Time-dependent differential equation

Important classes of polymeric flow processes are described by time-dependent differential equations. The most convenient method for solution of the time-... 

The analysis of steady-state and transient reactor behavior requires the calculation of reaction rates of neutrons with various materials. If the number density of neutrons at a point is n and their characteristic speed is v, a flux effective area of a nucleus as a cross section O, and a target atom number density N, a macroscopic cross section E = Na can be defined, and the reaction rate per unit volume is R = 0S. This relation may be appHed to the processes of neutron scattering, absorption, and fission in balance equations lea ding to predictions of or to the determination of flux distribution. The consumption of nuclear fuels is governed by time-dependent differential equations analogous to those of Bateman for radioactive decay chains. The rate of change in number of atoms N owing to absorption is as follows ... 

Volume of Monomer and Particle Phases. Additional time dependent differential equations were written for the volumes of the particle and aqueous phases. The volume of the monomer phase was calculated from the total emulsion volume and the volumes of the aqueous and polymer phases. This procedure allowed the determination of whether monomer droplets are present (Interval II) or absent (Interval III). 

Complex systems can often be represented by linear time-dependent differential equations. These can conveniently be converted to algebraic form using Laplace transformation and have found use in the analysis of dynamic systems (e.g., Coughanowr and Koppel, 1965, Stephanopolous, 1984 and Luyben, 1990). 

Now, note that the ACF (309) has the same formal structure as that used by Marechal in his peeling-off approach [83] of centrosymmetric cyclic dimers involving Fermi resonances this peeling-off ACF is, as for Eq. (309), the product of an ACF times a function that is the solution of a linear set of time-dependent differential equations having the structure of Eq. (306), but in which the matrix elements are constant. But, the Marechal procedure works in a way different from the present one ... 

So to obtain expectation values relevant to any particular experiment one needs an estimate of the density matrix at the time of measurement. For an NMR experiment, this typically requires the ability to estimate the time evolution of the density matrix for the pulse sequence used for the experiment. The time dependent differential equation that describes the time evolution of the density matrix, known as the Liouville-von Neumann equation is given by... 

The time-dependent differential equation describing the composition transformation in this still can easily be shown to be... 

W satisfies a system of linear (time-dependent) differential equations involving the matrix f (the Jacobian matrix of vector field) evaluated along the solution z(t). W may be seen as a fundamental matrix solution of the indicated Unear system. 

The time-dependent differential equation for the wave function W(x,t) bears the name of Schrodinger. For a one-dimensional case it is given by... 

The appropriate time-dependent differential equations for the com-bined-slowing-down diffusion theory may be obtained by recognizing that the time rate of change of the neutron density is given by the difference between the rates of disappearance and appearance of neutrons from sinks and sources. Thus we write... 

Now suppose that an amount of absorption 52a is added uniformly throughout the core and reflector at time / = 0 the one-velocity flux 0(r,O must now satisfy the following time-dependent differential equations ... 

Returning to our string example, we have in Eq. (1-14) a time-dependent differential equation. Suppose we wish to limit our consideration to standing waves that can be separated into a space-dependent amplitude function and a harmonic time-dependent function. Then... 

The computer programme is designed to solve a series of non-linear, simultaneous, time-dependent differential equations of the form A.z = h for the coordinates z. A and h are complicated functions of the vector z. Each step involves the inversion of the matrix A of dimensions x , where is of the order of 4 times the number of cells (=128, since the simulation is run as a closed semicircle of 32 cells, and the results are plotted with their mirror image to mimic a transverse section of an embryo). The programme used in the present series of investigations uses a new and more efficient procedure for performing the inversions of the matrix A. 

The process of ECM protein degradation was modeled by time-dependent ordinary differential equations. In the simulations, the concentration of MMPs was assumed to be at steady state, which meant that the production and degradation rates of the MMPs themselves occurred on a much larger time scale than those for the ECM. As a result, the time-dependent differential equation for the ECM proteins reduced to... 

In a separation process, ion exchange resin particles are generally used in a column. A complex time-dependent differential equation for mass balance in the column has to be combined with the diffusion flux expression for a resin particle, and other appropriate boundary and initial conditions, to determine the extent of separation. It is obvious from the preceding few paragraphs that the diffusion flux expressions are difficult to handle for resin particles. For ion exchange column analysis, practical approaches therefore utilize a linear-driving-force representation of the mass flux to a resin particle (Helfferich, 1962 Vermeulen et ai, 1973) this leads to the use of mass-transfer coefficients in resin particle flux expressions. 

U(t, to) is obtained by solving the time dependent differential equation (32) ... 

Many coupled time dependent differential equations exhibit types of solutions known as chaotie behavior . In principle, the solution of an initial value differential equation problem is eompletely determined by the differential equation and the set of initial eonditions. However, for some types of coupled nonlinear differential equations an extremely small change in the initial conditions produces a very distinguishably different time behavior. Such systems are said to exhibit chaotic behavior. One sueh system of equations is the Lorentz equations defined by ... 

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