Understanding the Boundary Value Problem
The boundary value problem given is defined by the following equation:
- y''(x) = f(x)
- Boundary conditions: y(0) = 0, y(1) = 0
This describes a second-order linear differential equation with specific conditions at the endpoints.
What is the Green's Function?
The Green's function, G(x, s), is a powerful tool used to solve inhomogeneous differential equations. It represents the influence of a point source at s on the solution at x, satisfying the same boundary conditions as the original differential equation.
Form of the Green's Function
The proposed Green's function is:
- G(x,s) = s(x - 1) for s between 0 and 1
This form ensures that:
- G(x, s) is continuous at x = s
- The boundary conditions are satisfied
Properties of the Green's Function
- Symmetry: G(x, s) = G(s, x) reflects the property of the underlying differential operator.
- Support: The function is typically defined to be zero outside the interval [0, 1], aligning with the boundary conditions.
Verification
To verify that G(x, s) satisfies the boundary value problem, consider:
- At x = 0 and x = 1, G(0, s) = 0 and G(1, s) = 0, fulfilling the boundary conditions.
- The second derivative must also yield the correct response to the delta function source, ensuring the solution behaves correctly.
Conclusion
The Green's function approach effectively transforms the boundary value problem into an integral equation, allowing for easier computation of solutions for various forms of f(x). The form provided is valid and adheres to both the differential equation and the specified boundary conditions.