413
Views
1
CrossRef citations to date
0
Altmetric
Articles

Collocation method based on shifted Chebyshev and radial basis functions with symmetric variable shape parameter for solving the parabolic inverse problem

&
Pages 369-387 | Received 31 Mar 2017, Accepted 10 Mar 2018, Published online: 16 May 2018

References

  • Dehghan M . Finding a control parameter in one-dimensional parabolic equations. Appl Math Comput. 2003;135:491–503.
  • Dehghan M . Parameter determination in a partial differential equation from the overspecified data. Math Comput Model. 2005;41:197–213.
  • Dehghan M . An inverse problem of finding a source parameter in a semilinear parabolic equation. Appl Math Model. 2002;25:743–754.
  • Mohebbi A , Abbasi M . A fourth-order compact difference scheme for the parabolic inverse problem with an overspecification at a point. Inverse Prob Sci Eng. 2015;23(3):457–478.
  • Cheng R . Determination of a control parameter in a one-dimensional parabolic equation using the moving least-square approximation. Int J Comput Math. 2008;85:1363–1373.
  • Deng ZC , Yu JN , Yang L . Optimization method for an evolutional type inverse heat conduction problem. J Phys A: Math Theor. 2008;41:035201(20pp)
  • Ma L , Wu Z . Radial basis functions method for parabolic inverse problem. Int J Comput Math. 2011;88(2):384–395.
  • Mohebbi A , Dehghan M . High-order scheme for determination of a control parameter in an inverse problem from the over-specified data. Comput Phys Commun. 2010;181:1947–1954.
  • Yousefi SA , Dehghan M . Legendre multiscaling functions for solving the onedimensional parabolic inverse problem. Numer Meth Part D E. 2009;25:1502–1510.
  • Cannon JR , Lin Y . Determination of parameter p(t) in Holder classes for some semilinear parabolic equations. Inverse Prob. 1988;4:595–606.
  • Cannon JR , Lin Y . Determination ofparameter p(t) in some quasi-linear parabolic differential equations. Inverse Prob. 1988;4:35–45.
  • Cannon JR , Lin Y , Wang S . Determination of source parameter in parabolic equations. Meccanica. 1992;27:85–94.
  • Dehghan M , Tatari M . Determination of a control parameter in a one-dimensional parabolic equation using the method of radial basis functions. Math Comput Model. 2006;44:1160–1168.
  • Yang L , Yu JN , Deng ZC . An inverse problem of identifying the coefficient of parabolic equation. Appl Math Model. 2008;32:1984–1995.
  • Nayroles B , Touzot G , Villon P . Generalizing the finite element method: diffuse approximation and diffuse elements. Comput Mech. 1992;10(5):307–318.
  • Belytschko T , Lu YY , Gu L . Element-free Galerkin methods. Int J Numer Meth Eng. 1994;37(2):229–256.
  • Atluri SN , Zhu T . New meshless local Petrov-Galerkin (MLPG) approach in computational mechanics. Comput Mech. 1998;22(2):117–127.
  • Hardy RL . Multiquadric equations of topography and other irregular surfaces. J Geophys Res. 1971;76:1905–1915.
  • Kansa EJ . Multiquadricsa scattered data approximation scheme with applications to computational fluid dynamics-I. Comput Math Appl. 1990;19:127–145.
  • Kansa EJ . Multiquadricsa scattered data approximation scheme with applications to computational fluid dynamics-II. Comput Math Appl. 1990;19:147–161.
  • Franke R . Scattered data interpolation: tests of some methods. Math Comp. 1982;48:181–200.
  • Foley TA . Near optimal parameter selection for multiquadric interpolation. J Appl Sci Comput. 1994;1:54–69.
  • Rippa S . An algorithm for selecting a good value for the parameter c in radial basis function interpolation. Adv Comput Math. 1999;11:193–210.
  • Fasshauer GE , Zhang JG . On choosing optimal shape parameters for RBF approximation. Numer Algorithms. 2007;45:345–368.
  • Flyer N , Lehto E . Rotational transport on a sphere: Local node refinement with radial basis functions. J Comput Phys. 2010;229(6):1954–1969.
  • Esmaeilbeigi M , Hosseini MM . A new approach based on the genetic algorithm for finding a good shape parameter in solving partial differential equations by Kansa’s method. Appl Math Comput. 2014;249:419–428.
  • Iurlaro L , Gherlone M . Sciuva MDi. Energy based approach for shape parameter selection in radial basis functions collocation method. Compos Struct. 2014;107:70–78.
  • Carlson RE , Foley TA . The parameter R2 in multiquadric interpolation. Comput Math Appl. 1991;21:29–42.
  • Hon YC , Mao X . An efficient numerical scheme for Burger’s equation. Appl Math Comput. 1998;95:37–50.
  • Kansa EJ , Hon YC . Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to elliptic partial differential equations. Comput Math Appl. 2000;39:123–137.
  • Sarra SA , Sturgill D . A random variable shape parameter strategy for radial basis function approximation methods. Eng Anal Bound Elem. 2009;33:1239–1245.
  • Xiang S , Wang KM , Ai YT , Sha YD , Shi H . Trigonometric variable shape parameter and exponent strategy for generalized multiquadric radial basis function approximation. Appl Math Model. 2012;36:1931–1938.
  • Golbabai A , Rabiei H . Hybrid shape parameter strategy for the RBF approximation of vibrating systems. Int J Comput Math. 2012;89(17):2410–2427.
  • Ranjbar M . A new variable shape parameter strategy for Gaussian radial basis function approximation methods. Ann U Craiova, Math Comput Sci Ser. 2015;42(2):260–272.
  • Micchelli CA . Interpolation of scattered data-distance matrices and conditionally positive definite functions. Constr Approx. 1986;2:11–22.
  • Carlson RE , Foley TA . Interpolation of track data with radial basis functions. Comput Math Appl. 1992;24:27–34.
  • Schaback R . Error estimates and condition numbers for radial basis function interpolation. Adv in Comput Math. 1995;3:251–264.
  • Dehghan M , Shakeri F . Method of lines solutions of the parabolic inverse problem with an overspecification at a point. Numer Algorithm. 2009;50:417–437.

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.