ORDINARY DIFFERENTIAL EQUATIONS AND DYNAMICAL SYSTEMS: PART 1

4.5.3 Data Uncertainty
Errors sometimes enter into an analysis because of uncertainty in the physical data on which a model is based. For instance, suppose we wanted to test the bungee jumper model by hav- ing an individual make repeated jumps and then measuring his or her velocity after a[r]

1. Given a value of ω , solve equation (15) for ψ . Discretizing the ∇ 2 operator turns this into a matrix equation of the form A ψ ~ = ~ ω where ψ ~ is a discretized vector rearrangement of ψ and similarly for ~ ω .
2. Discretize equation (14) so that we get another matrix equation[r]

scale,” International Journal of Nonlinear Di fferential Equations, vol. 7, pp. 97–104, 2002.
[3] D. R. Anderson, “Eigenvalue intervals for a two-point boundary value problem on a measure
chain,” Journal of Computational and Applied Mathematics, vol. 141, no. 1-2, pp. 57–64,[r]

Autonomous nonlinear differential equations constituted a system of ordinary differential equations, which often applied in different areas of mechanics, quantum physics, chemical engineering science, physical science, and applied mathematics. It is assumed that the secondorder autonomous nonlinear[r]

3. Step forward in time using a Runge-Kutta ordinary differential equation solver to get ω at a small time in the future.
This new value of ω is then used in step 1 and the process can continue indef- initely. We will examine various techniques for solving the matrix eq[r]

M. Benchohra, J. Henderson, and S. K. Ntouyas
Received 28 June 2007; Accepted 19 November 2007 Recommended by Kanishka Perera
Values of λ are determined for which there exist positive solutions of the system of dy- namic equations, u ΔΔ (t) + λa(t) f (v(σ (t))) = 0, v ΔΔ (t) + λb(t[r]

The methods that we shall develop here are applicable to equations with regular singular or ordinary points; we cannot, in general, use this same approach to find a solution in the neigh[r]

control the particular type and order of the solution of the Bessel equation which is described in the volume ‘The series solution of second order, ordinary differential equations and sp[r]

THE PATH CONSISTS OF CON-TINUOUS SEGMENTS REPRESENTING SOLUTIONS OF THE ORDINARY DIFFERENTIAL EQUATIONS 15.14.4.65 RAREFACTION WAVES, LINE SEGMENTS THAT CONNECT TWO POINTS U – AND U + SA[r]

18. Zhang, X, Feng, M, Ge, W: Existence results for nonlinear boundary-value problems with integral boundary conditions in Banach spaces. Nonlinear Anal. 69 , 3310 – 3321 (2008). doi:10.1016/j.na.2007.09.020
19. Anderson, DR, Tisdell, CC: Third-order nonlocal problems with sign-changing nonlineari[r]

x t = y 3 ϕ ( x , y , t ) + x 5 , y t = x 3 ϕ ( x , y , t ) + y 5 , where ϕ ( x , y , t ) is an arbitrary continuous function.
Let us show that the W = x 4 – y 4 satisfies the conditions of the Chetaev theorem. We have: 1. W > 0 for | x | > | y | , W = 0 for | x | = | y | .

11 Differential Equation Theory
110 Existence and uniqueness of solutions
A fundamental question that arises in scientific modelling is whether a given differential equation, together with initial conditions, can be reliably used to predict the behaviour of the trajectory at later times. We[r]

We do not pursue the formal properties of graphs, and of rooted trees in particular, because they are formulated in specialist books on this subject and are easily appreciated through examples and diagrams. In diagrammatic depictions of a directed graph, the vertices are represe[r]

(1)
in which the function y ( x ) is to be found.
In order to apply the Fourier transform technique (see Subsections 7.4-3, 10.4-1, and 10.4-2), we extend the domain of both conditions in Eq. (1) by formally rewriting them for all real values of x . This can be achiev[r]

[r]

w(x, t) = ψ 1 (t) + ψ 2 (t)x + ψ 3 (t)x 2 + ψ 4 (t)x 3 .
15.5.5-3. How to find linear subspaces invariant under a given nonlinear operator. The most difficult part in using the Titov–Galaktionov method for the construction of exact solutions to specific equations is to find li[r]

Rice, J.R. 1983, Numerical Methods, Software, and Analysis (New York: McGraw-Hill), § 9.2.
16.2 Adaptive Stepsize Control for Runge-Kutta
A good ODE integrator should exert some adaptive control over its own progress, making frequent changes in its stepsize. Usually the purpose of this ad[r]

those maximum values. A useful “trick” for getting constant fractional errors except
“very” near zero crossings is to set yscal[i] equal to | y[i] | + | h × dydx[i] | . (The routine odeint , below, does this.)
Here is a more technical point. We have to consider one additional possibility for ysc[r]

77
The time necessary to reach 99.9% of the equilibrium concentration of mutagenic tautomer in the system ( τ 99.9%) for these barriers falls within the range 3.84·10 -8 ÷ 2.13·10 -4 s, which is by orders smaller, except Cyt, than the time of an elementary act of one base pair replication (c[r]

Integration of Ordinary Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING ISBN 0-521-43108-5 Copyright C 1988-1992 by Cambridge University P[r]