Hypergeometric equation

If i is a linear function of z, then equations (33) and (34) may be reduced to the hypergeometric equation [20]. Subtracting equation (33) from equation (34) yields an expression for V that can be substituted into equation (33) to provide a second-order differential equation for P. With the subscript t identifying throat conditions, it may be shown that when V is linear in z, then... 

If rj 1 exceeds this interval, the spatial dependence of 37 (x) cannot be neglected. In this case the solution of Eq. (100) leads to a hypergeometric equation, which, due to its complexity, is not shown here. 

Straightforward verification shows that the change of variables u = Xmg2, F = exp(it/2)fm transforms (3.5.10) into a degenerate hypergeometric equation for the function F = F(u) [28], Therefore, the solution of the spectral problem... 

This equation is well known to mathematicians as the hypergeometric equation. 

Similar conclusions have been obtained by Wallach [12]. More detailed approximate results may be obtained by using only one delay group equation rather than six. In this case, elimination of C between the equations for N and C results in a second order differential equation for N which for constant dpjdt is a confluent hypergeometric equation. Results similar to those obtained by the above methods have been obtained by Elrod [13] using series expansions in inverse powers of a. 

The above equation is the confluent hypergeometric equation (the prototype) with linearly independent solutions tabulated in the literature [12] whereas Equation 3.76 resulted from a fluid flow problem with a parabolic velocity profile. 

In Example 7.9, this differential equation was shown to be the confluent hypergeometric equation ... 

Overall Reaction Rate Equation of Single-Route Complex Catalytic Reaction in Terms of Hypergeometric Series... 

As a result, it was obtained an analytical expression of the reaction rate in terms of hypergeometric series with no classical simplifications about the "limiting step" or the "vicinity of the equilibrium". The obtained explicit equation, "four-term equation", can be presented as follows in the Equation (77) ... 

In this chapter, we will try to answer the next obvious question can we find an explicit reaction rate equation for the general non-linear reaction mechanism, at least for its thermodynamic branch, which goes through the equilibrium. Applying the kinetic polynomial concept, we introduce the new explicit form of reaction rate equation in terms of hypergeometric series. 

Finally, we present the results of the case studies for Eley-Rideal and LH reaction mechanisms illustrating the practical aspects (i.e. convergence, relation to classic approximations) of application of this new form of reaction rate equation. One of surprising observations here is the fact that hypergeometric series provides the good fit to the exact solution not only in the vicinity of thermodynamic equilibrium but also far from equilibrium. Unlike classical approximations, the approximation with truncated series has non-local features. For instance, our examples show that approximation with the truncated hypergeometric series may supersede the conventional rate-limiting step equations. For thermodynamic branch, we may think of the domain of applicability of reaction rate series as the domain, in which the reaction rate is relatively small. 

We consider below the possibilities for simplification of overall reaction rate equations and introduce the main result of this chapter — the hypergeometric series for reaction rate. 

Recently Passare and Tsikh (2004) provided the detailed description of the domains of convergence of multi-dimensional hypergeometric series representing the roots of algebraic equations. They have relied on results of Birkeland obtained in 1920s. Birkeland found the Taylor series solutions to algebraic equations of the type... 

Figure 15 Overall reaction rate and its approximations step 2 is rate-limiting. Dots represent the exact reaction rate dependence, solid line is the first-term hypergeometric approximation, dashed line corresponds to the reaction-rate equation that assumes the limitation of step 2 and dash-dots represent the equilibrium approximation. Hypergeometric approximation survives the 100-times increase in rate-limiting stage kinetic parameters and it works when there is no rate-limiting step at all. Parameters r, = 5, fj = 15, rj = 10 t2 = 0.2, fj = 0-1 (a) rj = 2,fj = - (b) t2 = 20, = 10, (c).

Passare, M., and Tsikh, A., Algebraic equations and hypergeometric series, in "Legacy of Niels Henrik Abel The Abel Bicentennial", Oslo, Springer, June 3-8, 653-672 (2002). 

Note that Eq. (126) implies a nonzero initial velocity of the free boundary, in common with previous exact solutions, which were, however, selfsimilar. The present problem, while linear, is still in the form of a partial differential equation. However, it is readily solved by separation of variables, leading to an ordinary differential equation of the confluent hypergeometric form. The solution appears in terms of the confluent hypergeometric function of the first kind, defined by... 

C.J. van Duijn, Andro Mikelic, I.S. Pop, and Carole Rosier, Effective Dispersion Equations for Reactive Flows with Dominant Peclet and Damkohler Numbers Mark Z. Lazman and Gregory S. Yablonsky, Overall Reaction Rate Equation of Single-Route Complex Catalytic Reaction in Terms of Hypergeometric Series A.N. Gorban and O. Radulescu, Dynamic and Static Limitation in Multiscale Reaction Networks, Revisited... 

Unlike Xi, which in principle cannot be evaluated analytically at arbitrary a [90] for Xjnt an exact solution is possible for arbitrary values of the anisotropy parameter. Two ways were proposed to obtain quadrature formulas for Tjnt. One method [91] implies a direct integration of the Fokker-Planck equation. Another method [58] involves solving three-term recurrence relations for the statistical moments of W. The emerging solution for Tjnt can be expressed in a finite form in terms of hypergeometric (Kummer s) functions. Equivalence of both approaches was proved in Ref. 92. 

THE FUNCTIONS OF MATHEMATICAL PHYSICS, Harry Hochstadt. Comprehensive treatment of orthogonal polynomials, hypergeometric functions. Hill s equation, much more. Bibliography. Index. S22pp. 55 x 854. 65214-9 Pa. 8.95... 

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