First-order approximation differential equations

Let us first consider particles sufficiently large that diffusion may be neglected. For hindered settling the assumption corresponding to Kynch s model that the fall speed depends only on the particle concentration is that s = s(p). With these approximations Eq. (5.5.3) reduces to a first-order partial differential equation, which may be written... 

When distance is measured in AU units (lAU = 1.496 el3 m) and time is measured in years the value of K is approximately 40. This example illustrates two coupled second order differential equations or they may be considered a system of four first order coupled differential equations. Many solutions are possible depending on the set of initial conditions for the position and velocities. 

These coupled second-order partial differential equations do not have a closed-form solution. Accordingly, the approximate numerical technique of finite differences is employed. First, however, the boundary conditions must be prescribed in order to complete the formulation of the problem. Symmetry of the laminate about several planes permits reduction of the region of consideration to a quarter of the laminate cross section in the y-z plane at any value of x as shown in Figure 4-52. There, along the stress-free upper surface. 

Combining Eq. (24) with Eqs. (12) (14), Chen obtained f(a) for given N and c by the following numerical analysis. Firstly, a zero-th approximation to U(a) was calculated by numerical integration of Eq. (24) using a properly chosen zero-th approximation to f(a), secondly the calculated U(a) was used to obtain a first-order approximation to f(a) by solving the differential equation, Eq. (12), with Eqs. (13) and (14), and finally the process was iterated until the mean-square relative difference between the two successive approximations toT(a) became less than a prescribed small value. 

The solution, 6Ak, of this set of normal equations is a first-order approximation of the changes in Ak required to obtain the parameters Ak. If any 6Afc > e (error limit), Ak is replaced by Ak + 6Ak and the entire differential-correction procedure is repeated using these new estimates. 

This set of four partial differential equations, then, are the first-order approximations to the Boltzmann equation which satisfy the requirements (7.106), (7.111), (7.137), and (7.146). 

In general, two classes of spectral domain-decomposition methods have been proposed in the literature patching methods and variational methods. The difference is in the way how the interface conditions are imposed. To solve second-order partial differential equations as an exanple, the interface condition is typically enforced by requiring that the solution and its first normal derivative be continuous on each interface. In patching methods, the continuity conditions on each interface are discretized by enforcing them at selected points, and thus are satisfied exactly by any approximation. In variational methods, on the other hand, the continuity conditions are enforced implicitly or variationaUy with differential... 

From (5.56) one can obtain an integro-differential equation for operator What we need is the mean particle position, <(Tz>, and in order to find it two approximations are made. First, in taking the bath averages we assume free bath dynamics. Second, we decouple the bath and pseudospin averages, guided by perturbation theory. The result is a Langevin-like equation for the expectation <(T2> [Dekker 1987a Meyer and Ernst 1987 Waxman 1985],... 

For differential equations with periodic coefficients, the theorems are the same but the calculation of the characteristic exponents meets with difficulty. Whereas in the preceding case (constant coefficients), the coefficients of the characteristic equation are known, in the present case the characteristic equation contains the unknown solutions. Thus, one finds oneself in a vicious circle to be able to determine the characteristic exponents, one must know the solutions, and in order to know the latter, one must know first these exponents. The only resolution of this difficulty is to proceed by the method of successive approximations.11... 

The correction to the relaxing density matrix can be obtained without coupling it to the differential equations for the Hamiltonian equations, and therefore does not require solving coupled equations for slow and fast functions. This procedure has been successfully applied to several collisional phenomena involving both one and several active electrons, where a single TDHF state was suitable, and was observed to show excellent numerical behavior. A simple and yet useful procedure employs the first order correction F (f) = A (f) and an adaptive step size for the quadrature and propagation. The density matrix is then approximated in each interval by... 

Its modification gives the approximation of the first-order differential equation... 

The modeling of steady-state problems in combustion and heat and mass transfer can often be reduced to the solution of a system of ordinary or partial differential equations. In many of these systems the governing equations are highly nonlinear and one must employ numerical methods to obtain approximate solutions. The solutions of these problems can also depend upon one or more physical/chemical parameters. For example, the parameters may include the strain rate or the equivalence ratio in a counterflow premixed laminar flame (1-2). In some cases the combustion scientist is interested in knowing how the system mil behave if one or more of these parameters is varied. This information can be obtained by applying a first-order sensitivity analysis to the physical system (3). In other cases, the researcher may want to know how the system actually behaves as the parameters are adjusted. As an example, in the counterflow premixed laminar flame problem, a solution could be obtained for a specified value of the strain... 

This is identical to the first order Fade power series and gives a crude time delay approximation, when transformed back into differential equation form. 

It should be observed that k.3 was approximated to zero in the above treatment. The differential equations describing B and C concentrations are linear ones with respect to the participating concentrations. The expression for the A-concentration is inserted in eqs. (10b) and (10c) and the first order differential equations are solved with the initial conditions Cb=0 and Cc=0 at t=0. The solutions become... 

The partial differential equations representing material and energy balances of a reaction in a packed bed are rarely solvable by analytical means, except perhaps when the reaction is of zero or first order. Two examples of derivation of the equations and their analytical solutions are P8.0.1.01 and P8.01.02. In more complex cases analytical, approximations can be made (by "Collocation" or "Perturbation", for instance), but these usually are quite sophisticated to apply. Numerical solutions, on the other hand, are simple in concept and are readily implemented on a computer. Two such methods that are suited to nonlinear kinetics problems will be described. 

In order to describe the fluorescence radiation profile of scattering samples in total, Eqs. (8.3) and (8.4) have to be coupled. This system of differential equations is not soluble exactly, and even if simple boundary conditions are introduced the solution is possible only by numerical approximation. The most flexible procedure to overcome all analytical difficulties is to use a Monte Carlo simulation. However, this method is little elegant, gives noisy results, and allows resimulation only according to the method of trial and error which can be very time consuming, even in the age of fast computers. Therefore different steps of simplifications have been introduced that allow closed analytical approximations of sufficient accuracy for most practical purposes. In a first... 

For each occupied orbital y = 1,..., n, we have to solve the set of N -h 1 linear first-order differential equations (21). Unfortunately, it does not seem possible to obtain analytical solutions for any realistic choice of F(t) and, therefore, it is necessary to resort to finding approximate or numerical... 

The integrals in (28) and (54) are approximated by q Gaussian quadrature points, so that, for each quadrature energy point in (28), there is a set of 9 + 1 first-order differential equations to be solved, since... 

This set of first-order differential equations can be solved, approximately or numerically, for a specific system. The theory has been applied to Li scattered from Cs/W, and gives more satisfactory agreement with experiment than does the one-electron approach. 

This first order differential equation now governs the evolution of an initial field. For a finite step length the propagation is approximated by... 

The input data structure is very similar to the one in the module 1445. Two user routines are to be supplied. The first one starts at line 900 and evaluates the right hand sides of the differential equations. The second routine, starting at line 800, serves for computing the initial conditions at the current estimates of the parameters. If the initial estimates are parameter independent (we know them exactly), then this routine simply puts the known values into the variables YI(1),. .., YI(NY). The required partial derivatives are generated using divided differences approximation. In order to ease the use of the module a very simple example is considered here. 

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