Power-series solution

A power series solution of equation (G.27) yields a recursion formula relating i+4, Ui+2, and ak, which is too complicated to be practical. Accordingly, we make the further definition... 

Agee, L. J., 1978, Power Series Solutions of the Thermal-Hydraulic Conservation Equations, in Transient Two-Phase Flew, Proc. 2nd Specialists Meeting, OECD Committee for the Safety of Nuclear Installations, Paris, Vol. 1, pp. 385-410. (3)... 

When F and G in (1.92) are not constants, a useful approach is to try a power-series solution, a topic treated in most quantum-chemistry texts. 

Carrying out a power-series solution, one finds that the quadratically integrable normalized solutions of (1.132) are... 

If the radial equation is solved by using the more conventional power series solution,2 the two mathematically allowed solutions for R(r) have leading terms... 

A plain power-series solution is not practical the asymptotic behavior of j/ x) at large x requires instead a power series with a prefactor exp[—(co / ti)x2 / 2). ... 

Now, this equation looks perfectly amenable to plugging in a power series for u and chugging away. We have eliminated one degree of the variable and one of the constants. However, you will find, if you attempt a power series solution at this point, that you can write down a dependence among... 

For the free (unconfined) case, limr co R(r) = 0 and a = n, with n a positive integer, resulting in the usual energy eigenvalue formula E = —. However, for the CHA the wave functions must vanish at a finite value of the radial variable ro (Equation (4)), therefore, a cannot be an integer. Michels et al. proposed a power series solution/ (r) = Xj+ bsrs. The boundary conditions (4) can be written as /(ro) = bsro = 0-... 

For the inner region (r < ro) they propose a power series solution... 

The form of Eq. (5.1) suggests that for small shear rates a power series solution for xp(6, tj>) should be possible. Hence we try ... 

We can now evaluate this power series result and compare it to the result we obtained from the closed form solution. That is to say, if we had not recognized, as Mathematica did, that the power series solution could be recast as a log, then we might have simply used the solution we had. Let s compare this new solution with the previous one by making a fimction of it, evaluating t at each y and then plotting it against the previous results ... 

The general power-series solution of the differential equation is, thus, given hy... 

So far we have considered only cases where the potential energy V(ac) is a constant. Hiis makes the SchrSdinger equation a second-order linear homogeneous differential equation with constant coefficients, which we know how to solve. However, we want to deal with cases in which V varies with x. A useful approach here is to try a power-series solution of the SchrSdinger equation. 

Section 4.1 Power-Series Solution of Differential Equations 63... 

We might now attempt a power-series solution of (4.34). If we do now try a power series for of the form (4.4), we will find that it leads to a three-term recursion relation, which is harder to deal with than a two-term recursion relation like Eq. (4.15). We therefore modify the form of (4.34) so as to get a two-term recursion relation when we try a series solution. A substitution that will achieve this purpose is (see Problem 4.21) f(x) = e il/ x).Thus... 

To get a two-term recursion relation when we try a power-series solution, we make the following change of dependent variable ... 

Solution of the Radial Equation. We could now try a power-series solution of (6.64), but we would get a three-term rather than a two-term recursion relation. We therefore seek a substitution that will lead to a two-term recursion relation. It turns out that the proper substitution can be found by examining the behavior of the solution for large values of r. For large r, (6.64) becomes... 

We have seen in Chapter 3 that finite difference equations also arise in Power Series solutions of ODEs by the Method of Frobenius the recurrence relations obtained there are in fact finite-difference equations. In Chapters 7 and 8, we show how finite-difference equations also arise naturally in the numerical solutions of differential equations. 

Equation (6.147) is also valid for this same reaction as part of the two-step complex reaction in the domain of large time values, near the equilibrium of the complex reaction, as a result of the principle of detailed balance. Intuitively, this same relationship has to be valid in the domain of small time values in which the influence of the second reaction is insignificant. This can be checked using power series solutions to the kinetic equations, by successive derivation and substitution of initial values. Retaining the first two nonzero terms in the power series, Constales et al. (2012) found that... 

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