A spring-damping regularization of the Fourier sine series solution to the inverse Cauchy problem for a 3D sideways heat equation

References

• Beck JV, Blackwell B, Clair CRS. Inverse heat conduction: ill-posed problem. New York (NY): Wiley; 1985.
   Google Scholar
• Murio DA. The mollification method and the numerical solution of the inverse heat conduction problem by finite differences. Comput Math Appl. 1989;17:1385–1396. doi: 10.1016/0898-1221(89)90022-9
   Google Scholar
• Guo L, Murio DA, Roth C. A mollified space marching finite differences algorithm for the inverse heat conduction problem with slab symmetry. Comput Math Appl. 1990;19:75–89. doi: 10.1016/0898-1221(90)90196-Q
   Google Scholar
• Murio DA, Guo L. A stable space marching finite differences algorithm for the inverse heat conduction problem with no initial filtering procedure. Comput Math Appl. 1990;19:35–50. doi: 10.1016/0898-1221(90)90356-O
   Google Scholar
• Guo L, Murio DA. A mollified space-marching finite different algorithm for the two-dimensional inverse heat conduction problem with slab symmetry. Inverse Prob. 1991;7:247–259. doi: 10.1088/0266-5611/7/2/008
   Google Scholar
• Al-Khalidy N. On the solution of parabolic and hyperbolic inverse heat conduction problems. Int J Heat Mass Transf. 1998;41:3731–3740. doi: 10.1016/S0017-9310(98)00102-1
   Google Scholar
• Seidman TI, Eldén L. An ‘optimal filtering’ method for the sideways heat equation. Inverse Prob. 1990;6:681–696. doi: 10.1088/0266-5611/6/4/013
   Google Scholar
• Eldén L. Numerical solution of the sideways heat equation by difference approximation in time. Inverse Prob. 1995;11:913–923. doi: 10.1088/0266-5611/11/4/017
   Google Scholar
• Eldén L. Numerical solution of the sideways heat equation. In: Engl H, Rundell W, editors. Inverse problems in diffusion processes. Proceedings in applied mathematics. Philadelphia (PA): SIAM; 1995.
   Google Scholar
• Regińska T. Sideways heat equation and wavelets. J Comp Appl Math. 1995;63:209–214. doi: 10.1016/0377-0427(95)00073-9
   Google Scholar
• Regińska T, Eldén L. Solving the sideways heat equation by a wavelet-Galerkin method. Inverse Prob. 1997;13:1093–1106. doi: 10.1088/0266-5611/13/4/014
   Google Scholar
• Berntsson F. A spectral method for solving the sideways heat equation. Inverse Prob. 1999;15:891–906. doi: 10.1088/0266-5611/15/4/305
   Google Scholar
• Dorroh JR, Ru X. The application of the method of quasi-reversibility to the sideways heat equation. J Math Anal and Appl. 1999;15:503–519. doi: 10.1006/jmaa.1999.6462
   Google Scholar
• Eldén L, Berntsson F, Regińska T. Wavelet and Fourier methods for solving the sideways heat equation. SIAM J Sci Comput. 2000;21:2187–2205. doi: 10.1137/S1064827597331394
   Google Scholar
• Regińska T. Application of wavelet shrinkage to solving the sideways heat equation. BIT. 2001;41:1101–1110. doi: 10.1023/A:1021909816563
   Google Scholar
• Fu CL, Qiu CY, Zhu YB. A note on ‘sideways heat equation and wavelets’ and constant e∗. Comput Math Appl. 2002;43:1125–1134. doi: 10.1016/S0898-1221(02)80017-7
   Google Scholar
• Qiu CY, Fu CL, Zhu YB. Wavelets and regularization of the sideways heat equation. Comput Math Appl. 2003;46:821–829. doi: 10.1016/S0898-1221(03)90145-3
   Google Scholar
• Berntsson F. Sequential solution of the sideways heat equation by windowing of the data. Inverse Prob Sci Eng. 2003;11:91–103. doi: 10.1080/1068276021000048564
   Google Scholar
• Wang J. The multi-resolution method applied to the sideways heat equation. J Math Anal Appl. 2005;309:661–673. doi: 10.1016/j.jmaa.2004.11.025
   Google Scholar
• Xiong XT, Fu CL, Li HF. Fourier regularization method of a sideways heat equation for determining surface heat flux. J Math Anal Appl. 2006;317:331–348. doi: 10.1016/j.jmaa.2005.12.010
   Google Scholar
• Wang JR. Uniform convergence of wavelet solution to the sideways heat equation. Acta Math Sin-E. 2010;26:1981–1992. doi: 10.1007/s10114-010-7242-4
   Google Scholar
• Deng Y, Liu Z. New fast iteration for determining surface temperature and heat flux of general sideways parabolic equation. Nonl Anal Real World Appl. 2011;12:156–166. doi: 10.1016/j.nonrwa.2010.06.005
   Google Scholar
• Pakanen JE. A sequential one-dimensional solution for the sideways heat equation in a semi-infinite, finite and multilayer slab. Numer Heat Transfer B. 2012;61:471–491. doi: 10.1080/10407790.2012.687990
   Google Scholar
• Nguyen HT, Luu VCH. Two new regularization methods for solving sideways heat equation. J Inequal Appl. 2015;2015:65–81. doi: 10.1186/s13660-015-0564-0
   Google Scholar
• Chang CW, Liu CS, Chang J-R. A group preserving scheme for inverse heat conduction problems. Comput Model Eng Sci. 2005;10:13–38.
