(22h) An Efficient Method for Deriving Normalization Constants for Eigenfunctions of Sturm-Liouville Problems and Its Application to the Graetz Problem for Diffusive and Convection Heat/Mass Transfer

An Efficient Method for Deriving Normalization Constants for Eigenfunctions of Sturm-Liouville Problems and Its Application to the Graetz Problem for Diffusive and Convection Heat/Mass Transfer

Joel A. Paulson^a, T. Alan Hatton^a, and Richard D. Braatz^a,[1]

^aDepartment of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139

Sturm-Liouville (SL) problems arise in the solution methods to an important class of partial differential equations (PDEs) relevant in many industrial and scientific applications [1,2,3]. The importance of SL theory is rooted in the fact that the solutions of certain PDEs can be expanded as an infinite series of normalized eigenfunctions of the SL operator multiplied by spectral coefficients [4]. The spectral coefficients can be computed by matching the infinite series at the initial/boundary conditions. Although methods for determining the eigenfunctions (and corresponding eigenvalues) of the SL operator are well-known, few procedures are available for normalizing this set of eigenfunctions (i.e., ensuring the inner product of an eigenfunction with itself is unity). Most normalization procedures require a large amount of time and effort and may not yield an analytic expression in a simple and readily computable form.

This talk presents a new method for analytically determining the normalization constants for the eigenfunctions of regular, singular, and periodic SL problems. The resulting formula only requires derivatives with respect to the eigenvalue parameter, and is much simpler to use than the direct method (whereby the squared eigenfunction must be integrated) or the Greenâ??s function method (that requires the generation of the full spectral presentation of the SL operator) [5]. This new method offers a gateway for analytically determining normalization constants for some of the more complicated eigenfunctions arising in SL problems that have previously been elusive.

The power of this newly proposed method is demonstrated on the well-known Graetz problem that describes diffusive and convective heat or mass transfer through a pipe in the presence of fully developed laminar flow [4,5]. Combining the proposed method with some other mathematical results, we were able to derive, for the first time (to the best of the authorâ??s knowledge), an explicit expression for the orthonormal set of eigenfunctions for the Graetz problem that is valid and easily computable over the entire eigenspectrum.

References

[1] I. Stakgold. Boundary Value Problems of Mathematical Physics, volume 11. Macmillian, 1972.

[2] S. S. Bayin. Sturm-Liouville Theory. In Mathematical Methods in Science and Engineering (Chapter 8). John Wiley & Sons, Inc., Hoboken, NJ, 2006.

[3] M. A. Al-Gwaiz. Sturm-Liouville Theory and its Applications. Berlin, Springer, 2008.

[4] W. M. Deen. Analysis of Transport Pheomena (Topics in Chemical Engineering), volume 3. Oxford University Press, New York, NY, 1998.

[5] E. Papoutsakis, D. Ramkrishna, and H. C. Lim. The extended Graetz problem with Dirichlet wall boundary conditions. Applied Scientific Research, 36 (1980): 13-34.

[1]Corresponding author: braatz@mit.edu.