Please use radian mode and p = 3.142 Question 1 A function f(t) is defined as, f(t) = p -t for 0 < t < p Write down the odd extension of f(t) for -p < t < 0. Determine the Fourier sine series, and hence, calculate the Fourier series approximation for f(t) up to the 3rd harmonics when t = 1.11. Use p = 3.142. Give your answer to 3 decimal places. Answer : Question 2 The Fourier series expansion for the periodic function, f(t) = |sin t| is defined in its fundamental interval. Taking p = 3.142, calculate the Fourier cosine series approximation of f(t), up to the 6th harmonics when t = 2.16.

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Please use radian mode and p = 3.142
Question 1
A function f(t) is defined as,
f(t) = p - t for 0 < t < p
Write down the odd extension of f(t) for -p < t < 0.
Determine the Fourier sine series, and hence, calculate the Fourier series approximation for
f(t) up to the 3rd harmonics when t = 1.11. Use p = 3.142. Give your answer to 3 decimal
places.
Answer :
Question 2
The Fourier series expansion for the periodic function, f(t) = |sin t| is defined in its
fundamental interval. Taking p = 3.142, calculate the Fourier cosine series
approximation of f(t), up to the 6th harmonics when t = 2.16. Give your answer to 3
decimal places.
Answer :
Question 3
An infinite cosine series is given by,
Compute the sum to r = 3 and t = 0.94. Use p = 3.142. Give your answer to 3
decimal places.
Answer :Question 4
Consider the temperature distribution in a perfectly insulated rod of length L, where one end,
0
at x = 0, is maintained at a temperature of 0 C and the other end, at x = L, is insulated. This is
well modeled by the diffusion equation,
where a is a constant. It is subjected to boundary conditions,
t
The method of separation of variables, with ?(x, t) = X(x)T(t), is used to determine the
boundary-value problem satisfied by T(t) which is given by, T(t) = exp(-kt)
where k is in terms of a, L and n = 1,2,3,......
Calculate the value of T at t = 0.32, when a = 0.5 n = 1 and L = 1.61, giving your answer to 3 decimal
places. You may assume that p = 3.142.
Answer :
Question 5
Consider the temperature distribution in a perfectly insulated rod of length L, where one end,
0
at x = 0, is maintained at a temperature of 0 C and the other end, at x = L, is insulated. This is
well modeled by the diffusion equation,
where a is a constant. It is subjected to boundary conditions,
...

This is not exact solution of the problem above rather following is the discussion and basic concepts of Fourier series, which will help you to solve questions. A Fourier series is an expansion of a periodic function in terms of an infinite sum of cosines and sines. Fourier series make use of the orthogonality relationships of the cosine and sine functions. The computation and study of Fourier series is known as harmonic analysis and is extremely useful as a way to break up an arbitrary periodic function into a set of simple terms that can be plugged in, solved individually, and then recombined to obtain the solution to the original problem or an approximation to it to whatever accuracy is desired or practical. Examples of successive approximations to common functions using Fourier series are illustrated above. In particular, since the superposition principle holds for solutions of a linear homogeneous ordinary differential equation, if such an equation can be solved in the case of a single sinusoid, the solution for an arbitrary function is immediately available by expressing the original function as a Fourier series and then plugging in the solution for each sinusoidal component. In some special cases where the Fourier series can be summed in closed form, this technique can even yield analytic solutions. Any set of functions that form a complete orthogonal system have a corresponding generalized Fourier series analogous to the Fourier series. For example, using...

y of the roots of a Fourier-Bessel series. The computation of the (usual) Fourier series is based on the integral identities (1) (2) (3) (4) (5) for , where is the Kronecker delta. Using the method for a generalized Fourier series, the usual Fourier series involving sines and cosines is obtained by taking and . Since these functions form a complete orthogonal system over , the Fourier series of a function is given by (6) where (7) (8) (9) and , 2, 3, .... A Fourier series converges to the function (equal to the original function at points of continuity or to the average of the two limits at points of discontinuity) (10) if the function satisfies so-called dirichlet boundary conditioons. The coefficients can be expressed in terms of those in the Fourier series (27) (28) (29) For a function periodic in , these become (30) (31) These equations are the basis for the extremely important Fourier transform, which is obtained by transforming from a discrete variable to a continuous one as the length . The complex Fourier coefficient is implemented in as Fourier Coefficientn [ expr , t , n ]. We clearly have an even function here and so all we really need to do is compute the coefficients and they are liable to be a little messy because we’ll need to do integration by parts twice. We’ll leave most of the actual integration details to you to verify. The coefficients are then, The Fourier cosine series is then, Note that we’ll often strip out the from the series as we’ve done here because it will almost always be different from the other coefficients and it allows us to actually plug the coefficients into the series.

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