Question

In solving the heat equation using separation of variables, the eigenvalues correspond to ____.

• A. The Fourier coefficients
• B. The characteristic equation of the wave equation
• C. The eigenvalues of the Laplacian operator
• D. The energy of the system

Hint:

Separation of variables reduces the PDE to a boundary value problem X'' + lambda X = 0 for the spatial part.

Answer 1

The Correct Option is C

Step 1: Consider the one-dimensional heat equation \(u_t=\alpha^2u_{xx}\) on \(0<x<L\) with homogeneous boundary conditions.

Step 2: Assume a product solution \(u(x,t)=X(x)T(t)\) and substitute: \[X(x)T'(t)=\alpha^2X''(x)T(t)\]

Step 3: Divide both sides by \(\alpha^2X(x)T(t)\) to separate variables: \[\frac{T'(t)}{\alpha^2T(t)}=\frac{X''(x)}{X(x)}=-\lambda\] The left side depends only on \(t\) and the right only on \(x\), so both must equal a constant \(-\lambda\).

Step 4: This yields two ordinary differential equations: \(T'+\lambda\alpha^2T=0\) and \(X''+\lambda X=0\), the latter subject to the given boundary conditions.

Step 5: The boundary value problem \(X''+\lambda X=0\) with the boundary conditions has nontrivial solutions only for specific values of \(\lambda\); these are precisely the eigenvalues of the second-derivative (Laplacian) operator on that domain, with \(X(x)\) the corresponding eigenfunctions. \[\boxed{\text{The eigenvalues of the Laplacian operator}}\]