 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ii/paper-2/39c/b/solution

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

Let

u_{j} = (u_{1, j}, \dots, u_{m, j})^{T}

collect the unknowns in grid column

j

b_{j} = h^{2} (f_{1, j}, \dots, f_{m, j})^{T}

. The terms within one column form the block

B = [\frac{2}{3}, - \frac{10}{3}, \frac{2}{3}],

while coupling to either adjacent column contributes

C = [\frac{1}{6}, \frac{2}{3}, \frac{1}{6}] .

The zero boundary values remove all blocks outside the displayed range, so the nine-point finite-difference stencil becomes the block tridiagonal matrix system

B C C B ⋱ ⋱ ⋱ C C B u_{1} u_{2} ⋮ u_{m} = b_{1} b_{2} ⋮ b_{m} .

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