A meshless method for reconstructing the heat source and partial initial temperature in heat conduction

References

• Hadamard, J, 1923. Lectures on Cauchy's Problem in Linear Partial Differential Equations. New Haven, CT: Yale University Press; 1923.
   Google Scholar
• Han, H, Ingham, DB, and Yuan, Y, 1995. The boundary element method for the solution of the backward heat conduction equation, J. Comput. Phys. 116 (1995), pp. 292–299.
   Web of Science ®Google Scholar
• Lesnic, D, Elliott, L, and Ingham, D, 1998. An iterative boundary element method for solving the backward heat conduction problem using an elliptic approximation, Inverse Probl. Eng. 13 (1998), pp. 255–279.
   Google Scholar
• Mera, N, Elliott, L, Ingham, D, and Lesnic, D, 2001. An iterative boundary element method for solving the one dimensional backward heat conduction problem, Int. J. Heat Mass Transf. 44 (2001), pp. 1937–1946.
   Web of Science ®Google Scholar
• Mera, N, Elliott, L, and Ingham, D, 2002. An inversion method with decreasing regularization for the backward heat conduction problem, Numer. Heat Transfer B 42 (2002), pp. 215–230.
   Web of Science ®Google Scholar
• Jourhmane, M, and Mera, N, 2002. An iterative algorithm for the backward heat conduction problem based on variable relaxation factors, Inverse Probl. Eng. 10 (2002), pp. 293–308.
   Web of Science ®Google Scholar
• Kirkup, S, and Wadsworth, M, 2002. Solution of inverse diffusion problems by operator-splitting methods, Appl. Math. Model. 26 (2002), pp. 1003–1018.
   Web of Science ®Google Scholar
• Liu, J, 2002. Numerical solution of forward and backward problem for 2-D heat conduction equation, J. Comput. Appl. Math. 145 (2002), pp. 459–482.
   Web of Science ®Google Scholar
• Iijima, K, 2004. Numerical solution of backward heat conduction problems by a high-order lattice-free finite difference method, J. Chin. Inst. Eng. 27 (2004), pp. 611–620.
   Web of Science ®Google Scholar
• Liu, CS, 2004. Group preserving scheme for backward heat conduction problems, Int. J. Heat Mass Transf. 47 (2004), pp. 2567–2576.
   Web of Science ®Google Scholar
• Mera, NS, 2005. The method of fundamental solutions for the backward heat conduction problem, Inverse Probl. Sci. Eng. 13 (2005), pp. 65–78.
   Web of Science ®Google Scholar
• Liu, CS, Chang, CW, and Chang, JR, 2006. Past cone dynamics and backward group preserving schemes for backward heat conduction problems, CMES Comput. Model. Eng. Sci. 12 (2006), pp. 67–81.
   Web of Science ®Google Scholar
• Chiwiacowsky, LD, and de Campos Velho, HF, 2003. Different approaches for the solution of a backward heat conduction problem, Inverse Probl. Eng. 11 (2003), pp. 471–494.
   Google Scholar
• Yi, Z, and Murio, DA, 2004. Source term identification in 1-D IHCP, Comput. Math. Appl. 47 (2004), pp. 1921–1933.
   Web of Science ®Google Scholar
• Yi, Z, and Murio, DA, 2004. Identification of source terms in 2-D IHCP, Comput. Math. Appl. 47 (2004), pp. 1517–1533.
   Web of Science ®Google Scholar
• Yamamoto, M, and Zou, J, 2001. Simultaneous reconstruction of the initial temperature and heat radiative coefficient, Inverse Probl. 17 (2001), pp. 1181–1202, Special issue to celebrate Pierre Sabatier's 65th birthday (Montpellier, 2000).
   Web of Science ®Google Scholar
• Li, J, Yamamoto, M, and Zou, J, 2009. Conditional stability and numerical reconstruction of initial temperature, Commun. Pure Appl. Anal. 8 (2009), pp. 361–382.
   Web of Science ®Google Scholar
• Powell, MJD, 1987. "Radial basis functions for multivariable interpolation: A review". In: Mason, JC, and Cox, MG, eds. Algorithms for Approximation (Shrivenham, 1985). New York: Oxford University Press; 1987. pp. 143–167.
   Google Scholar
• Wu, ZM, 1992. Hermite-Birkhoff interpolation of scattered data by radial basis functions, Approx. Theory Appl. 8 (1992), pp. 1–10.
   Google Scholar
• Golberg, MA, and Chen, CS, 1999. "The method of fundamental solutions for potential, Helmholtz and diffusion problems". In: Golberg, MA, ed. Boundary Integral Methods: Numerical and Mathematical Aspects. Boston, MA: WIT Press/Computational Mechanics Publications; 1999. pp. 103–176.
