543
Views
3
CrossRef citations to date
0
Altmetric
Articles

A meshless method for solving 1D time-dependent heat source problem

&
Pages 51-82 | Received 23 Jan 2016, Accepted 16 Mar 2017, Published online: 26 Apr 2017

References

  • Yang F, Fu CL. The method of simplified Tikhonov regularization for dealing with the inverse time-dependent heat source problem. Comput Math Appl. 2010;60(5):1228–1236.
  • Liu J, Wang B, Liu Zh. Determination of a source term in a heat equation. Int J Comput Math. 2010;87(5):969–975.
  • Savateev EG. On problems of determining the source function in a parabolic equation. J Inverse Ill-Posed Prob. 1995;3(1):83–102.
  • Solov’ev VV. Solvability of the inverse problems of finding a source, using overdetermination on the upper base for a parabolic equation. Differ Equ. 1990;25:1114–1119.
  • Cannon JR, Zachman D. Parameter determination in parabolic partial differential equations from over-specified boundary data. Int J Eng Sci. 1982;20:779–788.
  • Prilepko AI, Solov’ev VV. Solvability theorems and Rote’s method for inverse problems for a parabolic equation. I Differ Equ. 1988;23:1230–1237.
  • Rundell W. The determination of an unknown non-homogeneous term in linear partial differential equations from overspecifie data. Appl Anal. 1980;10:231–242.
  • Cannon JR. Determination of an unknown heat source from overspecified data. SIAM J Numer Anal. 1968;5(2):275–286.
  • Cannon JR, Duchateau P. Structural identification of an unknown source term in heat equation. Inverse Prob. 1998;14(3):535–551.
  • Ling L, Yamamoto M, Hon YC, et al. Identification of source locations in two-dimentional heat equations. Inverse Prob. 2006;22(4):1289–1305.
  • Alves CJS, Calaco MJ, Leitao VMA, et al. Recovery the source term in a linear diffusion problem by the method of fundamental solutions. Inverse Prob Sci Eng. 2008;16(8):1005–1021.
  • Nili Ahmadabadi M, Arab M, Maalek Ghaini FM. The method of fundamental solutions for the inverse space-dependent heat source problem. Eng Anal Boundary Elem. 2009;33:1231–1235.
  • Yan L, Yang F-L, Fu C-L. A meshless method for solving an inverse spacewise-dependent heat source problem. J Comput Phys. 2009;228:123–136.
  • Farcas A, Lesnic D. The boundary-element method for the determination of a heat source dependent on one variable. J Eng Math. 2006;54:375–388.
  • Johansson BT, Lesnic D. Determination of a spacewise dependent heat source. J Comput Appl Math. 2007;209:66–80.
  • Johansson BT, Lesnic D. A variational method for identifying a spacewise-dependent heat source. IMA J Appl Math. 2007;72(6):748–760.
  • Johansson BT, Lesnic D. A procedure for determining a spacewise dependent heat source and the initial temperature. Appl Anal. 2008;87:265–276.
  • Yan L. C. L. Fu., F. F. Dou, A computational method for identifying a spacewise-dependent heat source, Int J Numer Methods. Biomed Eng. 2013;26(10):939–957.
  • Yang L, Yu J-N, Luo G-W, et al. Numerical identification of source terms for a two dimensional heat conduction problem in polar coordinate system. Appl Math Model. 2010;37(3):597–608.
  • Geng FZ, Lin YZ. Application of the variational iteration method to inverse heat source problems. Comput Math Appl. 2009;58(11–12):2098–2102.
  • Yang L, Dehghan M, Yu JN, et al. Inverse problem of time-dependent heat sources numerical reconstruction. Math Comput Simul. 2011;81(8):1656–1672.
  • Liu J, Wang B, Liu ZH. Determination of a source term in a heat equation. Int J Comput Math. 2010;87(5):969–975.
  • Yan L, Fu CL, Yang FL. The method of fundamental solutions for the inverse heat source problem. Eng Anal Boundary Elem. 2008;32(3):216–222.
  • Shahrezaee AM. A meshless Method for solving an inverse Time-dependent Heat Source Problem. J Sci Kharazmi Univer. 2013;13(2):363–372.
  • Liu CH. A two-stage LGSM to identify time-dependent heat source through an internal measurement of temperature. Int J Heat Mass Transfer. 2009;52:1635–1642.
  • Liu C-S. A Lie-group shooting method for reconstructing a past time-dependent heat source. Int J Heat Mass Transfer. 2012;55(5–6):1773–1781.
