# Boundary Value Problems/Lesson 4.1

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#### Sturm Liouville and Orthogonal Functions[edit | edit source]

The solutions in this BVP course will ALL be expressed as series built on orthogonal functions. Understanding that the simple problem with the boundary conditions and leads to solutions 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 as a substitution for .

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#### Fourier Series[edit | edit source]

From the above work, solving the problem:

with the boundary conditions and leads to an infinite number of solutions

. These are eigenfunctions with eigenvalues

## Homework Assignment from Powell's sixth edition Boundary Value Problems page 71.[edit | edit source]

### Project 1.2[edit | edit source]

This is a fourier series application problem.

You are given the piecewise defined function 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,

- Determine the value of the period: Ans. Period is 2860 μs. The time for a complete repetition of the waveform.
- Find the Fourier Series representation: .The video provides an explanation of the determining the coefficients

. The results are:

- Using 100 terms an approximation is;
- Shift right or left by an amount 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 . A plot follows: . It could also be shifted to the right by 1230 μs, that is is the new function.