Numerical BVP Methods

Differential Equations · Axiom Academy

Two ways to solve a boundary value problem when there's no formula for it: shoot from one end and correct your aim, or replace the whole equation with a grid of algebra. 1. Two Endpoints, No Starting Slope Consider a second-order equation on [a,b] with conditions fixed at both ends: You know and — but not y'(a) . Any single value of y'(a) you pick determines a unique curve leaving x=a (that part is just an IVP); the hard part is picking the ONE slope whose curve actually threads back through on the far side. The shooting method turns the BVP back into familiar territory: guess the missing slope, solve the resulting IVP, and see how far you land from the target. Adjust the guess and repeat. Guess an initial slope s for y'(a) . Solve the IVP forward to x=b (using any IVP solver — Euler, RK4, …). Compute the residual — how far off the landing was. If , stop. Otherwise adjust s (via a root-finder — bisection or secant/Newton on R ) and repeat from step 2. 3. Finite Differences: Discretize the Whole Domain Instead of marching an IVP, finite differences lay a grid across [a,b] and replace every derivative in the equation with an algebraic difference — turning the differential equation into ordinary algebra at each grid point, all at once.

This is the written version of the interactive lesson above. See the full Differential Equations course.