Gamma function equation solution

This integral is a special function related to the incomplete gamma function. The solution can be considered to be analytical even though the function may be unfamiliar. Figure 8.1 illustrates the behavior of Equation (8.8) as compared with CSTRs, PFRs, and laminar flow reactors with diffusion. 

Other integral transforms are obtained with the use of the kernels e" or xk among the infinite number of possibilities. The former yields the Laplace transform, which is of particular importance in the analysis of electrical circuits and the solution of certain differential equations. The latter was already introduced in the discussion of the gamma function (Section 5.5.4). 

This function is called the Bessel function of the first kind of order n. T(n +1) is the gamma function of n +1. From this, it can be seen that when n is a positive integer, J (x) starts off as x". When n = 0, Jo(0) = 1. When n is an integer, J (0) = 0. In all other cases, J is infinite at the origin. In many physical problems the solution to Bessel s equation must be defined (finite) and well-behaved at the origin, which eliminates all solutions except for those with integer values of n. It can also be shown that J satisfies the same recurrence relations as Dn, verifying that the functions are the same. 

The solution to the fractional diffusion equation is clearly dependent on fluctuations that have occurred in the remote past note the time lag k in the index on the fluctuations and the fact that it can be arbitrarily large. The extent of the influence of these distant fluctuations on the system response is determined by the relative size of the coefficients in the series. Using Stirling s approximation on the gamma functions determines the size of the coefficients in Eq. (25) as the fluctuations recede into the past, that is, as k — oo we obtain... 

The solution is a combination of exponential and Whittaker functions. In the literature this problem is usually left in terms of integrals (Gamma functions). However, Maple is able to solve the differential equation explicitly. Next, the constant C2 is found using the boundary condition bc2. 

The Ij Convolution, The function Ii(t) must by necessity be evaluated numerically using, for example, the closed Newton-Cotes fonmdae (the Trapezoidal Simpson s rules) for equally-spaced abscissae (the time axis is divided into N equal intervals of At). This can be done by direct nummcal solution of equation (4) or by using special algorithms e.g. the Gamma function (Hx)) ... 

In developing series solutions of differential equations and in other formal calculations it is often convenient to make use of properties of gamma and beta functions. The integral... 

A brief review of methods based on the integral adsorption Eq, (73) showed that they are attractive to evaluate the pore volume distribution. The analytical solution of this integral for sub-integral functions represented by the Dubinin-Astakhov equation and gamma-type... 

Band broadening effects such as dispersion and mass transfer resistance are represented by the number of tanks (or stages) N. This can be explained by evaluating the moments of the analytical solution of Equation 6.98. For linear isotherms and the injection of an ideal Dirac pulse of one component, this equation yields a gamma density function for the concentration profile. With the retention time Ir lin.i of Equation 6.49 one obtains the elution profile as the time dependence of the concentration in the last tank (k= N) ... 

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