application of Fourier series in automobile engineering .

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The Fourier series is named in honor of Jean-Baptiste Joseph Fourier (1768 to 1,830), who made important contributions to the study of trigonometric series, after preliminary by Leonhard Euler, Jean le Rond d'Alembert, and Daniel Bernoulli. [Nb 1] Introduced Fourier series in order to solve the equation of heat in a metal plate, publishing its first results in his 1807 on memory of heat propagation in solid bodies (Treaty on the propagation of heat in solids), and publish his analytical theory of heat (analytical theory of heat) in 1822. the first ideas to break down a periodic function in the sum of simple oscillating functions from the 3rd century BC, when ancient astronomers have proposed an empirical model of the planetary motions, based on deferential and epicycles.

The heat equation is a partial differential equation. Before the work of Fourier, no solution to the heat equation has been known in the general case, although specific solutions have been known if the heat source behaved in a simple way, in particular, if the heat source was a sinus or a cosine wave. These simple solutions are now sometimes called specific solutions. The idea of Fourier was to model a heat source like a complex superposition (or linear combination) of sine and cosine simple waves and to write the solution in the form of a superposition of the corresponding solutions. This superposition or linear combination is called the Fourier series.

From a modern point of view, the results of Fourier are somewhat informal due to the lack of a precise concept of function and integral in the early 19th century. Later, Peter Gustav Lejeune Dirichlet [1] and Bernhard Riemann [2] [3] [4] have expressed the results of Fourier with greater precision and formality.

Although the initial motivation was to solve the heat equation, it became clearer later that the same techniques can be applied to a wide range of mathematical problems and physical, and especially those involving linear constant coefficient differential equations, for which solutions are sine waves. The Fourier series has many of these applications in electrical engineering, vibration analysis, acoustics, optics, signal processing, imaging, quantum mechanics, econometrics, [5] theory of thin-walled shells, [6], etc.