# Basis functions of a differential equation, given boundary conditions

• rdfloyd
In summary, the conversation revolves around finding basis functions for a differential equation with specific boundary conditions. The speaker is struggling with understanding the concept and has received help from their professor and used their own common sense to come up with a solution. They mention using complex-exponentials instead of cos(kx) and sin(kx) as basis functions and question the purpose and meaning of the solution.
rdfloyd
First off, I've never taken a differential equations class. This is for my Math Methods for Physicists class, and we are on the topic of DE. Unfortunately, we didn't cover this much, so most of what I am about to show you comes from the professor giving me tips and my own common sense. I'd really like to learn this stuff, but I have no idea what I am doing.

## Homework Statement

Find the basis functions that result from the differential equation u''(x)=-k^2u(x) and these boundary conditions:

u(0)=u(L)
u'(0)=u'(L)

## The Attempt at a Solution

I don't know how to code in Latex yet, so forgive me for having to use a picture of my work thus far. Again, most of this is from other's help and common sense. I couldn't begin to explain what is going on here, or its uses.

Since the image is too big to be displayed, here is the link to it on my imgur account: http://imgur.com/D8B3U

Thanks!

Since $e^{ikx}= cos(kx)+ i sin(kx)$ and $e^{-ikx}= cos(kx)- i sin(kx)$, you can use as your "basis functions" cos(kx) and sin(kx) rather than $e^{ikx}$ and $e^{-ikx}$. (That is normally done when all your conditions are in terms of real numbers.)

I was originally going to do that (and basically ended up doing that anyways), but he told me I should use complex-exponentials instead.

What I ended up with was:

$u(x)=A \times Cos[(\frac{n \pi}{L}) x]$

I wish I had followed my gut instinct instead...Thanks for the help. I don't see a +reputation button (like on other forums), but I appreciate the help.I only have one more question: What was the purpose in all of this? What's the end result? I understand that the boundary conditions essentially setup a periodic function (looks a lot like the spring equation), but what does $u(x)$ really mean? And why doesn't B matter?

## 1. What are basis functions in a differential equation?

Basis functions are a set of mathematical functions used to represent the solution of a differential equation. They are usually chosen to be simple and well-behaved functions, such as polynomials or trigonometric functions, that can be combined to approximate the solution of the differential equation.

## 2. Why are basis functions important in solving differential equations?

Basis functions are important because they provide a way to approximate the solution of a differential equation. By using a combination of basis functions, we can break down a complex problem into simpler parts and find a solution that satisfies the given boundary conditions.

## 3. How do boundary conditions affect the choice of basis functions?

Boundary conditions play a crucial role in determining the appropriate basis functions for a given differential equation. They help to narrow down the set of possible functions that can be used to approximate the solution, and may also determine the form of the chosen functions.

## 4. Can different types of basis functions be used for the same differential equation?

Yes, different types of basis functions can be used to approximate the solution of a differential equation. The choice of basis functions depends on the specific problem at hand and the desired level of accuracy in the solution.

## 5. Are there any limitations to using basis functions in solving differential equations?

One limitation of using basis functions is that the accuracy of the solution depends on the number and types of functions chosen. In some cases, a large number of basis functions may be needed to obtain an accurate solution, which can be computationally expensive. Additionally, certain types of differential equations may not have a suitable set of basis functions for approximation.

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