# Partial DIfferential Equations / Eigenvalues

• DualCortex
In summary, the conversation discusses solving basic PDEs using the method of separation of variables and checking three cases to find eigenvalues of Sturm-Liouville problems. The question is raised if an S-L problem can have eigenvalues that span across all three cases and the response suggests checking all cases to avoid missing part of the solution. A link is provided for further clarification.
DualCortex
Hi, I'm barely a high school senior who is somewhat overwhelmed by a univ. course.
Anyway, we are just learning to solve some basic PDEs using the method of separation of variables.
With this method (and the questions we are given) we check three cases to find the eigenvalues of Sturm-Liouville problems ( which come out from the PDEs): when lambda is > 0, < 0, or = 0.
Up to know, I don't think we have seen any sample problem that has eigenvalues that apply to more than one of the cases. This is what I want to know, can an S-L problem have eigenvalues that span across those three cases?
If so, I'm guessing that if I check one of the cases and it does have eigenvalues that lead to a non-trivial solution, then I can safely ignore the other cases? Thank you so much for your time in advance.

Last edited:
Hello DualCortex, welcome to the forum,

You have to check all cases. If you don't do this you might forget part of the solution and the end result will certainly be incorrect. Consider p.e. the following:

Here it is important to study all cases. There are two of them which have non-trivial solutions and you need all of them for obtaining the solution using the boundary on the right hand side of the domain. In this particular example you will not be able to find one coefficient because the boundary conditions are all derivatives.

Hope this helps.

## 1. What are partial differential equations?

Partial differential equations (PDEs) are mathematical equations that describe how a function changes over multiple variables. They are used to model a wide range of physical phenomena, such as heat flow, fluid dynamics, and quantum mechanics.

## 2. What is the difference between ordinary differential equations and partial differential equations?

The main difference between ordinary differential equations (ODEs) and PDEs is that ODEs involve only one independent variable, while PDEs involve multiple independent variables. This means that the solution to a PDE is a function of multiple variables, rather than just one.

## 3. What are eigenvalues and eigenvectors in the context of PDEs?

In PDEs, eigenvalues and eigenvectors are used to solve linear systems of equations. Eigenvalues are the values that satisfy a characteristic equation, while eigenvectors are the corresponding non-zero solutions. They are important in PDEs because they allow us to find solutions to equations that would otherwise be difficult to solve.

## 4. Why are eigenvalues important in PDEs?

Eigenvalues are important in PDEs because they help us to understand the behavior of a system. They can tell us about the stability of a solution and how it will change over time. In addition, they are used to transform a PDE into a set of simpler equations, making it easier to solve.

## 5. How are PDEs used in real-world applications?

PDEs are used in a wide range of real-world applications, including engineering, physics, finance, and biology. They can be used to model and predict the behavior of complex systems, such as the flow of fluids, the spread of diseases, and the behavior of stock prices. PDEs are also used in computer graphics and image processing to create realistic simulations and animations.

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