Separation of Variables and the Heat Equation
Assume the solution is a product of one-variable functions and the equation splits into ordinary ones. The boundary conditions pick out which modes are allowed, and the initial profile is matched by a series in those modes.
What a learner can do afterwards
- Separate a boundary value problem into two ordinary equations
- Identify the eigenvalues that the boundary conditions force
- Match a given initial profile by computing its coefficients
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Assume the solution is a product u(x,t) equals X(x) times T(t). Insert this into u_t equals u_xx to get X times T prime equals X double prime times T. Divide by XT so each side depends on one variable only. Both sides must equal one shared constant, called negative L. That yields the pair X double prime plus L X equals 0 for space and T prime plus L T equals 0 for time. With conductivity k the split carries k on the time side only: X double prime over X equals T prime over kT.
Pin a rod of length pi at zero temperature at both ends. Then X(0) and X(pi) are zero, which forces the shapes sin(nx) with eigenvalues L equals n squared. The first three eigenvalues are 1, 4, and 9. Each time factor decays like e to the negative Lt, so the n equals 3 mode dies much faster than the n equals 1 mode. High modes always decay fastest.
The initial profile is matched by adding the modes with the right coefficients. Distinct sine modes are orthogonal, so each coefficient is read off directly: for f(x) equals 2 sin x plus 5 sin 3x, the sin 3x coefficient is 5. A steady state has u_t zero, so u_xx is zero and the profile is the straight line joining the end values. On a rod of length 2 with ends 0 and 20, the midpoint settles at 10.
Insulated ends change the mode family: zero slope at the ends gives cosines instead of sines. On a rod of length 2 the eigenvalues are (n pi over 2) squared, so the smallest positive one is pi squared over 4. The slowest decay always belongs to the smallest eigenvalue, which on the pi rod with zero ends is rate 1.
Products split the equation into two ordinary ones, the ends pick the modes, and the start profile sets their sizes.
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