BibTex RIS Kaynak Göster
Yıl 2017, Cilt: 7 Sayı: 1, 33 - 50, 01.06.2017

Öz

Kaynakça

  • Cannon,J.R. and Van de Hoek,J., (1982), The one phase stefan problem subject to energy, J. Math. Anal. Appl., 86, pp.281-292.
  • Cannon,J.R., Eteva,S.P., and Van de Hoek,J., (1987), A Galerkin procedure for the diffusion equation subject to the specification of mass, SIAM J. Numer. Anal., 24, pp.499-515.
  • Cannon,J.R., (1963), The solution of the heat equation subject to the specification of energy, Quart. Appl. Math., 21, pp.155-160.
  • Capasso,V. and Kunisch,K., (1988), A reaction-diffusion system arising in modeling manevironment diseases, Quart. Appl. Math., 46, pp.431-450.
  • Shidfar,A., Pourgholi,R., and Ebrahimi,M., (2006), A numerical method for solving of a nonlinear inverse diffusion problem, Comput. Math. Appl., 52, pp.1021-1030.
  • Dehghan,M., (2001), An inverse problem of finding a source parameter in a semilinear parabolic equation, Appl. Math. Model., 25, pp.743-754.
  • Dehghan,M., (2002), Numerical techniques for a parabolic equation subject to an overspecified bound- ary condition, Appl. Math. Comput. 132, pp.299-313.
  • Dehghan,M., (2003), Numerical solution of one-dimensional parabolic inverse problem, Appl. Math. Comput., 136, pp.333-344.
  • Tatari,M. and Dehghan,M., (2007), Identifying a control function in parabolic partial differential equations from overspecified boundary data, Computers and Mathematics with Applications, 53, pp.1933-1942.
  • Beck,J.V., Blackwell,B., and St.Clair,C.R., (1985), Inverse Heat Conduction: IllPosed Problems
  • Wiley-Interscience, NewYork. Beck,J.V. and Murio,D.C., (1986), Combined function specification-regularization procedure for solu- tion of inverse heat condition problem, AIAA J., 24, pp.180-185.
  • Cabeza,J.M.G., Garcia,J.A.M., and Rodriguez,A.C., (2005), A Sequential Algorithm of Inverse Heat
  • Conduction Problems Using Singular Value Decomposition, International Journal of Thermal Sciences, , pp.235-244. Molhem,H. and Pourgholi,R., (2008), A numerical algorithm for solving a one-dimensional inverse heat conduction problem, Journal of Mathematics and Statistics, 4(1), pp.60-63.
  • Pourgholi,R., Azizi,N., Gasimov,Y.S., Aliev,F., and Khalafi,H.K., (2009), Removal of Numerical Insta- bility in the Solution of an Inverse Heat Conduction Problem, Communications in Nonlinear Science and Numerical Simulation, 14(6), pp.2664-2669.
  • Pourgholi,R. and Rostamian,M., (2010), A numerical technique for solving IHCPs using Tikhonov regularization method, Applied Mathematical Modelling, 34(8), pp.2102-2110.
  • Pourgholi,R., Rostamian,M., and Emamjome,M.,(2010), A numerical method for solving a nonlinear inverse parabolic problem, Inverse Problems in Science and Engineering, 18(8), pp.1151-1164.
  • Tadi,M., (1997), Inverse Heat Conduction Based on Boundary Measurement, Inverse Problems, 13, pp.1585-1605.
  • Shidfar,A., Zolfaghari,R., and Damirchi,J., (2009), Application of Sinc-collocation method for solving an inverse problem,Journal of computational and Applied Mathematics, 233, pp.545-554.
  • Cant-Paz,E., (1995), A summary of research on parallel genetic algorithms, IlliGAL Report No 95007
  • University of Illinois. Pourgholi,R., Dana,H., and Tabasi,S.H., (2014), Solving an inverse heat conduction problem using ge- netic algorithm: Sequential and multi-core parallelization approach, Applied Mathematical Modelling, Volume 38, Issues 78.
  • Pourgholi,R., Molai,A.A., and Houlari,T., (2013), Resolution of an inverse parabolic problem using
  • Sinc-Galerkin method, TWMS J. App. Eng. Math.,V2(3), pp.160-181. Lund,J. and Bowers,K., (1991), Sinc Methods for Quadrature and Differential Equations, Siam, Philadelphia, PA.
  • Stenger,F., (1979), A Sinc-Galerkin method of solution of boundary-value problems, Math. Comp., , pp.85-109.
  • Whittaker,E.T., (1915), On the functions which are represented by the expansions of the interpolation theory, Proc. Roy. Soc. Edinburg, 35, pp.181-194.
  • Whittaker,J.M., (1935), Interpolation Function Theory, in: Cambridge Tracts in Mathematics and Mathematical Physics, Vol.33, Cambridge University Press, London.
  • Koonprasert,S. and Bowers,K., (2004), The Fully Sinc-Galerkin Method for Time-Dependent Bound- ary Conditions, Numerical Method for Partial Differential Equations, 20(4), pp.494-526.
  • Hansen,P.C., (1992), Analysis of discrete ill-posed problems by means of the L-curve, SIAM Rev., 34, pp.561-80.
  • Lawson,C.L. and Hanson,R.J., (1995), Solving Least Squares Problems, Philadelphia, PA:SIAM.
  • Tikhonov,A.N. and Arsenin,V.Y., (1977), On the solution of Ill-posed problems, New York, Wiley.
  • Tikhonov,A.N., and Arsenin,V.Y., (1977), Solution of Ill-Posed Problems, V.H.Winston and Sons, Washington, DC.
  • Martin,L., Elliott,L., Heggs,P.J., Ingham,D.B., Lesnic,D., and Wen,X., (2006), Dual Reciprocity
  • Boundary Element Method Solution of the Cauchy Problem for Helmholtz-type Equations with Vari- able Coefficients, Journal of sound and vibration, 297, pp.89-105. Elden,L., (1984), A Note on the Computation of the Generalized Cross-validation Function for Ill- conditioned Least Squares Problems, BIT, 24, pp.467-472
  • Golub,G.H., Heath,M., and Wahba,G., (1979), Generalized Cross-validation as a Method for Choosing a Good Ridge Parameter, Technometrics, 21(2), pp.215-223.
  • Wahba,G., (1990), Spline Models for Observational Data, CBMS-NSF Regional Conference Series in
  • Applied Mathematics, Vol.59, SIAM, Philadelphia. Holland,J.H., (1975), Adaptation in Natural and Artificial System, University of Michigan Press, Ann Arbor.
  • Liu,F-B., (2008), A modified genetic algorithm for solving the inverse heat transfer problem of esti- mating plan heat source, International Journal of Heat and Mass Transfer, 51, pp.3745-3752.
  • Reza Pourgholi, for the photograph and short biography, see TWMS J. Appl. and Eng. Math., V.2, No.2, 2012.

