Heat Transfer Through Concentric Cylinders

Why this equation has no closed form

The governing equation for steady radial conduction through a cylindrical wall, with a convective boundary on the outside, gives ln(r₂/r₁) = 2.2185·ln(0.264/(r₂·ln(r₂/r₁))) once the known quantities (heat flow, conductivity, convection coefficient) are folded into those two constants. The unknown r₂ appears three times, once buried inside the innermost log — there's no algebraic move that isolates it. This is the same situation as the Root Finding lab's classic example, just dressed up in real engineering units instead of an abstract polynomial.

The same equation, five different walks to the same root

Bisection brackets the root and halves the interval — slow but unconditionally safe. Newton's method uses the local slope (numerically estimated here) to leap toward the root — fast, but needs a derivative. Secant approximates that slope from two nearby points instead of computing it — almost as fast, no derivative needed. Fixed-point iteration rearranges the equation into r₂ = g(r₂) and just repeats it — simple, but only if the rearrangement happens to converge (try rearranging the equation a different way and it can diverge immediately, as the original notebook this is based on found).

Steffensen's method: the fixed-point speedup

The convergent rearrangement here is real but slow — plain fixed-point iteration takes 88 steps to reach a tolerance of 0.001. Steffensen's method takes the same iteration g(r₂) and applies Aitken's Δ² acceleration to it: compute two ordinary steps, then extrapolate past both using how far apart they are. The same convergent g, accelerated this way, reaches the same tolerance in 3 steps — turning a borderline-impractical method into one of the fastest options here, without ever needing a derivative. That's the direct answer to "did Steffensen's method help?"