• Alem, Leila (Laboratory of Applied Mathematics Badji Mokhtar University) ;
  • Chorfi, Lahcene (Laboratory of Applied Mathematics Badji Mokhtar University)
  • 투고 : 2017.06.20
  • 심사 : 2018.03.09
  • 발행 : 2018.07.31


We consider an ill-posed problem for the heat equation $u_{xx}=u_t$ in the quarter plane {x > 0, t > 0}. We propose a new method to compute the heat flux $h(t)=u_x(1,t)$ from the boundary temperature g(t) = u(1, t). The operator $g{\mapsto}h=Hg$ is unbounded in $L^2({\mathbb{R}})$, so we approximate h(t) by $h_{\delta}(t)=u_x(1+{\delta},\;t)$, ${\delta}{\rightarrow}0$. When noise is present, the data is $g_{\epsilon}$ leading to a corresponding heat $h_{{\delta},{\epsilon}}$. We obtain an estimate of the error ${\parallel}h-h_{{\delta},{\epsilon}}{\parallel}$, as well as the error when $h_{{\delta},{\epsilon}}$ is approximated by the trapezoidal rule. With an a priori choice rule ${\delta}={\delta}({\epsilon})$ and ${\tau}={\tau}({\epsilon})$, the step size of the trapezoidal rule, the main theorem gives the error of the heat flux as a function of noise level ${\epsilon}$. Numerical examples show that the proposed method is effective and stable.


inverse problems;ill-posed problem;stable approximation;error estimate


  1. L. Chorfi and L. Alem, Stable algorithm for identifying a source in the heat equation, Electron. J. Differential Equations 2015 (2015), No. 267, 14 pp.
  2. L. Elden, Approximations for a Cauchy problem for the heat equation, Inverse Problems 3 (1987), no. 2, 263-273.
  3. L. Elden, F. Berntsson, and T. Reginska, Wavelet and Fourier methods for solving the sideways heat equation, SIAM J. Sci. Comput. 21 (2000), no. 6, 2187-2205.
  4. C.-L. Fu, X.-T. Xiong, and P. Fu, Fourier regularization method for solving the surface heat flux from interior observations, Math. Comput. Modelling 42 (2005), no. 5-6, 489- 498.
  5. R. Gorenflo and S. Vessella, Abel Integral Equations, Lecture Notes in Mathematics, 1461, Springer-Verlag, Berlin, 1991.
  6. C. W. Groetsch, A numerical method for some inverse problems in nonlinear heat transfer, Comput. Math. Appl. (Athens) 1 (1994), 49-59.
  7. C. W. Groetsch and M. Hanke, A general framework for regularized evaluation of unstable operators, J. Math. Anal. Appl. 203 (1996), no. 2, 451-463.
  8. D. N. Hao, H. J. Reinhardt, and A. Schneider, Numerical solution to a sideways parabolic equation, Internat. J. Numer. Methods Engrg. 50 (2001), no. 5, 1253-1267.<1253::AID-NME81>3.0.CO;2-6
  9. D. A. Murio, The Mollification Method and the Numerical Solution of Ill-posed Problems, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1993.
  10. G. Talenti and S. Vessella, A note on an ill-posed problem for the heat equation, J. Austral. Math. Soc. Ser. A 32 (1982), no. 3, 358-368.
  11. V. S. Vladimirov, Equations of Mathematical Physics, Translated from the Russian by Audrey Littlewood. Edited by Alan Jeffrey. Pure and Applied Mathematics, 3, Marcel Dekker, Inc., New York, 1971.