Boundary Value Problems/Lesson 4.1

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Sturm Liouville and Orthogonal Functions
The solutions in this BVP course will ALL be expressed as series built on orthogonal functions. Understanding that the simple problem $$ X'' + {\lambda}^2 X = 0 $$ with the boundary conditions $$ \alpha_1 X(a) + \alpha_2 X'(a) =0 $$ and $$ \beta_1 X(b) + \beta_2 X'(b) =0 $$ leads to solutions $$ X(x) $$ that are orthogonal functions is crucial. Once this concept is grasped the majority of the work in this course is repetitive. In the following notes think of the function $$ \Phi (x) $$ as a substitution for $$ X(x) $$. TO SEE ALL OF THE PAGES DOUBLE CLICK ON THE FIRST PAGE. THEN YOU WILL BE ABLE TO DOWNLOAD NOTES. THESE WILL BE CONVERTED FOR THE WIKI AALD (at a later date)

Fourier Series
From the above work, solving the problem: $$ X'' + {\lambda}^2 X = 0 $$ with the boundary conditions $$  X(0) =0 $$ and $$  X(L)  =0 $$ leads to an infinite number of solutions  $$ X_n(x)= \Phi_n(x) = sin \left ( \frac{n \pi}{L} x \right ) $$. These are eigenfunctions with eigenvalues $$ \lambda_n = \frac{n \pi}{L} $$

Project 1.2
This is a fourier series application problem. You are given the piecewise defined function $$ f(t) $$ shown in the following graph.



The positive unit pulse is 150 μs in duration and is followed by a 100 μs interval where f(t) =0. Then f(t) is a negative unit pulse for 150 μs once again returning to zero. This pattern is repeated every 2860 μs. We will attempt to represent f(t) as a Fourier series,

. The results are:$$ a_0 = 0 $$ $$ a_n = \frac {\sin \left( {\frac {15}{143}}\,n\,\pi \right) +\sin \left( {\frac {25}{143}}\,n\,\pi  \right) -\sin \left( {\frac {40}{143}}\,n\,\pi  \right)  } {n \pi } $$ $$ b_n= \frac {- \left( -1+\cos \left( {\frac {15}{143}}\,n\,\pi  \right) +\cos \left( {\frac {25}{143}}\,n\,\pi  \right) -\cos \left( {\frac {40}{ 143}}\,n\,\pi \right)  \right)} {n \pi} $$
 * 1) Determine the value of the period: Ans. Period is 2860 μs. The time for a complete repetition of the waveform.
 * 2) Find the Fourier Series representation: $$ f(t) = a_0 + \sum_{n=1}^{\infty} a_n cos(n \pi t / a ) + b_n sin(n \pi t / a) $$ .The video provides an explanation of the determining the coefficients $$ a_0,a_n, b_n $$
 * 1) Using 100 terms an approximation is;[[Image:Plotproj1_2_approx.jpg]]
 * 2) Shift $$f(t)$$ right or left by an amount $$b$$ such that the resulting periodic function is an odd function. Here is a plot of shifting it to the left half way between the +1 and -1 pulses. This is a shiift of b= 200 μs. The new funnction is $$ f(t+200) $$. A plot follows: [[Image:proj1_2shiftleft.jpg]]. It could also be shifted to the right by 1230 μs, that is $$ f(t-1230)$$ is the new function.