Fourier series: Visualizing Joseph Fourier’s imaginative discovery via fea and time-frequency decomposition

Fourier analysis is one of the most utilized mathematical techniques across a wide range of disciplines that covers from physics and engineering to biology, economics, oceanography, and other areas. However, the geniality of Joseph Fourier in the process of his discovery is not well known and recognized. This paper attempts to reproduce some of the original experiments carried out that Joseph Fourier conducted via the use of finite element analysis (FEA). The authors expect that the results will bring more insight into the understanding of this revolutionary technique as well as illustrate the fundamental need for an integration of physical intuition and mathematics for the progress of scientific developments.