Solutions to equations

In analogy to equation Bl.5.3. we can write the steady-state solution to equation B 1.5.10 for tlie SFIG process as... 

This Liouville-space equation of motion is exactly the time-domain Bloch equations approach used in equation (B2.4.13). The magnetizations are arrayed in a vector, and anything that happens to them is represented by a matrix. In frequency units (1i/2ti = 1), the fomial solution to equation (B2.4.26) is given by equation (B2.4.27) (compare equation (B2.4.14H. 

You can order the molecular orbitals that are a solution to equation (47) according to their energy. Electrons populate the orbitals, with the lowest energy orbitals first. Anormal, closed-shell, Restricted Hartree Fock (RHF) description has a maximum of two electrons in each molecular orbital, one with electron spin up and one with electron spin down, as shown ... 

This is a more difficult equation to solve than that for the solubility of Pb(I03)2 in distilled water, and its solution is not immediately obvious. A rigorous solution to equation 6.34 can be found using available computer software packages and spreadsheets. 

Analytieal solutions to equation 4.32 for a single load applieation are available for eertain eombinations of distributions. These coupling equations (so ealled beeause they eouple the distributional terms for both loading stress and material strength) apply to two eommon eases. First, when both the stress and strength follow the Normal distribution (equation 4.38), and seeondly when stress and strength ean be eharaeterized by the Lognormal distribution (equation 4.39). 

The outlet eoneentration from a maximum mixedness reaetor is found by evaluating the solution to Equation 9-34 at = 0 =... 

C (0). The analytieal solution to Equation 9-34 is rather eomplex for reaetion order n > 1, the (-r ) term is usually non-linear. Using numerieal methods, Equation 9-34 ean be treated as an initial value problem. Choose a value for = C (0) and integrate Equation 9-34. If C (A.) aehieves a steady state value, the eorreet value for C (0) was guessed. Onee Equation 9-34 has been solved subjeet to the appropriate boundary eonditions, the eonversion may be ealeulated from Caouc = Ca(0). 

This solution for (X) in which all the unknowns are zero is called the trivial solution. A nontrivial solution to Equation (A.27) exists, therefore, only when matrix [A] is singular, that is, when A = 0. 

The various solutions to Equation 3 correspond to different stationary states of the particle (molecule). The one with the lowest energy is called the ground stale. Equation 3 is a non-relativistic description of the system which is not valid when the velocities of particles approach the speed of light. Thus, Equation 3 does not give an accurate description of the core electrons in large nuclei. 

Refer to equation 5, which relates Ny to the parameters in the reactor. For the continuous reactor these parameters are evaluated at t = t. However, the solution to equation 5 is complicated by the fact that Ny is not only on the left hand side, but Ny also appears in the expression for Rj p as a first power. Newton s method of convergence is used to solve equation 5 for the continuous reactor. 

Just as the reactions are consecutive, solutions to this set can be carried out consecutively. The equation for component A depends only on a and can be solved directly. The result is substituted into the equation for component B, which then depends only on b and t and can be solved. This procedure is repeated until the last, stable component is reached. Assuming component D is stable, the solutions to Equations (2.21) are... 

A solution to Equation (8.12) together with its boundary conditions gives a r, z) at every point in the reactor. An analytical solution is possible for the special case of a first-order reaction, but the resulting infinite series is cumbersome to evaluate. In practice, numerical methods are necessary. 

The first step in developing the numerical method is to And a formal solution to Equation (8.63). Observe that Equation (8.63) is variable-separable ... 

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 output tracer signal is attenuated and shows a phase shift, but there is no change in frequency. All solutions to Equations (15.45) and (15.46) have these characteristics. Differentiate Pm tot — tot co tot to show that the maximum deviation occurs when cot tot =—tot. Some trigonometry then shows that the maximum deviation is... 

For simplicity let us assume there is no cyclization. When f2 1, one can derive an analytical solution to Equation 11. 

If the coefficients involved in equation (44) are constant, that is, Ai = a, Ci — c and B, b, then particular solutions can be found in explicit form. This can be done by attempting particular solutions to equation (45) in the form j/j, =, while the number q 0 remains as yet unknown. 

It is well-known that the general solution to equation (3) is of the form... 

That is to say, the meaning of stability of scheme (21) is that a solution (21) depends continuously on the right-hand side and this dependence is uniform in the parameter h. This implies that a small change of the right-hand side results in a small change of the solution. If the scheme is solvable and stable, it is correct. Note that the uniqueness of the scheme (21) solution is a consequence of its solvability and stability and, hence, we might get rid of the uniqueness requirement in condition (1). Indeed, assume to the contrary that there were two solutions to equation (21), say and By the linearity property of the operator A, their difference = y — yf should satisfy the homogeneous equation... 

Some a priori estimates. We now consider several simplest a priori estimates for a solution to equation (21), the form of which depends on the subsidiary information on the operator of a scheme. These estimates are typical for difference elliptic problems. 

Assuming this to be the case, it is required to find a solution to equation (12) subject to the periodicity condition (x + l) = u(x), which is equivalent to the requirements... 

Conditions (27) and (28) together provide the existence of a unique solution to equation (26). By virtue of property 1) condition (27) can be replaced by... 

The stability of a solution to equation (11) with respect to perturbations of the right-hand side / and perturbations of the operator A is called strong stability. The problem statement here is as follows with regard to the equations... 

Theorem Let u be a solution to equation (11) and u be a solution to equation (14), where A, A and Aq are self-adjoint positive operators for which the inverse operators exist. If condition (18) and the inequality A > CjTo, Cj > 0 hold, then the estimates are valid ... 

Indeed, by introducing x = x/a and denoting once again x by x we obtain (3). Where searching a solution to equation (2) on the segment 0 < x < I, it is sensible to pass to the dimensionless variables... 

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