Convergence and Divergence of Approximations in terms of the Derivatives of Heat Kernel

2014/09/08 by Jaywan Chung, Chung, Jaywan
Computer Science · Mathematics · #41A30 / 35K05 / 34L10 / 33C45 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Approximation and Integration #Numerical methods in inverse problems #math.AP #math.FA #msc:33C45 #msc:34L10 #msc:35K05 #msc:41A30

paper · pdf · doi:10.48550/arxiv.1409.2289

10 pages

arxiv created 2014/09/08 · openalex publication_date 2014/09/08 · arxiv updated 2014/09/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider an approximate solution to the heat equation which consists of the derivatives of heat kernel. Some conditions in the initial value, under which the approximation converges to the solution of the heat equation or diverges when the number of terms of the approximation goes to infinity with a fixed time t, will be given. For example, when the initial data is a Gaussian e^-|\bf x|²/4t₀, the approximation converges when t>t₀. But if t<t₀, it diverges to infinity. Also the L^∞-error estimate will be given and the meaning of the approximation will be clarified by comparing with eigenfunction expansion.

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