Tutorial 11: Fourier Series Expansion
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- The function is to be represented by a Fourier series expansion over the finite interval . Obtain a suitable (a) full-range series expansion (b) half-range sine series expansion (c) half-range cosine series expansion
Solution
Question (a)
For full-range series expansion,
Question (b)
For half-range sine series,
From
When is odd,
When is even,
Question (c)
Half-range cosine:
- Sketch the graphs of: (a) full-range series expansion (b) half-range sine series expansion (c) half-range cosine series expansion
for in Q1 for . Draw and label the period and the finite interval on each graph.
Solution
The original function:
Question (a)
Question (b)
Question (c)
- The temperature distribution at a distance , measured from one end, along a bar of length 10 inch is given by:
Express as a Fourier series expansion consisting of sine terms only.
Solution
where
Consider
Consider
Since
- Suppose a uniform beam of length is simply supported at and at . If the load per unit length is given by , then the differential equation for the deflection is
where and are constants.
(a) Expand in a half-range sine series.
(b) Find a particular solution of the differential equation.
Solution
Question (a)
Question (b)
From
Let