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Fourier series expansion - calculus

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A function is defined as followed:

see attachment

Where f(t+2)=f(t) that is, f(t) has period 2.
i) Draw a plot of the function f(t). Comment fully on whether the function is even or odd or none of these.
ii) Find the first four non-zero coefficients for the Fourier series expansion of the function f(t)
iii) Using eg. excel, draw a plot of the partial sum of the Fourier series which uses just there first four non-zero terms and comment on it in relation to your plot (i)

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Solution Preview

The function looks like:

This function is neither even function since , nor it is an odd function since
For example, while
However we can write the function as:
And is an odd function and it looks like

The general expansion of a function over the symmetric interval is given by:

Solution Summary

The solution shows how to utilize parity to simplify the calculations of Fourier series coefficients.

See Also This Related BrainMass Solution

Temperature Distribution Exposed to a Hot Gas Stream

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Please see the attached file for the fully formatted problem.

L .M. Chiappetta and D.R. Sobel ("Temperature distribution within a hemisphere exposed to a hot gas stream," SIAM Review 26, 1984, p. 575?577) analyze the steady- state temperature in the rounded tip of a combustion-gas sampling probe. The tip is approximately hemispherical iii shape, with radius a. Its outer surface is exposed to hot gases at a temperature Tc, and its base is cooled by water at temperature Tw circulating inside the probe. Let the temperature inside the tip he T. Because of the symmetry of the problem, there is no azinnitha.1 variation in T. In other words, if we are using spherical coordinates, T = T(r, 0), where U is the colatitude. So, T is governed by the differential equation
inside the domain ... (the upper hemisphere with radius a). Let us use a Dirichlet condition at the base of the tip.
and a Robin boundary condition at the outer surface of the tip,
As usual, we will also assume that tile temperature is bounded at the origin and along the north pole. Let .... and solve for it using separation of variables.
Hints: You will need to figure out the orthogonality condition for the eigenfunctions, but you can assume that tile eigenfunctions will he orthogonal with respect to the appropriate weight function because the eigenvalue problem is in Sturm-Liouville form.

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