Question

Find the GF for \(\mathbf{L}_{x}=d^{2} / d x^{2}+k^{2}\) with BCs \(u(0)=u(a)=0\).

Short answer

The Green's function for \(\mathbf{L}_{x}=d^{2} / d x^{2}+k^{2}\) with boundary conditions \(u(0)=u(a)=0\) is \(G(x, x')=\frac{1}{k}\sin(kx_<)\sin[k(a-x_>)].

Step-by-step solution

Define the Differential Operator

The given differential operator is \(\mathbf{L}_{x}=d^{2} / d x^{2}+k^{2}\). This will act on an unknown function \(u(x)\) that we will need to determine.

Construct the Homogeneous Solution

We first solve the homogeneous equation \(\mathbf{L}_{x}u=0\), which simplifies to \(\frac{d^{2}u}{dx^{2}}+k^{2}u=0\). The solution to this homogeneous equation is \(u(x)=A\sin(kx) + B\cos(kx)\) where A and B are constants that will be determined by the boundary conditions.

Apply the Boundary Conditions

We then apply the given boundary conditions \(u(0)=0\) and \(u(a)=0\). When \(x=0\), the equation becomes \(B=0\), hence \(u(x)=A\sin(kx)\). Substituting \(a\) for \(x\), the equation becomes \(0=A\sin(ka)\), leading to two possible scenarios: either \(A=0\) which would imply the trivial solution, or \(\sin(ka)=0\), which means that \(ka=n\pi\) where \(n\) is an integer.

Formulate the Green's Function

Finally, the Green's function can be written as \(G(x, x')=\frac{1}{k}\sin(kx_<)\sin[k(a-x_>)],\) where \(x_<\) is the minimum of \(x\) and \(x'\) and \(x_>\) is the maximum of \(x\) and \(x'\). The factor of \(1/k\) is to normalize the function.

Key concepts

Differential Operator

When tackling problems involving differential equations, understanding the role of a differential operator is crucial. In the context of our exercise, the differential operator is denoted by \( \mathbf{L}_{x}=\frac{d^{2}}{dx^{2}}+k^{2} \). This operator tells us how to act on a function \(u(x)\) to obtain another function or to determine its properties.

The operator is composed of two parts: the second derivative with respect to \(x\), which provides a measure of how the function's rate of change is accelerating or decelerating, and the term \(k^{2}\), which adds a fixed value times the function itself to the result. This combination is common in physical problems, such as those describing oscillations or wave behavior under specific conditions.

To solve an equation using this differential operator, one must generally find a function which, when operated on by \( \mathbf{L}_{x}\), yields a particular solution. This search for a function is central to establishing the Green's Function for the problem, thus providing a powerful tool to tackle more complex boundary value problems.

Homogeneous Solution

A homogeneous solution plays a key role in solving differential equations and is found by setting the non-homogeneous part to zero. For our exercise, solving the homogeneous equation \( \mathbf{L}_{x}u=0 \) leads to \( \frac{d^{2}u}{dx^{2}}+k^{2}u=0 \).

The solution to this type of second-order linear differential equation is a combination of sine and cosine functions, \( u(x)=A\sin(kx) + B\cos(kx)\), where \(A\) and \(B\) are constants determined by boundary conditions. Why sine and cosine? Because they inherently satisfy the properties of this equation for any \(k\) value due to their periodic and oscillatory nature.

Homogeneous solutions are particularly important because they form the complementary part of the general solution to the differential equation and are essential in constructing Green's Function. They incorporate the behavior of the system without any external forces or influences, hence 'homogeneous', referring to the system's self-contained dynamics.

Boundary Conditions

Boundary conditions (BCs) are critical to solving physical and mathematical problems uniquely. They describe constraints that solutions must meet at the boundaries of the domain under consideration. In our lesson, the BCs are \(u(0)=u(a)=0\).

Applying these BCs to the homogeneous solution \( u(x)=A\sin(kx) + B\cos(kx)\) guides us in discovering the values of \(A\) and \(B\). When \(x=0\), the cosine component disappears, leaving us with \(B=0\). The sine term, however, leads to a condition for \(A\) when \(x=a\), which directs us to the frequencies or modes (\(k\) values) the system can naturally adopt. This is encapsulated in the condition \(\sin(ka)=0\), which implies that \(ka=n\pi\) where \(n\) is an integer, restricting \(k\) to specific quantized values.

Boundary conditions are a powerful concept because they ensure the uniqueness of solutions and provide physical meaning to mathematical problems. They differentiate between multitude solutions to a differential equation, confining the search to one that is not only mathematically correct but also physically applicable.