How to Find Green Function for y'' + 1/4y = f(x) with Boundary Conditions?

In summary, the student is trying to find a Green function for the equation y''+1/4y=f(x). The function must satisfy the boundary conditions y(0)=y(pi) = 0 and the equation must have a general solution.
  • #1
spock0149
31
0

Homework Statement



Hey folks,

I need to find a Green function for the equation:

y'' +1/4y = f(x)

With boundary conditions y(0)=y(pi) = 0


The Attempt at a Solution



I tried some combination of solutions that look like sin(kx) and sin(k-pi)
and looked at the strum liouville equation too and meassed with a Wronkskian.

I was just wandering if there was an easier way to do this?
 
Physics news on Phys.org
  • #2
anyone?
:)
 
  • #3
pde problems are usually messy and tedious, which i have so far avoided and even prof. mathwonk himself admitted he is lacking in. perhaps you should post in the differential equations forum where there are some pde's experts to assist you.
 
Last edited:
  • #4
spock0149 said:

Homework Statement



Hey folks,

I need to find a Green function for the equation:

y'' +1/4y = f(x)

With boundary conditions y(0)=y(pi) = 0


The Attempt at a Solution



I tried some combination of solutions that look like sin(kx) and sin(k-pi)
and looked at the strum liouville equation too and meassed with a Wronkskian.

I was just wandering if there was an easier way to do this?

Do you mean y"+ (1/4)y ? (Not y"+ 1/(4y).)

What is the DEFINITION of Green's function?

The Green's function for this problem, G(x,t), must satisfy:

Gxx+ (1/4)G= 0 for all [itex]x\ne t[/itex].
G(0,t)= 0, G([itex]\pi[/itex],t)= 0
G is continuous at x= t.
Derivative of G at t, from the right, minus derivative of G at t,from the left, must equal 1.

The general solution of y"+ (1/4)y= 0 is A cos((1/2)x+ B sin((1/2)x) so Green's function must be of the form
[tex]G(x,t)= \left{\begin{array}{c}A cos((1/2)x)+ B sin((1/2)x if x< t \\C cos((1/2)x)+ D sin((1/2)x) if x> t \end{array}\right[/itex]

G(0,t)= A= 0, G([itex]\pi[/itex],t)= D= 0

G(t,t)= A cos((1/2)t)+ B sin((1/2)t= C cos((1/2)t)+ D sin((1/2)t)

-(1/2)C sin((1/2)t)+ (1/2)D cos((1/2)t)+ (1/2)A sin((1/2)t)- (1/2)B cos((1/2)t)= 1

Solve for A, B, C, D.
 

1. What is a Green function?

A Green function is a mathematical function that represents the response of a linear differential equation to a specific forcing function. It is used to solve boundary value problems and can be thought of as a propagator of information.

2. Why do we need Green functions?

Green functions are necessary for solving certain types of differential equations, particularly boundary value problems. They provide a systematic method for finding solutions to these equations and can be used to model physical systems and phenomena.

3. How do we find Green functions?

The process of finding Green functions involves solving the given differential equation and using boundary conditions to determine the form of the function. This can be done analytically or numerically, depending on the complexity of the problem.

4. What are some applications of Green functions?

Green functions have a wide range of applications in physics, engineering, and other scientific fields. They are used to solve problems in electromagnetism, fluid dynamics, quantum mechanics, and more. They are also commonly used in image and signal processing.

5. Are there any limitations to using Green functions?

While Green functions are a powerful tool for solving certain types of problems, they do have some limitations. They are typically only applicable to linear differential equations and may not provide accurate solutions for highly nonlinear systems. Additionally, finding Green functions for complex problems can be a challenging and time-consuming task.

Similar threads

  • Calculus and Beyond Homework Help
Replies
1
Views
664
  • Calculus and Beyond Homework Help
Replies
3
Views
119
  • Calculus and Beyond Homework Help
Replies
8
Views
1K
  • Calculus and Beyond Homework Help
Replies
8
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
873
  • Calculus and Beyond Homework Help
Replies
8
Views
995
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
6
Views
505
  • Calculus and Beyond Homework Help
Replies
5
Views
499
Back
Top