The Discrete Sine (DST) and Cosine (DCT) Transforms for Boundary Value Problems

Abstract

In the preceding chapter, the discrete Fourier transform was introduced and was first used in a series—type solution of difference equations with periodic conditions that are the compatible conditions with the series form of the DFT. Then we used the DFT as a direct transform for reducing the difference equation to an algebraic equation as well as involving periodic boundary conditions, or even if there is a jump discontinuity at the end points instead of the zero jump for the (typical) periodic conditions. These series solutions, as well as the direct operational sum method of the DFT, were illustrated in Example 4.8 for solving a traffic flow problem. As we introduced briefly in Section 2.3, integral as well as finite (complex) exponential, sine, and cosine Fourier transforms are all used to algebraize differential operators, but they differ in accommodating different boundary conditions. So, as it may be expected, the discrete sine and cosine Fourier transforms are also compatible with different boundary conditions. This topic was discussed at the end of Section 2.4 in light of their respective operational pairs (2.127), (2.140), and (2.144), (2.143). There, only the important initial step of their algebraizing the difference operator E+E ⁻¹ in G _n+1+G _n-1 = (E+E ⁻¹)G _n was illustrated in that discussion, and in Examples 2.12–2.14 for problems to be completed for their final solution in the second section of this chapter. This delay was necessary in order to have more tools of the operational as well as the sequence pairs developed for the discrete sine and cosine transforms. The emphasis in Section 2.4 was on the direct DST and DCT transforms operational sum calculus method of reducing difference equations, of even order along with accommodating the boundary conditions that are compatible with either the DST or DCT transforms. As we did in Section 4.4 for the use of the DFT in solving difference equations, with periodic boundary conditions we will, in Section 5.2, use the above DST and DCT for a series solution method as well as a direct operational sum calculus method. But, since the latter method was well discussed in Section 2.4, we will start the same and other problems with the series solution type method of using the DST and the DCT transforms. So in the following section we will establish the discrete sine and cosine transforms, and illustrate them with a few examples. Of course, the reason for beginning our study with the DFT, which seems to apply only to a very specialized kind of problems, is that once the DFT has been presented, the discrete sine and cosine transforms appear with very little additional work.