Chapter 3: Fourier Series

The heat equation was used to illustrate the method of separation of variables in Sec. 1.6. Separation of variables produced product solutions to the partial differential equation describing heat conduction in a uniform, one-dimensional rod. The product solutions were chosen to satisfy the Dirichlet boundary conditions selected for Example 1.12. To find a solution to a chosen initial condition the Principle of Superposition is employed. This necessitates expanding the initial condition function as a series of eigenfunctions (in the example, as a series of trigonometric sine functions). This produces a formal solution of an initial boundary value problem for the heat equation. As seen in the examples of Sec. 1.6, if the initial condition function is a finite linear combination of sine functions, then by equating coefficients of the sine functions, the solution to the initial boundary value problem of the one-dimensional heat equation is found in closed form. French mathematician and physicist/engineer, Jean Baptiste Joseph Fourier is known for initiating the investigation of the representation of functions as trigonometric series (now called Fourier series) and the application of this method to problems of heat transfer. In fact, Fourier essentially developed the procedure of separation of variables and presented many concrete examples of trigonometric expansions in connection with boundary value problems related to heat conduction problems. German mathematician Peter Dirichlet is believed to be the first person to describe sufficient conditions that guarantee the convergence of a Fourier series. The work of Fourier and Dirichlet leads to the important branch of mathematics known as Fourier analysis, or more generally, harmonic analysis today.