Boundary Value Problem + Green's Function

In summary, a boundary value problem is a mathematical problem with specified conditions at the boundaries of the problem domain. Green's function is a mathematical tool used to solve these types of problems by converting them into integral equations. This method offers advantages such as handling complex boundary conditions and providing a systematic approach for scientists and engineers. Real-world applications of Green's function include solving problems in physics, engineering, and finance.
  • #1
sassie
35
0
Consider the BVP

y''+4y=f(x) (0[tex]\leq[/tex]x[tex]\leq[/tex]1)
y(0)=0 y'(1)=0

Find the Green's function (two-sided) for this problem.

Working: So firstly, I let y(x)=Asin2x+Bcos2x

Then using the boundary conditions,

Asin(2.0)+Bcos(2.0)=0 => B=0

y'(x)=2Acos(2x)-2Asin(2x)
y'(0)=2A=0 => A=0

But is this right? How can I derive a Green's function (two-sided) from this? Please help.
 
Physics news on Phys.org
  • #2
Please ignore. I figured out what I did wrong.
 

Related to Boundary Value Problem + Green's Function

What is a boundary value problem?

A boundary value problem is a mathematical problem that involves finding a solution to a differential equation subject to specified boundary conditions. These conditions are usually given at the endpoints or boundaries of the problem domain.

What is Green's function?

Green's function is a mathematical tool used to solve boundary value problems by converting them into integral equations. It represents the response of a system to an impulse or delta function input and can be used to find the solution to a differential equation with specified boundary conditions.

How is Green's function used to solve boundary value problems?

Green's function is used to solve boundary value problems by converting them into integral equations. The solution to the differential equation can then be found by solving the integral equation using the properties of Green's function.

What are the advantages of using Green's function to solve boundary value problems?

One of the main advantages of using Green's function is its ability to handle complex boundary conditions that may be difficult to solve using other methods. It also provides a systematic and efficient approach to solving boundary value problems, making it a useful tool for scientists and engineers.

What are some real-world applications of Green's function in solving boundary value problems?

Green's function is used in a variety of fields, such as physics, engineering, and finance, to solve boundary value problems. Some examples include calculating the electric potential in a conductor, analyzing the flow of fluids in a pipe, and predicting the price of financial derivatives.

Similar threads

I Green's function for 2-D Laplacian within square/rectangular boundary
    • Differential Equations
Replies
5
Views
587
I Understanding relationship between heat equation & Green's function
    • Differential Equations
Replies
6
Views
2K
A Green's function for Sturm-Louiville ODE
    • Differential Equations
Replies
3
Views
1K
I Understanding Euler Method: Finding Initial Condition of y(0)=1
    • Differential Equations
Replies
7
Views
2K
A Green's Function Approach for ODE with Boundary Conditions - Why the Difference?
    • Differential Equations
Replies
1
Views
2K
A 3D Heat equation with elementary Dirichlet BC
    • Differential Equations
Replies
1
Views
3K
I A question about boundary conditions in Green's functions
    • Differential Equations
Replies
1
Views
2K
A Replacing a non-harmonic function with a harmonic function
    • Differential Equations
Replies
1
Views
2K
A Improper boundary in non-linear ODE (pseudospectral methods)
    • Differential Equations
Replies
4
Views
1K
General solution vs particular solution
    • Differential Equations
2
Replies
52
Views
2K

Hot Threads

Recent Insights

Back
Top