Solving a Cauchy problem for the heat equation using cubic smoothing splines

References

• Beck JV, Blackwell B, Clair SR. Inverse heat conduction. Ill-posed problems. New York: Wiley; 1985.
   Google Scholar
• Eldén L, Berntsson F, Regińska T. Wavelet and Fourier methods for solving the sideways heat equation. SIAM J Sci Comput. 2000;21(6):2187–2205.
   Web of Science ®Google Scholar
• Gardarein J-L, Gaspar J, Corre Y, et al. Inverse heat conduction problem using thermocouple deconvolution application to the heat flux estimation in a tokamak. Inverse Probl Sci Eng. 2013;21(5):854–864.
   Web of Science ®Google Scholar
• Jahedi M, Berntsson F, Wren J, et al. Transient inverse heat conduction problem of quenching a hollow cylinder by one row of water jets. Int J Heat Mass Transf. 2018;117(Supplement C):748–756.
   Google Scholar
• Li D, Wells MA. Effect of subsurface thermocouple installation on the discrepancy of the measured thermal history and predicted surface heat flux during a quench operation. Metallurg Mater Trans. 2005;36B:343–354.
   Google Scholar
• Taler J, Weglowski B, Pilarczyk M. Monitoring of thermal stresses in pressure components using inverse heat conduction methods. Int J Numer Meth Heat Fluid Flow. 2017;27(3):740–756.
   Web of Science ®Google Scholar
• Wikström P, Blasiak W, Berntsson F. Estimation of the transient surface temperature, heat flux and effective heat transfer coefficient of a slab in an industrial reheating furnace by using an inverse method. Steel Res Int. 2007;78(1):63–70.
   Web of Science ®Google Scholar
• Berntsson F. An inverse heat conduction problem and improving shielded thermocouple accuracy. Numer Heat Transfer, Part A: Appl. 2012;61(10):754–763.
   Web of Science ®Google Scholar
• Xiong X-T, Fu C-L. A spectral regularization method for solving surface heat flux on a general sideways parabolic. Appl Math Comput. 2008;197(1):358–365.
   Web of Science ®Google Scholar
• Carasso AS. Slowly divergent space marching schemes in the inverse heat conduction problem. Numer Heat Transfer, Part B. 1993;23:111–126.
   Web of Science ®Google Scholar
• Guo L, Murio DA, Roth C. A mollified space marching finite differences algorithm for the inverse heat conduction problem with slab symmetry. Computers Math Appl. 1990;19:75–89.
   Web of Science ®Google Scholar
• Mejía CE, Murio DA. Numerical solution of generalized IHCP by discrete mollification. Computers Math Appl. 1996;32:33–50.
   Web of Science ®Google Scholar
• Berntsson F. A spectral method for solving the sideways heat equation. Inverse Probl. 1999;15:891–906.
   Web of Science ®Google Scholar
• Blevins LG, Pitts WM. Modeling of bare and a spirated thermocouples in compartment fires. Fire Saf J. 1999;33:239–259.
   Web of Science ®Google Scholar
• Kemp SE, Annaheim S, Rossi RM, et al. Test method for characterising the thermal protective performance of fabrics exposed to flammable liquid fires. Fire Mater. 2016;41:750–767.
   Web of Science ®Google Scholar
• Karimi M, Rezaee A. Regularization of the Cauchy problem for the Helmholtz equation by using Meyer wavelet. J Comput Appl Math. 2017;320:76–95.
   Web of Science ®Google Scholar
• Regińska T. Sideways heat equation and wavelets. J Comput Appl Math. 1995;63:209–214.
   Web of Science ®Google Scholar
• Foadian S, Pourgholi R, Hashem Tabasi S. Cubic b-spline method for the solution of an inverse parabolic system. Appl Anal. 2018;97(3):438–465.
   Web of Science ®Google Scholar
• Reinsch CH. Smoothing by spline functions. Numer Math. 1971;16(5):451–454.
   Web of Science ®Google Scholar
• Isakov V. Inverse problems for partial differential equations. New York: Springer; 1998.
   Google Scholar
• Eldén L. Solving the sideways heat equation by a ‘method of lines’. J Heat Transfer, Trans ASME. 1997;119:406–412.
   Web of Science ®Google Scholar
• Gustafsson B, Kreiss H-O, Oliger J. Time dependent problems and difference methods. New York: Wiley Interscience; 1995.
   Google Scholar
• Anderson GD, Vamanamurthy MK, Vuorinen MK. Conformal invariants, inequalities, and quasiconformal maps. New York: John Wiley & Sons, Inc.; 1997.
   Google Scholar
• Ragozin DL. Error bounds for derivative estimates based on spline smoothing of exact or noisy data. J Approx Theor. 1983;37(4):335–355.
   Web of Science ®Google Scholar
• Schoenberg IJ. Spline interpolation and the higher derivatives. Proc Nat Acad Sci USA. 1964;51:24–28.
   PubMed Web of Science ®Google Scholar
• Agmon S. Lectures on elliptic boundary value problems. Princeton (NJ): D. Van Nostrand Co.; 1965.
   Google Scholar