这是一篇来自美国的关于常微分方程、误差分析及精度改进的数学代写

Assignment

Consider a numerical analysis of a differential equation,

y ′′ = −y, (1)

with initial conditions for:

y = 0 and y′ = 1 at x = 0

The exact solution is known to be

y = sin x.

Conduct numerical analysis for 0 ≤ x ≤ 2π using various methods and analyze the following items:

• errors in amplitude A (= √y2 + y′2) and phase ϕ (= tan−1(y/y′)) at x = 2π.
• accuracy improvement by Richardson extrapolation and/or introducing higher-order methods.
• discuss on stability of the methods.
• work load, coding efforts, etc.

The report should be in A4 size and include a description of the problem, methods, analysis with fifigures and tables, discussion, and conclusions. MATLAB code written in live-script (.mlx) should be presented as an appendix.