  • Hazanee A, Ismailov MI, Lesnic D, et al. An inverse time-dependent source problem for the heat equation. Appl Numer Math. 2013;69:13–33.
  • Yang F, Guo HZ, Li XX. The method of central difference for the inverse time-dependent heat source problem. Appl Math Comput. 2011;218(7):3025–3034.
  • Ma Y-J, Fu C-L, Zhang Y-X. Identification of an unknown source depending on both time and space variables by a variational method. Appl Math Model. 2013;36(10):5080–5090.
  • Mierzwiczak M, Kolodziej JA. Application of the method of fundamental solutions and radial basis functions for inverse transient heat source problem. Comput Phys Commun. 2010;181:2035–2043.
  • Kolodziej JA, Mierzwiczak M, Cialkowski M. Application of the method of fundamental solutions and radial basis functions for inverse heat source problem in case of steady-state. Int Commun Heat Mass Transfer. 2010;37(2):121–124.
  • Yang CY. A sequential method to estimate the strength of the heat source based on symbolic computation. Int J Heat Mass Transfer. 1998;41(14):2245–2252.
  • Yang CY. Solving the two-dimensional inverse heat source problem through the linear least-squares error method. Int J Heat Mass Transfer. 1998;41(2):393–398.
  • Akpofure E. Taigbenu. Inverse solutions of temperature, heat flux and heat source by the Green element method, Appl Math Model. 2015;39(2):667–681.
  • Liu C-S. Finding unknown heat source in a nonlinear Cauchy problem by the Lie-group differential algebraic equations method. Eng Anal Boundary Elem. 2015;50:148–156.
  • Shi C, Wang C, Wei T. Numerical solution for an inverse heat source problem by an iterative method. Appl Math Comput. 2014;244:577–597.
  • Yang F, Fu C-L. A mollification regularization method for the inverse spatial-dependent heat source problem. J Comput Appl Math. 2014;255:555–567.
  • Hào DN, Thanh PX, Lesnic D, et al. Determination of a source in the heat equation from integral observations. J Comput Appl Math. 2014;264:82–98.
  • Marin L. A meshless method for the stable solution of singular inverse problems for two-dimensional Helmholtz-type equations. Eng Anal Boundary Elem. 2010;34:274–288.
  • Kupradze VD. Potential methods in elasticity. In: Nauka M, editor. Prilozhenia teorii funktsii v mekhanike sploshnoi sredy. Trudy Mezhdunarodn. Simposiuma, Tbilisi, 1963. Solid mechanics. Vol. 1, 1965. p. 211–216
  • Mathon R, Johnston RL. The approximate solution of elliptic boundaryvalue problems by fundamental solutions. SIAM J Numer Anal. 1977;14:638–650.
  • Bogomolny A. Fundamental solutions method for elliptic boundary value problems. SIAM J Numer Anal. 1985;22:644–69.
  • Fairweather G, Karageorghis A. The method of fundamental solutions for elliptic boundary value problems. Adv Comput Math. 1998;9:69–95.
  • Golberg MA, Chen CS. The method of fundamental solutions for potential, Helmholtz and diffusion problems. In: Golberg MA, editor. Boundary integral methods -- Numerical and Mathematical Aspects. Computational Mechanics Publications; 1998. p. 103–176.
  • Chen CS, Hon YC, Schaback RA. Scientific computing with radial basis functions, preprint. Department of Mathematics: University of Southern Mississippi, Hattiesburg (MS), USA; 2005.
  • Ushijima T, Chiba F. A fundamental solution method for the reduced wave problem in a domain exterior to a disc. J Comput Appl Math. 2003;152:545–557.
  • Shigeta T, Young DL. Method of fundamental solutions with optimal regularization techniques for Cauchy problem of the Laplace equation with singular points. J Comput Phys. 2009;228:1903–1915.
  • Airweather G, Karageorghis A. The method of fundamental solutions for numerical solutions of the biharmonic equation. J Comput Phys. 1987;69:434–459.
  • Balakrishnan K, Ramachandran PA. A particular solution for non-linear Poisson problems in heat and mass transfer. J Comput Phys. 1999;150:239–267.
  • Schaback R. Adaptive numerical solution of MFS systems. A Plenary Talk at the First International Workshop on the Method of Fundamental Solutions (MFS2007), Ayia Napa, Cyprus, June 11–13. 2007.
  • Hon YC, Wei T. A fundamental solution method for inverse heat conduction problem. Eng Anal Boundary Elem. 2004;28:489–495.
