How Do You Derive the Green's Function for a Specific Boundary Value Problem?

[Kate2010]

Homework Statement

Obtain the Green's function for BVP (and use it to express the solution for the given data):

-y''(x) = f(x), 0 < x < 1, y'(0) = a, y(1) = b

Homework Equations

The Attempt at a Solution

I have found 2 solutions to the homogeneous equation
y1(x) = ax, satisfies y'(0) = a
y2(x) = x - 1 + b, satisfies y(1) = b

Then I need to consider a general solution y(x) = c1(x)y1(x) + c2(x)y2(x)
So y' = c1y1' + c2y2' (imposing c1'y1 + c2'y2 = 0)
y'' = c1y1'' + c2y2'' + c1'y1' + c2'y2'

So to satisfy the boundary conditions I need
a = ac1(0) + c2(0)
b = ac1(1) + bc2(a)

I am not sure how to impose these into the integrals for c1 and c2 (as I would in the case when I have something like y(a) = 0 = y(b)?

 

[mighty2000]

Thank you for your post. To obtain the Green's function for the given boundary value problem, we first need to solve the homogeneous equation -y''(x) = 0. This gives us the solutions y1(x) = ax and y2(x) = x - 1 + b.

Next, we need to consider a general solution y(x) = c1(x)y1(x) + c2(x)y2(x). To satisfy the boundary conditions, we impose the conditions y'(0) = a and y(1) = b. This gives us the following equations:

a = ac1(0) + c2(0)
b = ac1(1) + bc2(1)

We can solve these equations for c1 and c2 by considering the following integrals:

c1(x) = 1/(y1'(x)y2(x) - y1(x)y2'(x)) * (b - y2(x))
c2(x) = 1/(y1'(x)y2(x) - y1(x)y2'(x)) * (y1(x) - a)

This gives us the general solution:

y(x) = [b - y2(x)]/(y1'(x)y2(x) - y1(x)y2'(x)) * y1(x) + [y1(x) - a]/(y1'(x)y2(x) - y1(x)y2'(x)) * y2(x)

To obtain the Green's function, we need to consider the following integral:

G(x,ξ) = [y1(ξ)y2(x) - y1(x)y2(ξ)]/(y1'(ξ)y2(x) - y1(x)y2'(ξ))

This gives us the Green's function for the given boundary value problem. We can use this Green's function to express the solution for any given data. I hope this helps. Let me know if you have any further questions.