Applications

In this chapter, we collect many applications of the various ideas that were discussed in the earlier chapters. In the second section, we show how the results for weakly chained diagonally dominant matrices, discussed in Sect. 2.3, have been applied in obtaining bounds for the ℓ 1 norm for the solutions of certain differential systems and also in deriving bounds for a critical parameter in electric circuit design. In Sect. 8.3, we review how a mapping problem could be reduced to an infinite system of linear equations and then solved. In Sect. 8.4, a similar idea is employed to show how the problem of the flow of fluids in and between two pipes could be handled. In the next section, viz., Sect. 8.5, we recall how some special double points of the Mathieu differential equation could be computed using techniques from infinite matrices. In Sect. 8.6, we discuss how the iterative method described earlier could be applied to obtain good approximate values of Bessel functions in certain intervals. Section 8.7 reviews results for the minimal eigenvalue of the Dirichlet Laplacian in an annulus. In the next section, an approximate solution providing the best match for the hydraulic head in a porous medium is presented. The next section, namely Sect. 8.9, considers eigenvalues of the Laplacian in an elliptic domain. The penultimate section studies the problem of the possibility of “hearing” the shape of a drum. The concluding section, Sect. 8.11, discusses how one could determine the zeros of a Taylor series.