The nonlinear decay ,

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Ordinary differential equations · Numerical and computational mathematics

Error, Chance and the Machine, Ch Ordinary differential equations: order, stability and stiffness

Problem

The nonlinear decay y′=−y3y' = -y^3, y(0)=2y(0) = 2, is advanced by one backward Euler step of size h=0.1 sh = 0.1\,\mathrm{s}. This requires solving the implicit equation g(y1)≡y1+hy13−y0=0g(y_1) \equiv y_1 + h y_1^3 - y_0 = 0 for y1y_1 by Newton's method, starting from the initial guess y1(0)=y0=2y_1^{(0)} = y_0 = 2. ((a)) Write down g′(y1)g'(y_1). ((b)) Perform two Newton iterations and give y1y_1 to four significant figures, quoting the residual g(y1)g(y_1) as a check.

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