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- W1971011623 abstract "The authors consider the problem of determining the temperature distribution<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$uleft( {x,t} right)$ id=E1><mml:mi>u</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>on the half-line<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$x = 0$ id=E2><mml:mi>x</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math>,<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$t > 0$ id=E3><mml:mi>t</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:math>, from measurements at an interior point, for all<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$t > 0$ id=E4><mml:mi>t</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:math>. As is well-known, this is an ill-posed problem Using the Tikhonov method, the authors give a regularized solution, and assuming the (unknown) exact solution is in<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$H^alpha left( mathbb{R} right)$ id=E5><mml:msup><mml:mi>H</mml:mi><mml:mi>α</mml:mi></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>ℝ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>,<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$alpha > 0$ id=E6><mml:mi>α</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:math>. They give an error estimate of the order<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=${1 mathord{left/ {vphantom {1 {left( {{{1n1} mathord{left/ {vphantom {{1n1} varepsilon }} right. kern-nulldelimiterspace} varepsilon }} right)^alpha }}} right. kern-nulldelimiterspace} {left( {{{1n1} mathord{left/ {vphantom {{1n1} varepsilon }} right. kern-nulldelimiterspace} varepsilon }} right)^alpha }}$ id=E7><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mi>n</mml:mi><mml:mn>1</mml:mn></mml:mrow><mml:mo>/</mml:mo><mml:mi> ϵ </mml:mi></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mi>α</mml:mi></mml:msup></mml:mrow></mml:mrow></mml:math>for<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$varepsilon to {mathrm{0}}$ id=E8><mml:mi>ϵ </mml:mi><mml:mo>→</mml:mo><mml:mtext>0</mml:mtext></mml:math>, where<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$varepsilon > 0$ id=E9><mml:mi>ϵ </mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:math>is a bound on the measurement error." @default.
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- W1971011623 date "1995-01-01" @default.
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- W1971011623 title "Surface temperature determination from Borehole measurements: Regularization and error estimates" @default.
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- W1971011623 doi "https://doi.org/10.1155/s0161171295000755" @default.
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