   Google Scholar
• Chang CW, Liu CH, Wang CC. Review of computational schemes in inverse heat conduction problems. Smart Sci. 2018;6:94–103. doi: 10.1080/23080477.2017.1408987
   Google Scholar
• Yu Y, Xu D, Hon YC. Reconstruction of inaccessible boundary value in a sideways parabolic problem with variable coefficients – forward collocation with finite integration method. Eng Anal Bound Elem. 2015;61:78–90. doi: 10.1016/j.enganabound.2015.07.007
   Google Scholar
• Liu CS. Homogenized functions to recover H(t)/H(x) by solving a small scale linear system of differencing equations. Int J Heat Mass Transf. 2016;101:1103–1110. doi: 10.1016/j.ijheatmasstransfer.2016.05.133
   Google Scholar
• Liu CS. To recover heat source G(x)+H(t) by using homogenized function and solving rectangular differencing equations. Numer Heat Transfer B. 2016;69:351–363. doi: 10.1080/10407790.2015.1125211
   Google Scholar
• Liu CS, Chen W, Lin J. A homogenized function to recover wave source by solving a small scale linear system of differencing equations. Comput Model Eng Sci. 2016;111:421–435.
   Google Scholar
• Liu CS, Qiu L, Wang F. Nonlinear wave inverse source problem solved by a method of m-order homogenization functions. Appl Math Lett. 2019;91:90–96. doi: 10.1016/j.aml.2018.11.025
   Google Scholar
• Liu CS, Chang CW. Solving the inverse conductivity problems of nonlinear elliptic equations by the superposition of homogenization functions method. Appl Math Lett. 2019;94:272–278. doi: 10.1016/j.aml.2019.03.017
   Google Scholar
• Liu CS, Chen YW, Chang JR. Solving a nonlinear convection-diffusion equation with source and moving boundary both unknown by a family of homogenization functions. Int J Heat Mass Transf. 2019;138:25–31. doi: 10.1016/j.ijheatmasstransfer.2019.04.026
   Google Scholar
• Liu CS, Chang CW. Solving the 3D Cauchy problems of nonlinear elliptic equations by the superposition of a family of 3D homogenization functions. Eng Anal Bound Elem. 2019;105:122–128. doi: 10.1016/j.enganabound.2019.04.001
   Google Scholar
• Liu CS, Lin J, Qiu L. Solving the higher-dimensional nonlinear inverse heat source problems by the superposition of homogenization functions method. Int J Heat Mass Transf. 2019;141:651–657. doi: 10.1016/j.ijheatmasstransfer.2019.07.007
   Google Scholar
• Liu CS, Qiu L, Lin J. Simulating thin plate bending problems by a family of two-parameter homogenization functions. Appl Math Model. 2020;79:284–299. doi: 10.1016/j.apm.2019.10.036
   Google Scholar
• Liu CS, Kuo CL. A spring-damping regularization and a novel Lie-group integration method for nonlinear inverse Cauchy problems. Comput Model Eng Sci. 2011;77:57–80.
   Google Scholar
• Liu CS, Kuo CL, Liu D. The spring-damping regularization method and the Lie-group shooting method for inverse Cauchy problems. Comput Mater Contin. 2011;24:105–123.
   Google Scholar
• Liu CS, Chang CW. Analytic series solutions of 2D forward and backward heat conduction problems in rectangles and a new regularization. Inv Prob Sci Eng. 2020; Available from https://doi.org/10.1080/17415977.2020.1719086
   Google Scholar