Final answer:
Fourier series expansions can be applied to both boundary value problems and initial value problems, and are a valuable tool in mathematical analysis for decomposing periodic signals into sines and cosines.
Step-by-step explanation:
The correct statement is b. Fourier series expansions can be used both on boundary value problems and initial value problems. Fourier series are a tool in mathematical analysis, specifically in the study of periodic functions, and are widely utilized in various fields like physics, engineering, and signal processing. This method decomposes a periodic function or a signal into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines, which are called the Fourier series.
The expansions can be applied to boundary value problems, which typically occur in the study of physical problems where the solution is required to satisfy certain conditions at the boundaries of the domain. Additionally, they can also be used in the analysis of initial value problems, where you need to determine the future behavior of a system based on its initial state.