How Do You Derive Coefficients for Expanding f(x) Using Sine Waves?

  • Thread starter ChaseRLewis
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In summary, when teaching oneself PDE and understanding Separation of Variables, it may seem simple until reaching the point of understanding the orthogonality of sin functions. According to orthogonality, each constant becomes An = 2 ∫01 f(x)sin(nπx)dx, where f(x) is not well described and is simply labeled as "solved". It is likely that f(x) represents the initial condition, with f(0) being the initial value at 0x and f(1) being the initial value at 1x, with no values in between. Overall, the relation appears to be finding coefficients for expanding f(x) as a linear combination of sine waves.
  • #1
ChaseRLewis
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Teaching myself PDE so I'm understanding Separation of variables it's pretty simple up to the point you need to realize the orthogonality of the sin functions. So based on orthogonality it states that each constant becomes

An = 2 ∫01 f(x)sin(nπx)dx

that f(x) isn't described very well and they just say "solved" so my guess is that f(x) is the initial condition such that

f(0) =initial at 0 x
f(1) = initial at 1 x
and nothing in between.
 
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  • #2
f(x) is an arbitrary function ... it looks like the relation is finding the coefficents for expanding f(x) as a linear combination of sine waves.
 
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