   Google Scholar
• Fairweather, G, and Karageorghis, A, 1998. The method of fundamental solutions for elliptic boundary value problems, Adv. Comput. Math. 9 (1998), pp. 69–95.
   Web of Science ®Google Scholar
• Chen, W, and Hon, YC, 2003. Numerical investigation on convergence of boundary knot method in the analysis of homogeneous Helmholtz, modified Helmholtz, and convection-diffusion problems, Comput. Methods Appl. Mech. Eng. 192 (2003), pp. 1859–1875.
   Web of Science ®Google Scholar
• Chen, W, and Tanaka, M, 2002. A meshless, integration-free, and boundary-only RBF technique, Comput. Math. Appl. 43 (2002), pp. 379–391.
   Web of Science ®Google Scholar
• Kupradze, VD, and Aleksidze, MA, 1964. The method of functional equations for the approximate solution of certain boundary-value problems, Ž. Vyčisl. Mat. Mat. Fiz. 4 (1964), pp. 683–715.
   Google Scholar
• Mathon, R, and Johnston, RL, 1977. The approximate solution of elliptic boundary-value problems by fundamental solutions, SIAM J. Numer. Anal. 14 (1977), pp. 638–650.
   Web of Science ®Google Scholar
• Kress, R, and Mohsen, A, 1986. On the simulation source technique for exterior problems in acoustics, Math. Methods Appl. Sci. 8 (1986), pp. 585–597.
   Google Scholar
• Karageorghis, A, and Fairweather, G, 1987. The method of fundamental solutions for the numerical solution of the biharmonic equation, J. Comput. Phys. 69 (1987), pp. 434–459.
   Web of Science ®Google Scholar
• Chen, CS, 1995. The method of fundamental solutions for nonlinear thermal explosions, Commun. Numer. Methods Eng. 11 (1995), pp. 675–681.
   Web of Science ®Google Scholar
• Hon, YC, and Wei, T, 2004. A fundamental solution method for inverse heat conduction problem, Eng. Anal. Boundary Elem. 28 (2004), pp. 489–495.
   Web of Science ®Google Scholar
• Ling, L, Hon, YC, and Yamamoto, M, 2005. Inverse source identification for Poisson equation, Inverse Probl. Sci. Eng. 13 (2005), pp. 433–447.
   Web of Science ®Google Scholar
• Ling, L, Yamamoto, M, Hon, YC, and Takeuchi, T, 2006. Identification of source locations in two-dimensional heat equations, Inverse Probl. 22 (2006), pp. 1289–1305.
   Web of Science ®Google Scholar
• Yan, L, Fu, CL, and Yang, FL, 2008. A fundamental solution method for inverse heat conduction problem, Eng. Anal. Boundary Elem. 28 (2008), pp. 489–495.
   Google Scholar
• Yan, L, Yang, FL, and Fu, CL, 2009. A meshless method for solving an inverse spacewise-dependent heat source problem, J. Comput. Phys. 228 (2009), pp. 123–136.
   Web of Science ®Google Scholar
• Shidfar, A, Darooghehgimofrad, Z, and Garshasbi, M, 2009. Note on using radial basis functions and Tikhonov regularization method to solve an inverse heat conduction problem, Eng. Anal. Boundary Elem. 33 (2009), pp. 1236–1238.
   Web of Science ®Google Scholar
• Tikhonov, AN, and Arsenin, VY, Solutions of Ill-posed Problems, Scripta Series in Mathematics. Washington, DC: V. H. Winston & Sons, John Wiley & Sons, New York, 1977 (Translated from the Russian, Preface by translation editor F. John.).
   Google Scholar
• Isakov, V, 1990. "Inverse Source Problems". In: Mathematical Surveys and Monographs. Vol. 34. Providence, RI: American Mathematical Society; 1990.
   Google Scholar
• Hon, YC, and Wei, T, 2005. The method of fundamental solution for solving multidimensional inverse heat conduction problems, CMES Comput. Model. Eng. Sci. 7 (2005), pp. 119–132.
   Web of Science ®Google Scholar
• Hansen, PC, 1998. Rank-deficient and Discrete Ill-posed Problems: Numerical Aspects of Linear Inversion. Philadelphia, PA: SIAM Monographs on Mathematical Modeling and Computation, Society for Industrial and Applied Mathematics (SIAM); 1998.
   Google Scholar
• Wahba, G, and Wendelberger, J, 1980. Some new mathematical methods for variational objective analysis using splines and cross validation, Monthly Weather Rev. 108 (1980), pp. 1122–1143.
   Web of Science ®Google Scholar
• Hansen, PC, 1994. Regularization tools: A Matlab package for analysis and solution of discrete ill-posed problems, Numer. Algorithms 6 (1994), pp. 1–35.
   Google Scholar