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In this paper, for the purpose of model reduction the analytical expressions of proper orthogonal decomposition (POD) modes are derived for the heat equation with boundary control. The autocorrelation function of the latter is viewed as the kernel of a self adjoint compact operator, and the POD modes and corresponding eigenvalues are computed by solving homogeneous integral equations of the second kind. The computed POD modes are compared to the modes obtained from snapshots. The explicit computation of the POD modes and eigenvalues allow the computation of different n-widths approximations for the heat equation, including the linear, Kolmogorov, Gelfand, and Bernstein n-widths.