDDs [9,10,17] to the case of P.D.E.s Our idea is to look for a wider space Y ⊃ X such that a solution u : [a, b] → Y is a Lipschitz function (with respect to the weaker norm of Y )...

Ngày tải lên: 29/10/2015, 14:19

... (2.5) is a solution of (2.1) if λ is a root of the characteristic equation (2.3) Roots of the characteristic equation have to be positive for the functions (2.5) to be defined To show the solutions ... proof is complete Automorphic functions A function p ∈ C0 (᏶) is called automorphic over Φ provided p = p ◦ Φ While constant functions are automorphic with respect to any Φ, there are examples of ... −2, , where Φ−1 is the inverse element to the element Φ in S according to multiplication ◦ in group Ᏻ Linear functional equations of the kth order with constant coefficients Let a j ∈ R, j = 0,1,2,...

Ngày tải lên: 22/06/2014, 22:20

... 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 ... Journal of Inequalities and Applications For the special case that n and p 1, various problems on the solutions of 1.1 , such as the existence of periodic solutions, bifurcations of periodic solutions, ... Methods & Applications, vol 35, no 4, pp 457–474, 1999 J Li and X.-Z He, “Multiple periodic solutions of differential delay equations created by asymptotically linear Hamiltonian systems,” Nonlinear...

Ngày tải lên: 22/06/2014, 02:20

... odeint are drivers Section 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 ... ODE integrator that implemented this idea was developed by Bulirsch and Stoer, and so extrapolation methods are often called Bulirsch-Stoer methods Predictor-corrector methods store the solution ... types of methods can be applied to any initial value problem Each comes with its own set of debits and credits that must be understood before it is used 710 Chapter 16 Integration of Ordinary Differential...

Ngày tải lên: 15/12/2013, 04:15

... increments with proper 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 ... sequence of steps in 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 ... efficiency are not mere tens of percents or factors of two; they can sometimes be factors of ten, a hundred, or more Sometimes accuracy may be demanded not directly in the solution itself, but in...

Ngày tải lên: 15/12/2013, 04:15

... either no solution, or else more than one solution vector x In the latter event, the solution space consists of a particular solution xp added to any linear combination of (typically) N − M vectors ... Much of the sophistication of complicated linear equation-solving packages” is devoted to the detection and/or correction of these two pathologies As you work with large linear sets of equations, ... occur during the solution In fact, the preceding item can be viewed as the special case where the loss of significance is unfortunately total 34 Chapter Solution of Linear Algebraic Equations at...

Ngày tải lên: 15/12/2013, 04:15

... two rows of A and the corresponding rows of the b’s and of 1, does not change (or scramble in any way) the solution x’s and Y Rather, it just corresponds to writing the same set of linear equations ... scrambles the order of the rows in the solution If we this, we will need to unscramble the solution by restoring the rows to their original order Gauss-Jordan elimination uses one or more of the above ... the solution set is unchanged and in no way scrambled if we replace any row in A by a linear combination of itself and any other row, as long as we the same linear combination of the rows of the...

Ngày tải lên: 15/12/2013, 04:15

... new factorization to solve the next set of linear equations The LU decomposition is complicated to update because of pivoting However, QR turns out to be quite simple for a very common kind of update, ... succession of linear systems each of which differs only slightly from its predecessor Instead of doing O(N ) operations each time to solve the equations from scratch, one can often update a matrix factorization ... } The next routine, qrsolv, is used to solve linear systems In many applications only the part (2.10.4) of the algorithm is needed, so we separate it off into its own routine rsolv Sample page...

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

... powers of h, 724 Chapter 16 Integration of Ordinary Differential Equations } CITED REFERENCES AND FURTHER READING: Gear, C.W 1971, Numerical Initial Value Problems in Ordinary Differential Equations ... 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 effort ... low-accuracy solution of a set of equations A second warning is that the techniques in this section are not particularly good for differential equations that have singular points inside the interval of...

Ngày tải lên: 24/12/2013, 12:16

... 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 ... are then extrapolated to zero stepsize The sequence of separate attempts to cross the interval H is made with increasing values of n, the number of substeps Bulirsch and Stoer originally proposed ... 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...

Ngày tải lên: 24/12/2013, 12:16

... 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 are stored in ... 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 ... 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 should also use the more efficient sequence of stepsizes suggested...

Ngày tải lên: 24/12/2013, 12:16

... generalized to sets of linear equations with constant coefficients: y = −C · y (16.6.9) yn+1 = (1 − Ch) · yn (16.6.10) Now a matrix An tends to zero as n → ∞ only if the largest eigenvalue of A has ... name predictor-corrector is often loosely used to denote all these methods We suspect that predictor-corrector integrators have had their day, and that they are no longer the method of choice ... gives In general this is some nasty set of nonlinear equations that has to be solved iteratively at each step Suppose we try linearizing the equations, as in Newton’s method: yn+1 = yn + h f(yn )...

Ngày tải lên: 24/12/2013, 12:16

... experience with predictor-corrector routines, and they see no reason to make a precipitous change of habit It is not a bad idea for you to be familiar with the principles involved, and even with the ... 16 Integration of Ordinary Differential Equations For multivalue methods the basic data available to the integrator are the first few terms of the Taylor series expansion of the solution at 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...

Ngày tải lên: 24/12/2013, 12:16

... Westlake, J.R 1968, A Handbook of Numerical Matrix Inversion and Solution of Linear Equations (New York: Wiley) Suppose we are able to write the matrix A as a product of two matrices, L·U=A (2.3.1) ... solving What is the advantage of breaking up one linear set into two successive ones? The advantage is that the solution of a triangular set of equations is quite trivial, as we have already seen ... backsubstitution The combination of Gaussian elimination and backsubstitution yields a solution to the set of equations The advantage of Gaussian elimination and backsubstitution over Gauss-Jordan elimination...

Ngày tải lên: 24/12/2013, 12:16

... column in the reduction free_vector(vv,1,n); 48 Chapter Solution of Linear Algebraic Equations To summarize, this is the preferred way to solve the linear set of equations A · x = b: float **a,*b,d; ... the loop of the above fragment and (e.g.) divide by powers of ten, to keep track of the scale separately, or (e.g.) accumulate the sum of logarithms of the absolute values of the factors and ... America) Determinant of a Matrix 50 Chapter Solution of Linear Algebraic Equations A quick-and-dirty way to solve complex systems is to take the real and imaginary parts of (2.3.16), giving A·x−C·y=b...

Ngày tải lên: 24/12/2013, 12:16

... of a Solution to Linear Equations A δx x+ b δx δb A−1 Figure 2.5.1 Iterative improvement of the solution to A · x = b The first guess x + δx is multiplied by A to produce b + δb The known vector ... and r[1 n] are input vectors and are not modified { unsigned long j; float bet,*gam; 52 Chapter Solution of Linear Algebraic Equations In that case, the solution of the linear system by LU decomposition ... δb The linear set with this right-hand side is inverted, giving δx This is subtracted from the first guess giving an improved solution x 2.5 Iterative Improvement of a Solution to Linear Equations...

Ngày tải lên: 24/12/2013, 12:16

... savings is not just a factor “7/8”; it is that factor at each hierarchical level of the recursion In total it reduces the process of matrix multiplication to order N log2 instead of N What about all ... 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America) c22 = Q1 + Q3 − Q2 + Q6 104 Chapter Solution of Linear Algebraic Equations CITED REFERENCES AND FURTHER READING: ... 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, is strictly prohibited To order...

Ngày tải lên: 24/12/2013, 12:16