Separating the variables

In order to And a solution to the differential equation (2.160) whilst accounting for the initial and boundary conditions, the method of separating the variables or product solution is used3 

The functions F and G each depend on only one variable, and satisfy the following differential equation from (2.160) 

3The application of the Laplace transformation delivers the same result. The inverse transformation from the frequency to the time region requires the use of the inversion theorem, see 2.3.2. In order to avoid this in this case the simple, classical product solution is applied. 

The left hand side of (2.164) depends on the (dimensionless) time f+, the right hand side on the position coordinate r+ the variables are separated. The equality demanded by (2.164) is only possible if both sides of (2.164) are equal to a constant —fi2. This constant /( is known as the separation parameter. With this the following ordinary differential equations are produced from (2.164) 

Products of their solutions with the same value of the separation parameter /t give solutions to the heat conduction equation (2.160). 

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Separating the variables and integrating between the limits shown below yields Eqs. (2-6), (2-7), and (2-8) as equivalent forms of the integrated first-order rate equation. 

The stoichiometric coefficient 2 is commonly omitted from the rate expression because there is only one reactant species, so no ambiguity exists.) Separating the variables and integrating between the usual limits gives Eq. (2-15). 

If the Hamiltonian would be the sum of one-electron operators only, one could easily separate the variables in the basic Schrodin-ger equation (Eq. II. 1), and the total wave function 0 would then be the product of N one-particle functions each one being an... 

After separating the variables, (3.14.3.1) was solved by integration and the initial conditions were implemented (/0 = 0, S = S0). The resulting expression is... 

The most accurate method of measuring the influence of temperature on reaction rate is to separate the variables by first determining isothermal rate curves at a series of different temperatures and expressing each set of observations in the form... 

After separating the variables, this equation can be solved by integration between the limits (fo = 0, [A]0) and (t, [A],). The integrated expression is... 

Upon substituting this product in (15) and separating the variables we obtain... 

Separating the variables s and t to different sides and integrating from... 

In order to solve the wave equation for the hydrogen atom, it is necessary to transform the Laplacian into polar coordinates. That transformation allows the distance of the electron from the nucleus to be expressed in terms of r, 9, and (p, which in turn allows the separation of variables technique to be used. Examination of Eq. (2.40) shows that the first and third terms in the Hamiltonian are exactly like the two terms in the operator for the hydrogen atom. Likewise, the second and fourth terms are also equivalent to those for a hydrogen atom. However, the last term, e2/r12, is the troublesome part of the Hamiltonian. In fact, even after polar coordinates are employed, that term prevents the separation of variables from being accomplished. Not being able to separate the variables to obtain three simpler equations prevents an exact solution of Eq. (2.40) from being carried out. 

It is noted that (ip1)2 + ip2 k2. The symmetry is broken by separating the variables and this eliminates the quantum-mechanical equivalence of proton and electron. Only electrostatic interaction, V = 4pfoT remains in the electronic wave equation. 

Equation (12) can be solved analytically, by separating the variables according to standard procedures. Because of the nature of the Coulomb potential it is necessary to transform to spherical polar coordinates first i.e. 

Basically, there are two different ways to decompose a 2S-MILP (see Figure 9.10). The scenario decomposition separates the 2S-MILP by the constraints associated to a scenario, whereas the stage decomposition separates the variables into first-stage and second-stage decisions. For both approaches, the resulting subproblems are MILPs which can be solved by standard optimization software. 

This equation represents Cp for a single, pure substance. Separating the variables yields... 

Writing the equation in this way tells us that if we know the enthalpy of the system, we also know the temperature dependence of G -i-T. Separating the variables and defining Gj as the Gibbs function change at Ti and similarly as the value of G2 at T2, yields... 

Next, we multiply together the two T terms, rearrange and separate the variables, to... 

Separating the variables (i.e. rearranging the equation) and indicating the limits, we obtain... 

Having derived the symmetry relations between the expansion parameters in equation (55), we can proceed to fit the expansions through the ab initio dipole moment values. The expansion parameters in the expressions for and fiy are connected by symmetry relations since these two quantities have E symmetry in and so and fiy must be fitted together. The component ji, with A" symmetry, can be fitted separately. The variables p in equation (55) are chosen to reflect the properties of the potential surface, rather than those of the dipole moment surfaces. Therefore, the fittings of fi, fiy, fifi require more parameters than the fittings of the MB dipole moment representations. We fitted the 14,400 ab initio data points using 77 parameters for the component and 141 parameters for fi, fiy. The rms deviations attained were 0.00016 and 0.0003 D, respectively. 

After obtaining derivatives for both sides, separating the variables, and integrating one obtains... 

If the reaction is carried out iso thermally, fe is a constant and, separating the variables... 

To further reduce the computational burden, an attempt was made to separate the variables. To see how this may be implemented, let us consider Eq. (A.l), which enforces the minimum negativity constraint. Note that it may also be written ... 

This net symmetric regauging operation successfully separates the variables, so that two inhomogeneous wave equations result to yield the new Maxwell... 

A value of a = 1 can also be obtained in some cases where a reactant B is involved, under experimental conditions where [B] = [B]e, a concentration much larger than [A]. We can rearrange Equation 15-5 to separate the variables, so that only [A] is on the lefthand side of the equation and only t on the righthand side ... 

This equation may be integrated by separating the variables and multiplying each side by an exponential factor ... 

Note that N is constantly reducing in magnitude as a function of time. Rearrangement of Equation (3.2) to separate the variables gives... 

Separating the variables involved and integrating the equations between Mpo and 0, the distance the particle traveled in the first decelerating stage, i.e., the maximum depth of penetration into the opposed stream, xmax, can be obtained as... 

Assuming uniaxial symmetry of the time-dependent solution and separating the variables in Eq. (4.125) in the form... 

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