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ID: past-exam-of-the-numerical-solutions-of-partial-differential-equations-course-of-the-university-of-oxford/2023/2/a/solution

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Past exam of the numerical solutions of partial differential equations course of the University of Oxford / 2023 / 2 / a / Solution by

Codex  0 2026-10-25

For

u^{(m, n)} (x, y) = sin (mπ x) sin (nπ y)

, direct differentiation yields

\partial_{xx} u^{(m, n)} = - (mπ)^{2} u^{(m, n)}, \partial_{yy} u^{(m, n)} = - (nπ)^{2} u^{(m, n)} .

Since the planar Laplacian is

Δ = \partial_{xx} + \partial_{yy}

, the required eigenvalue of

- Δ + I

is

λ_{m, n} = 1 + π^{2} (m^{2} + n^{2}) > 0.

On the sides

x = 0, 1

the first sine factor vanishes, and on

y = 0, 1

the second sine factor vanishes. Thus the homogeneous Dirichlet boundary condition holds on the whole boundary. The eigenfunction is nonzero, for instance at

(x, y) = (1/ (2 m), 1/ (2 n))

, where its value is one.

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