399
Views
2
CrossRef citations to date
0
Altmetric
Original Articles

A class of multistep numerical difference schemes applied in inverse heat conduction problem with a control parameter

&
Pages 887-942 | Received 06 Jun 2017, Accepted 12 Jun 2018, Published online: 01 Aug 2018

References

  • Hon YC, Wei T. A fundamental solution method for inverse heat conduction problem. Eng Anal Bound Elem. 2004;28:489–495. doi: 10.1016/S0955-7997(03)00102-4
  • Hansen PC. Regularization tools: a Matlab package for analysis and solution of discrete ill-posed problems. Numer Algorithms. 1994;6:1–35. doi: 10.1007/BF02149761
  • Cheng J, Yamamoto M. One new strategy for a priori choice of regularizing parameters in Tikhonov's regularization. Inverse Probl. 2000;16:31–38. doi: 10.1088/0266-5611/16/4/101
  • Tikhonov AN, Arsenin VY. Solutions of ill-posed problems. Washington (DC), New York: V.H. Winston & Sons, John Wiley & Sons; 1977.
  • Dehghan M. Numerical solution of one-dimensional parabolic inverse problem. Appl Math Comput. 2003;136:333–344.
  • Dehghan M. Finding a control parameter in one-dimensional parabolic equations. Appl Math Comput. 2003;135:491–503.
  • Dehghan M. An inverse problem of finding a source parameter in a semilinear parabolic equation. Appl Math Model. 2001;25:743–754. doi: 10.1016/S0307-904X(01)00010-5
  • Cannon JR, Lin YP, Xu SZ. Numerical procedures for the determination of an unknown coefficient in semi-linear parabolic differential equations. Inverse Probl. 1994;10:227–243. doi: 10.1088/0266-5611/10/2/004
  • Dehghan M. Parameter determination in a partial differential equation from the overspecified data. Math Comput Model. 2005;41:196–213. doi: 10.1016/j.mcm.2004.07.010
  • Mitchell AR, Griffiths DF. The finite difference method in partial differential equations. New York: Wiley; 1980.
  • Tatari M, Dehghan M. A method for solving partial differential equations via radial basis functions: application to the heat equation. Eng Anal Bound Elem. 2010;34:206–212. doi: 10.1016/j.enganabound.2009.09.003
  • Chen W, Tanaka M. A meshless, integration-free, and boundary-only RBF technique. Comput Math Appl. 2002;43:379–391. doi: 10.1016/S0898-1221(01)00293-0
  • Dehghan M, Tatari M. Determination of a control parameter in a one-dimensional parabolic equation using the method of radial basis function. Math Comput Model. 2006;44:1160–1168. doi: 10.1016/j.mcm.2006.04.003
  • Ma LM, Wu ZM. Radial basis functions method for parabolic inverse problem. Int J Comput Math. 2011;88(2):384–395. doi: 10.1080/00207160903452236
  • Arghand M, Amirfakhrian M. A meshless method based on the fundamental solution and radial basis function for solving an inverse heat conduction. Adv Math Phys. 2015;2015:1–8. Article ID 256726. doi: 10.1155/2015/256726
  • Hon YC, Wei T. The method of fundamental solution for solving multidimensional inverse heat conduction problems. Comput Model Eng Sci. 2005;7(2):119–132.
  • Dong CF, Sun FY, Meng BQ. A method of fundamental solutions for inverse heat conduction problems in an anisotropic medium. Eng Anal Bound Elem. 2007;31:75–82. doi: 10.1016/j.enganabound.2006.04.007
  • Shen SY. A numerical study of inverse heat conduction problems. Comput Math Appl. 1999;38:173–188. doi: 10.1016/S0898-1221(99)00248-5
  • Johansson T, Lesnic D. Determination of a spacewise dependent heat source. J Comput Appl Math. 2007;209:66–80. doi: 10.1016/j.cam.2006.10.026
  • Dehghan M. Implicit solution of a two-dimensional parabolic inverse problem with temperature overspecification. J Comput Anal Appl. 2001;3(4):383–398.
  • Dehghan M. Finite difference schemes for two-dimensional parabolic inverse problem with temperature overspecification. Int J Comput Math. 2000;75:339–349. doi: 10.1080/00207160008804989
  • Chantasiriwan S. An algorithm for solving multidimensional inverse heat conduction problem. Int J Heat Mass Transf. 2001;44:3823–3832. doi: 10.1016/S0017-9310(01)00037-0
  • Dehghan M. Determination of a control function in three-dimensional parabolic equations. Math Comput Simul. 2003;61:89–100. doi: 10.1016/S0378-4754(01)00434-7
  • Dehghan M, Abbaszadeh M, Mohebbi A. The numerical solution of nonlinear high dimensional generalized Benjamin–Bona–Mahony–Burgers equation via the meshless method of radial basis function. Comput Math Appl. 2014;68:212–237. doi: 10.1016/j.camwa.2014.05.019
