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 Incoming links: Uniform forcing of a periodic heat equation

Past exam of the year 2 of the earth sciences (geology) course of the University of Oxford / 2025 / Paper 3 / 7 / h / Solution 2026-10-25

A spatially uniform source changes only the zero Fourier mode. Its exact integrated increment over a time step beginning at

t

is

Δ q = \int_{t}^{t + Δ t} 0.1 e^{- s /10} d s = e^{- t /10} - e^{- (t + Δ t) /10} .

The unnormalised two-dimensional discrete Fourier transform stores a constant field

q

n_{x} n_{y} q

in element (1,1).^[ref] Keep the existing homogeneous exponential multiplication and add this source increment to that element:

for i = 1:nt
    t = (i-1)*dt;
    u_hat = u_hat .* exp(-alpha*K2*dt);
    dq = exp(-t/10) - exp(-(t+dt)/10);
    u_hat(1,1) = u_hat(1,1) + nx*ny*dq;
end

This replacement uses the original variable names and MATLAB indexing. The uniform forcing of a periodic heat equation is integrated exactly in time here. No source multiplier is applied to the other modes, and the zero-mode homogeneous multiplier equals one. Adding only dq to the unnormalised coefficient would underestimate the physical temperature increment by the factor

n_{x} n_{y}

.

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