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A method for recovering the spatially-dependent coefficient \(q(x)\) in the parabolic equation \(w_{t}-w_{xx}+q(x)w=0\) , \(x\in (0,L)\) , \(t>0\) , from a knowledge of the boundary data \(w(0,t)\) , \(w_{x}(0,t)\) , \(w(L,t)\) and under the condition \(w(x,0)=0\) , is developed. It is based on Neumann series of Bessel functions (NSBF) representations for solutions of the related Sturm-Liouville equation. With the aid of the Laplace transform and NSBF representations, the inverse problem is reduced to a system of linear algebraic equations for the NSBF coefficients. The coefficient \(q(x)\) is recovered from an arithmetic combination of the first two unknowns of this system. The approach leads to an efficient numerical algorithm. Numerical efficiency is illustrated by test examples.

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Approximate Reconstruction of a One-Dimensional Parabolic Equation from Boundary Data

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  • 发表日期：

    2025年

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  • Vladislav V. Kravchenko

摘要

A method for recovering the spatially-dependent coefficient \(q(x)\) in the parabolic equation \(w_{t}-w_{xx}+q(x)w=0\) , \(x\in (0,L)\) , \(t>0\) , from a knowledge of the boundary data \(w(0,t)\) , \(w_{x}(0,t)\) , \(w(L,t)\) and under the condition \(w(x,0)=0\) , is developed. It is based on Neumann series of Bessel functions (NSBF) representations for solutions of the related Sturm-Liouville equation. With the aid of the Laplace transform and NSBF representations, the inverse problem is reduced to a system of linear algebraic equations for the NSBF coefficients. The coefficient \(q(x)\) is recovered from an arithmetic combination of the first two unknowns of this system. The approach leads to an efficient numerical algorithm. Numerical efficiency is illustrated by test examples.