A measure thoretic approach for studying inverse problems of the heat equation and the wave equation

<p style='text-indent:20px;'>We consider a heat equation and a wave equation in one spatial dimension. This article deals with the inverse problem of determining the size of the spatial interval from some extra boundary information on the solution. Under several different circumstances, we prove uniqueness, non-uniqueness and some size estimates. Moreover, we numerically solve the inverse problems and compute accurate approximations of the size. This is illustrated with several satisfactory numerical experiments.</p>

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A Moment Method on Inverse Problems for the Heat Equation

Inverse problems and related topics ◽

10.1201/9780429187841-6 ◽

2019 ◽

pp. 81-88

Author(s):

Mishio Kawashita ◽

Yaroslav Kurylev ◽

Hideo Soga

Keyword(s):

Inverse Problems ◽

Heat Equation ◽

Moment Method

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Solution of Inverse Problems for Wave Equation with a Nonlinear Coefficient

Computational Mathematics and Mathematical Physics ◽

10.1134/s0965542521090049 ◽

2021 ◽

Vol 61 (9) ◽

pp. 1511-1520

Author(s):

A. V. Baev

Keyword(s):

Wave Equation ◽

Inverse Problems ◽

Nonlinear Coefficient

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Rough estimates on the scalar heat kernel

Hypoelliptic Laplacian and Orbital Integrals (AM-177) ◽

10.23943/princeton/9780691151298.003.0013 ◽

2011 ◽

Author(s):

Jean-Michel Bismut

Keyword(s):

Wave Equation ◽

Heat Equation ◽

Heat Kernel ◽

Heat Kernels ◽

Hypoelliptic Operator ◽

Uniform Bounds ◽

Standard Heat ◽

Probabilistic Construction

This chapter establishes rough estimates on the heat kernel rb,tX for the scalar hypoelliptic operator AbX on X defined in the preceding chapter. By rough estimates, this chapter refers to just the uniform bounds on the heat kernel. The chapter also obtains corresponding bounds for the heat kernels associated with operators AbX and another AbX over ̂X. Moreover, it gives a probabilistic construction of the heat kernels. This chapter also explains the relation of the heat equation for the hypoelliptic Laplacian on X to the wave equation on X and proves that as b → 0, the heat kernel rb,tX converges to the standard heat kernel of X.

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