In this paper, we employ linear combinations of n heat kernels to approximate solutions to the heat equation. We show that such approximations are of order O(t(1/2p-2n+1/2)) in Lp-norm, 1 ≤ p ≤ ∞, as t → ∞. For positive solutions of the heat equation such approximations are achieved using the theory of truncated moment problems. For general sign-changing solutions these type of approximations are obtained by simply adding an auxiliary heat kernel. Furthermore, inspired by numerical computations, we conjecture that such approximations converge geometrically as n → ∞ for any fixed t > 0.
Bibliographical notePublisher Copyright:
© 2009 Society for Industrial and Applied Mathematics.
- Approximation of an integral formula
- Asymptotics convergence rates
- Heat equation
- Heat kernel