 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ii/paper-2/40e/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

With

u = (u_{1}, \dots, u_{n})^{T}

and

b = h^{2} (f_{1}, \dots, f_{n})^{T}

, the system is

A u = b, A = - 2 1 1 - 2 ⋱ 1 ⋱ 1 ⋱ - 2 1 1 - 2 .

Its diagonal part is

D = - 2 I

u^{(ν + 1)} = u^{(ν)} + ω D^{- 1} (b - A u^{(ν)}) .

Componentwise,

u_{i}^{(ν + 1)} = (1 - ω) u_{i}^{(ν)} + \frac{ω}{2} (u_{i - 1}^{(ν)} + u_{i + 1}^{(ν)} - h^{2} f_{i}),

with

u_{0}^{(ν)} = u_{n + 1}^{(ν)} = 0

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