Asymptotic Approximations and Expansions

Many problems in transport and chemical reaction engineering are nonlinear and cannot be solved analytically. A powerful approach to solve such problems lies in the method of matched asymptotic expansions that often provide analytical expressions for the solution. The method is based on an expansion whose convergence is based on concepts somewhat different from that usually understood. An example is considered below to clarify the nature of such expansions. Such expansions can be used in the solution of nonlinear equations for limiting values of parameters associated with the problem. Several examples are available in the chemical engineering literature (Leal, 1992, 2007 Been, 1998 Varma and Morbidelli, 1997). 

A detailed treatment of this example is given by Erdelyi (1956). Let 

The foregoing sum is dehnitely divergent because of . However, let us proceed without regard for the divergence to invoke the well-known result from the theory of gamma functions, 

Disregarding conditions required for interchanging the summation and integration, one obtains 

The alternating series within the integrand on the right-hand side of Equation (2.258) can now be summed exactly to write from Equation (2.259) 

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