chapter 8, Fourier Series, Eigenvalue Problems, and Green's Function

Excerpt

Historical Comments: Joseph Fourier (1768–1830) was one of several famous French mathematicians who flourished during the time of Napoleon. In 1794, Napoleon offered Fourier the chair of mathematics at the Ecole Normale in Paris, He left this position in 1798 to accompany Napoleon and a group of other scientists to Egypt, where he remained for four years, establishing the scientific institute of Cairo. Fourier returned to France in 1802 to become prefect of the department of Isere at Grenoble in the French Alps.

The theory widely known as Fourier series is credited to Fourier, who came across such representations in his classic studies of heat conduction. His basis papers, presented to the Academy of Sciences in Paris in 1807 and 1811, were criticized by the referees (most strongly by Lagrange) for a lack of rigor and consequently were not then published. Fourier was called to Paris by the Academy of Sciences in 1816, whereupon he succeeded Laplace as president of the board of the Ecole Polytechnique. When publishing the classic Théorie analytique de la Chaleur in 1822, he also incorporated his earlier work that was previously rejected. Fourier died in Paris on May 16, 1830.

Leonhard Euler (1707–1783) solved the first eigenvalue problem when he developed a model for describing the “buckling” modes of a vertical column. However, the general theory of eigenvalue problems for second-order DEs, commonly called the Sturm-Liouville theory, originated in the work of Jacques C. F. Sturm (1803–1855), a professor of mechanics at the Sorbonne, and Joseph Liouville (1809–1882), a professor of mathematics at the College de France.

The method of Green's function is named for George Green (1793–1841), who gained recognition for his work on reflection and refraction of sound and light waves. He also extended the work of Poisson in the theory of electricity and magnetism.

In this chapter we introduce the method of Fourier series for the analysis of periodic waveforms (e.g., power signals). This approach reduces the signal being studied to a spectral representation in which the distribution of power is found to be concentrated at specific frequencies that are harmonically related to a fundamental frequency. In addition, we discuss the related notion of eigenvalue problem for homogeneous boundary value problems, the eigenfunctions of which are used to develop generalized Fourier series. Last, the method of Green's function is introduced for solving nonhomogeneous problems (including eigenvalue problems). By representing the Green's function in a “bilinear” representation, we amalgamate the theory of Fourier series and eigenvalue problems with that of the Green's function method.