Prove Sturm-Liouville differential operator is self adjoint.

In summary, the homework statement is to prove Sturm-Liouville differential operator is self adjoint when subjected to Dirichlet, Neumann, or mixed boundary conditions.
  • #1
shreddinglicks
214
6
Member warned that some effort must be shown on homework questions

Homework Statement


Prove Sturm-Liouville differential operator is self adjoint when subjected to Dirichlet, Neumann, or mixed boundary conditions.

Homework Equations


l = -(d/dx)[p(x)(d/dx)] + q(x)

The Attempt at a Solution


I have no idea. If someone can give me a place to start that would be awesome.
 
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  • #2
shreddinglicks said:

Homework Statement


Prove Sturm-Liouville differential operator is self adjoint when subjected to Dirichlet, Neumann, or mixed boundary conditions.

Homework Equations


l = -(d/dx)[p(x)(d/dx)] + q(x)

The Attempt at a Solution


I have no idea. If someone can give me a place to start that would be awesome.
I haven't looked at that stuff for years, but my first suggestion to you is to give a more complete statement of the problem. By stating clearly what the problem is, you may see a way to solve it. For example, in the relevant equations you should list the various boundary conditions. Also give the definition of what it means for this differential operator to be self-adjoint. That will surely involve an integral or two. In other words, write down exactly what you need to prove. Once you carefully do that, if my memory serves me correctly, an integration by parts and using the BC's may solve your problem.
 
  • #3
LCKurtz said:
I haven't looked at that stuff for years, but my first suggestion to you is to give a more complete statement of the problem. By stating clearly what the problem is, you may see a way to solve it. For example, in the relevant equations you should list the various boundary conditions. Also give the definition of what it means for this differential operator to be self-adjoint. In other words, write down exactly what you need to prove. Once you carefully do that, if my memory serves me correctly, an integration by parts and using the BC's may solve your problem.

This is the complete statement of the problem that is on my homework sheet.

I looked over some examples I found through Google and came out with what is attached.

I'm going to do some research into the boundary conditions later. I need a little break from studying.
 

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  • #4
shreddinglicks said:
This is the complete statement of the problem that is on my homework sheet.

The homework helpers here don't have access to your class notes. You should post relevant definitions here.

I looked over some examples I found through Google and came out with what is attached.

I'm going to do some research into the boundary conditions later. I need a little break from studying.

That looks like the sort of thing I was suggesting.
 
  • #5
LCKurtz said:
The homework helpers here don't have access to your class notes. You should post relevant definitions here.
That looks like the sort of thing I was suggesting.
I wish I had class notes, my teacher did not cover this and it's not in any textbook I have. If it seems I am on the right page then that's a relief.
 

Related to Prove Sturm-Liouville differential operator is self adjoint.

What is a Sturm-Liouville differential operator?

A Sturm-Liouville differential operator is a type of differential operator that is used to solve certain types of differential equations, particularly those with boundary value conditions. It is commonly used in mathematical physics and engineering.

What does it mean for a differential operator to be self adjoint?

A differential operator is said to be self adjoint if it is equal to its adjoint, which is essentially its transpose. This means that the operator behaves the same way when applied to a function and its complex conjugate.

Why is it important for the Sturm-Liouville differential operator to be self adjoint?

The self adjointness of the Sturm-Liouville differential operator is important because it allows us to use techniques from linear algebra to solve differential equations. This makes the solution process more efficient and leads to more accurate results.

How can we prove that the Sturm-Liouville differential operator is self adjoint?

The Sturm-Liouville differential operator can be proven to be self adjoint by using Green's identity, which is a theorem in vector calculus. By applying this identity to the operator and its adjoint, we can show that they are equal and therefore, the operator is self adjoint.

What are the applications of the Sturm-Liouville differential operator?

The Sturm-Liouville differential operator has many applications in mathematical physics and engineering, particularly in solving boundary value problems. It is also used in areas such as quantum mechanics, heat transfer, and signal processing.

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