  • Dehghan M, Mohammadi V. The numerical solution of Cahn–Hilliard (CH) equation in one, two and three-dimensions via globally radial basis functions (GRBFs) and RBFs-differential quadrature (RBFs-DQ) methods. Eng Anal Bound Elem. 2015;51:74–100. doi: 10.1016/j.enganabound.2014.10.008
  • Lakestani M, Dehghan M. The use of Chebyshev cardinal functions for the solution of a partial differential equation with an unknown time-dependent coefficient subject to an extra measurement. J Comput Appl Math. 2010;235:669–678. doi: 10.1016/j.cam.2010.06.020
  • Mohebbi A, Dehghan M. High-order scheme for determination of a control parameter in an inverse problem from the over-specified data. Comput Phys Comm. 2010;181:1947–1954. doi: 10.1016/j.cpc.2010.09.009
  • Dehghan M. Fourth-order techniques for identifying a control parameter in the parabolic equations. Int J Eng Sci. 2002;40(4):433–447. doi: 10.1016/S0020-7225(01)00066-0
  • Dehghan M, Shakeri F. Method of lines solutions of the parabolic inverse problem with an overspecification at a point. Numer Algorithms. 2009;50(4):417–437. doi: 10.1007/s11075-008-9234-3
  • Shamsi M, Dehghan M. Determination of a control function in three-dimensional parabolic equations by Legendre pseudospectral method. Numer Methods Partial Differ Equ. 2012;28(1):74–93. doi: 10.1002/num.20608
  • Saadatmandia A, Dehghan M. Computation of two time-dependent coefficients in a parabolic partial differential equation subject to additional specifications. Int J Comput Math. 2010;87(5):997–1008. doi: 10.1080/00207160802253958
  • Cannon JR, Duchateau P. An inverse problem for an unknown source in heat equation. J Math Anal Appl. 1980;75:465–485. doi: 10.1016/0022-247X(80)90095-5
  • Cannon JR, Lin Y, Wng S. Determination of source parameter in parabolic equation. Meccanica. 1992;27:85–94. doi: 10.1007/BF00420586
  • Macbain JA, Bendar JB. Existence and uniqueness properties for the one-dimensional magnetotellurics inversion problem. J Math Phys. 1986;27:645–649. doi: 10.1063/1.527219
  • Macbain JA. Inversion theory for a parametrized diffusion problem. SIAM J Appl Math. 1987;47:1386–1391. doi: 10.1137/0147091
  • Cannon JR, Lin Y. An inverse problem of finding a parameter in a semi-linear heat equation. J Math Anal Appl. 1990;145:470–484. doi: 10.1016/0022-247X(90)90414-B
  • Cannon JR, Lin YP. Determination of a parameter p(t) in some quasi-linear parabolic differential equations. Inverse Probl. 1988;4:35–45. doi: 10.1088/0266-5611/4/1/006
  • Cannon JR, Lin YP. Determination of parameter p(t) in Ho¨lder classes for some semilinear parabolic equations. Inverse Probl. 1988;4:595–606. doi: 10.1088/0266-5611/4/3/005
  • Li CJ. A kind of multistep finite difference methods for arbitrary order linear boundary value problem. Appl Math Comput. 2008;196:858–865.
  • Hanke M, Scherzer O. Inverse problems light: numerical differentiation. Am Math Mon. 2001;108(6):512–521. doi: 10.1080/00029890.2001.11919778
  • Wei T, Li M. High order numerical derivatives for one-dimensional scattered noisy data. Appl Math Comput. 2006;175:1744–1759.
  • Zhang YF, Li CJ. A Gaussian RBFs method with regularization for the numerical solution of inverse heat conduction problems. Inverse Probl Sci Eng. 2016;24(9):1606–1646. doi: 10.1080/17415977.2015.1131825
  • Engl HW, Hanke M, Neubauer A. Regularization of inverse problems. Dordrecht/Boston/London: Kluwer Academic Publishers; 1996.
  • Wang YB, Jia XZ, Cheng. J. A numerical differentiation method and its application to reconstruction of discontinuity. Inverse Probl. 2002;18:1461–1476. doi: 10.1088/0266-5611/18/6/301
  • Hào DN, Chuong LH, Lesnic. D. Heuristic regularization methods for numerical differentiation. Comput Math Appl. 2012;63:816–826. doi: 10.1016/j.camwa.2011.11.047
  • Süli E, Mayers D. An introduction to numerical analysis. Cambridge (UK): Cambridge University Press; 2003.
  • Adams RA. Sobolev spaces, pure and applied mathematics. Vol. 65. New York/London: Academic Press; 1975.
  • de Boor C. Spline toolbox for use with matlab. Natick: The MathWorks, Inc.; 1992.

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.