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Application of the inverse method to estimate the time-varying heat sources on infinite cylinders.
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This paper estimates the heat source of infinite cylinders at different time and in various locations by inverse method. First, equally divide time and space in the formula and discretize them using the second-order finite difference method. Next, substitute into the quantities of the known thermal conductivity, boundary conditions, initial conditions, as well as the measured temperatures of every location and time. However, there are errors in the measured temperatures of every location and time. In this paper, the measured temperatures are the results of adding random errors of 1%, 3%, and 5% to the analytical solution, respectively. Using these noisy measured temperatures to inverse calculate the heat sources results in more error in the estimated values than the exact heat source solution. Consequently, Savitzky-Gollay digital filter method is adopted to smooth the measured temperatures by weight averaging seven or eleven measured points, respectively, prior to substituting the values into the discretized equations for the inverse calculation. The result demonstrates that comparing to the unsmoothed estimated heat source value, the smoothed heat source value is much more precise. In addition, the research finds that Savitzky-Gollay digital filter method is more effective to estimate the heat source when there are multiple measured points. In the digital filter method, inverse calculation gives more accuracy after smoothing eleven measured points than seven measured points. Nevertheless, if the amount of measured points is not large, smoothing seven measured points would be accurate enough.
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