Constant coefficient

Viscoelastic parameters Ki, K2 and q of the model are determined by fitting the experimental curve V(t) to the sum of cosines with constant coefficients ... 

Taking the natural logaritlnn of (A3.1.54), we see that In+ In has to be conserved for an equilibrium solution of the Boltzmaim equation. Therefore, Incan generally be expressed as a linear combination with constant coefficients... 

The method shown affords easy generalization to higher order coupling in the important case where a single mode is engaged, that is, G i = g i = (l/i /2) e . Then the two off-diagonal terms derived above are, after physics-based constant coefficients have been affixed, in the upper right comer... 

From the derivation of the method (4) it is obvious that the scheme is exact for constant-coefficient linear problems (3). Like the Verlet scheme, it is also time-reversible. For the special case A = 0 it reduces to the Verlet scheme. It is shown in [13] that the method has an 0 At ) error bound over finite time intervals for systems with bounded energy. In contrast to the Verlet scheme, this error bound is independent of the size of the eigenvalues Afc of A. 

The procedure we followed in the previous section was to take a pair of coupled equations, Eqs. (5-6) or (5-17) and express their solutions as a sum and difference, that is, as linear combinations. (Don t forget that the sum or difference of solutions of a linear homogeneous differential equation with constant coefficients is also a solution of the equation.) This recasts the original equations in the foiin of uncoupled equations. To show this, take the sum and difference of Eqs. (5-21),... 

Neither of these equations tells us which spin is on which electron. They merely say that there are two spins and the probability that the 1, 2 spin combination is ot, p is equal to the probability that the 2, 1 spin combination is ot, p. The two linear combinations i i(l,2) v /(2,1) are perfectly legitimate wave functions (sums and differences of solutions of linear differential equations with constant coefficients are also solutions), but neither implies that we know which electron has the label ot or p. 

Generally, Oijki depends on x. The isotropic solid is characterized by the constant coefficients Oijki of the form... 

Rate Equations with Concentration-Independent Mass Transfer Coefficients. Except for equimolar counterdiffusion, the mass transfer coefficients appHcable to the various situations apparently depend on concentration through thej/g and factors. Instead of the classical rate equations 4 and 5, containing variable mass transfer coefficients, the rate of mass transfer can be expressed in terms of the constant coefficients for equimolar counterdiffusion using the relationships... 

All these results generalize to homogeneous linear differential equations with constant coefficients of order higher than 2. These equations (especially of order 2) have been much used because of the ease of solution. Oscillations, electric circuits, diffusion processes, and heat-flow problems are a few examples for which such equations are useful. 

Substitution If it is possible to rearrange a difference equation so that it takes the form af oy, o + hf 1 -1- cfy, = ( )(x) with a, b, c constants, then the substitution =fxyx reduces the equation to one with constant coefficients. 

The number of independent rate equations is the same as the number of independent stoichiometric relations. In the present example. Reactions (1) and (2) are reversible reactions and are not independent. Accordingly, C,. and C, for example, can be eliminated from the equations for and which then become an integrable system. Usually only systems of linear differential equations with constant coefficients are solvable analytically. 

The unsteady material balances of tracer tests are represented by linear differential equations with constant coefficients that relate an input function Cj t) to a response function of the form... 

In general, consider a system whose output is x t), whose input is y t) and contains constant coefficients of values a, h, c,..., z. If the dynamics of the system produce a first-order differential equation, it would be represented as... 

Equations (2.9), (2.10) and (2.11) are linear differential equations with constant coefficients. Note that the order of the differential equation is the order of the highest derivative. Systems described by such equations are called linear systems of the same order as the differential equation. For example, equation (2.9) describes a first-order linear system, equation (2.10) a second-order linear system and equation (2.11) a third-order linear system. 

For a first order reaction (-r ) = kC, and Equation 8-147 is then linear, has constant coefficients, and is homogeneous. The solution of Equation 8-147 subject to the boundary conditions of Danckwerts and Wehner and Wilhelm [23] for species A gives... 

Linear differential equations with constant coefficients can be solved by a mathematical technique called the Laplace transformation . Systems of zero-order or first-order reactions give rise to differential rate equations of this type, and the Laplaee transformation often provides a simple solution. 

Systems of reversible first-order reactions lead to sets of simultaneous linear differential equations with constant coefficients. A solution may be obtained by means of a matrix formulation that is widely used in quantum mechanics and vibrational... 

The preceding two equations are examples of linear differential equations with constant coefficients and their solutions are often found most simply by the use of Laplace transforms [1]. 

Now we can increase the BOD removal efficiency witli decreasing organic loading. Wliat would be the amount of oxygen required per kg BOD removal, with 90% removal efficiency Given data for constant coefficient a = 0.5 and b = 0.3... 

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... 

Summing the question of the variational equations, one must say that in relatively simple systems with constant coefficients, there is no particular difficulty in carrying out these calculations. But in more... 

Suppose we have a simple differential equation with constant coefficients... 

For both of these cases, Eqs. (13)—(15) constitute a system of two linear ordinary differential equations of second order with constant coefficients. The boundary conditions are similar to those used by Miyauchi and Vermeulen, which are identical to those proposed by Danckwerts (Dl). The equations may be transformed to a dimensionless form and solved analytically. The solutions may be recorded in dimensionless diagrams similar to those constructed by Miyauchi and Vermeulen. The analytical solutions in the present case are, however, considerably more involved algebraically. 

The constant coefficient for float C arises from turbulence promotion, and for this reason the coefficient is also substantially independent of the fluid viscosity. The meter can be made relatively insensitive to changes in the density of the fluid by selection of the density of the float, pf. Thus the flowrate for a given meter will be independent of p when dG/dp = 0. 

Equation (9.14) is a linear ODE with constant coefficients. An analytical solution is possible when the reactor is isothermal and the reaction is first order. The general solution to Equation (9.14) with = —ka is... 

The ODEs are linear with constant coefficients. They can be converted to a single, second order ODE, much like Equation (11.22), if an analytical solution is desired. A numerical solution is easier and better illustrates what is necessary for anything but the simplest problem. Convert the independent variable to dimensionless form, = z/L. Then... 

Equation (15.34) is the system model. It is a linear PDE with constant coefficients and can be converted to an ODE by Laplace transformation. Define... 

The second-order difference equations with constant coefficients. If... 

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