  • Marin L, Lesnic D. The method of fundamental solutions for the Cauchy problem in two-dimensional linear elasticity. Int J Solids Struct. 2004;41:3425–3438.
  • Marin L. A meshless method for solving the Cauchy problem in threedimensional elastostatics. Comput Math Appl. 2005;50:73–92.
  • Marin L. Numerical solutions of the Cauchy problem for steady-state heat transfer in two-dimensional functionally graded materials. Int J Solids Struct. 2005;42:4338–4351.
  • Marin L, Lesnic D. The method of fundamental solutions for the Cauchy problem associated with two-dimensional Helmholtz-type equations. Comput Struct. 2005;83:267–278.
  • Marin L. A meshless method for the numerical solution of the Cauchy problem associated with three-dimensional Helmholtz-type equations. Appl Math Comput. 2005;165:355–374.
  • Jin B, Zheng Y. A meshless method for some inverse problems associated with the Helmholtz equation. Comput Methods Appl Mech Eng. 2006;195:2270–2280.
  • Jin B, Marin L. The method of fundamental solutions for inverse source problems associated with the steady-state heat conduction. Int J Numer Methods Eng. 2007;69:1570–1589.
  • Young DL, Jane SJ, Fan CM, et al. The method of fundamental solutions for 2D and 3D Stokes problems. J Comput Phys. 2006;211:1–8.
  • Golberg MA. The method of fundamental solutions for Poissons equations. Eng Anal Boundary Elem. 1995;16:205–213.
  • Golberg MA, Muleshkov AS, Chen CS, et al. Polynomial particular solutions for certain partial differential operators. Numer Methods Partial Differ Equ. 2002;19(1):112–133.
  • Young DL, Tsai CC, Murugesan K, et al. Time-dependent fundamental solutions for homogeneous diffusion problems. Eng Anal Boundary Elem. 2004;29:1463–1473.
  • Gu Y, Chen W, Zhang C, et al. A meshless singular boundary method for three-dimensional inverse heat conduction problems in general anisotropic media. Int J Heat Mass Transfer. 2015;84:91–102.
  • Gu Y, Gao H, Chen W, et al. Fast-multipole accelerated singular boundary method for large-scale three-dimensional three-dimensional potential problems. Int J Heat Mass Transfer. 2015;90:291–301.
  • Cannon JR. The one-dimensional heat equation. Reading (MA): Addison Wesley; 1984.
  • Marin L. A meshless method for the stable solution of singular inverse problems for two-dimensional Helmholtz-type equations. Eng Anal Boundary Elem. 2010;34:274–288.
  • Tikhonov AN, Arsenin VY. Methods for slving ill-posed problems. Moscow: Nauka; 1986.
  • Tautenhahn U, Hämarik U. The use of monotonicity for choosing the regularization parameter in ill-posed problems. Inverse Prob. 1999;15:1487–1505.
  • Morozov VA. Methods for solving incorrectly posed problems. New York (NY): Springer; 1984.
  • Hansen PC. Rank-deficient and discrete ill-posed problems. Philadelphia: SIAM; 1998.
  • Golub G, Heath M, Wahba G. Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics. 1979;21(2):215–223.
  • Iserles A. A first course in the numerical analysis of differential equations. Cambridge University Press; 1996.
  • Kreyszig E. Introductory Functional Analysis with Applications. New York (NY): Wiley; 1978.
  • Canuto C, Hussaini MY, Quarteroni A, et al. Spectral methods: fundamentals in single domains. Berlin: Springer; 2006.
  • Mashayekhi S, Ordokhani Y, Razzaghi M. A hybrid functions approach for the Duffing equation. Phys Scr. 2013;88(2):025002.
  • Johansson BT, Lesnic D, Reeve T. A meshless method for an inverse two phase one-dimensional linear Stefan problem. Inverse Prob Sci Eng. 2013;21:17–33.
  • Wei T, Wang JC. Simultaneous determination for a space-dependent heat source and the initial data by the MFS. Eng Anal Boundary Elem. 2012;36:1848–1855.
  • Wei T, Hon YC, Ling L. Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators. Eng Anal Boundary Elem. 2007;31(4):373–385.
  • Dong CF, Sun FY, Meng BQ. A method of fundamental solutions for inverse heat conduction problems in an anisotropic medium. Eng Anal Boundary Elem. 2007;31:75–82.
  • Reeve T, Tomas Johansson B. The method of fundamental solutions for a time-dependent two-dimensional Cauchy heat conduction problem. Eng Anal Boundary Elem. 2013;37:569–578.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.