Solution

Each coordinate of the transformed vectors evolves independently because

D

and

E

are diagonal matrices. Reorder the unknowns by collecting the

k

th components across all grid columns:

v_{k} = ((v_{1})_{k}, \dots, (v_{m})_{k})^{T}, c_{k} = ((c_{1})_{k}, \dots, (c_{m})_{k})^{T} .

(1)

Then system (3) splits into the

m

uncoupled systems

Λ_{k} v_{k} = c_{k}, k = 1, \dots, m,

(2)

where

Λ_{k} = [e_{k}, d_{k}, e_{k}] = d_{k} e_{k} e_{k} d_{k} ⋱ ⋱ ⋱ e_{k} e_{k} d_{k} .

(3)

Thus the original $m^{2} \times m^{2}$ problem reduces to one discrete sine transform followed by $m$ independent tridiagonal solves and an inverse sine transform.

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Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 39C / d / Solution

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