COMBINING GENETIC ALGORITHM AND SINC-GALERKIN METHOD FOR SOLVING AN INVERSE DIFFUSION PROBLEM

Yıl 2017, Cilt: 7 Sayı: 1, 33 - 50, 01.06.2017

Öz

A numerical approach combining the use of a genetic algorithm with the solution of the Sinc-Galerkin method is proposed for the determination of an unknown time-dependent diffusivity a t in an inverse diffusion problem IDP . At the beginning of the numerical algorithm, Sinc-Galerkin method is employed to solve the direct diffusion problem. The present approach is to rearrange the matrix forms of the governing equations. Then, the genetic algorithm is adopted to find the solution of IDP. The genetic algorithm used in this work is not a classical genetic algorithm. Instead, the application of the genetic algorithm to this discrete-time optimal control problem is called a real-valued genetic algorithm RVGA . Some numerical experiments con rm the utility of this algorithm as the results are in good agreement with the exact data. Results show that a reasonable estimation can be obtained by combining the genetic algorithm and Sinc-Galerkin method within a CPU with clock speed 2.7 GHz.

Kaynakça

  • Cannon,J.R. and Van de Hoek,J., (1982), The one phase stefan problem subject to energy, J. Math. Anal. Appl., 86, pp.281-292.
  • Cannon,J.R., Eteva,S.P., and Van de Hoek,J., (1987), A Galerkin procedure for the diffusion equation subject to the specification of mass, SIAM J. Numer. Anal., 24, pp.499-515.
  • Cannon,J.R., (1963), The solution of the heat equation subject to the specification of energy, Quart. Appl. Math., 21, pp.155-160.
  • Capasso,V. and Kunisch,K., (1988), A reaction-diffusion system arising in modeling manevironment diseases, Quart. Appl. Math., 46, pp.431-450.
  • Shidfar,A., Pourgholi,R., and Ebrahimi,M., (2006), A numerical method for solving of a nonlinear inverse diffusion problem, Comput. Math. Appl., 52, pp.1021-1030.
  • Dehghan,M., (2001), An inverse problem of finding a source parameter in a semilinear parabolic equation, Appl. Math. Model., 25, pp.743-754.
  • Dehghan,M., (2002), Numerical techniques for a parabolic equation subject to an overspecified bound- ary condition, Appl. Math. Comput. 132, pp.299-313.
  • Dehghan,M., (2003), Numerical solution of one-dimensional parabolic inverse problem, Appl. Math. Comput., 136, pp.333-344.
  • Tatari,M. and Dehghan,M., (2007), Identifying a control function in parabolic partial differential equations from overspecified boundary data, Computers and Mathematics with Applications, 53, pp.1933-1942.
  • Beck,J.V., Blackwell,B., and St.Clair,C.R., (1985), Inverse Heat Conduction: IllPosed Problems
  • Wiley-Interscience, NewYork. Beck,J.V. and Murio,D.C., (1986), Combined function specification-regularization procedure for solu- tion of inverse heat condition problem, AIAA J., 24, pp.180-185.
  • Cabeza,J.M.G., Garcia,J.A.M., and Rodriguez,A.C., (2005), A Sequential Algorithm of Inverse Heat
  • Conduction Problems Using Singular Value Decomposition, International Journal of Thermal Sciences, , pp.235-244. Molhem,H. and Pourgholi,R., (2008), A numerical algorithm for solving a one-dimensional inverse heat conduction problem, Journal of Mathematics and Statistics, 4(1), pp.60-63.
  • Pourgholi,R., Azizi,N., Gasimov,Y.S., Aliev,F., and Khalafi,H.K., (2009), Removal of Numerical Insta- bility in the Solution of an Inverse Heat Conduction Problem, Communications in Nonlinear Science and Numerical Simulation, 14(6), pp.2664-2669.
  • Pourgholi,R. and Rostamian,M., (2010), A numerical technique for solving IHCPs using Tikhonov regularization method, Applied Mathematical Modelling, 34(8), pp.2102-2110.
  • Pourgholi,R., Rostamian,M., and Emamjome,M.,(2010), A numerical method for solving a nonlinear inverse parabolic problem, Inverse Problems in Science and Engineering, 18(8), pp.1151-1164.
  • Tadi,M., (1997), Inverse Heat Conduction Based on Boundary Measurement, Inverse Problems, 13, pp.1585-1605.
  • Shidfar,A., Zolfaghari,R., and Damirchi,J., (2009), Application of Sinc-collocation method for solving an inverse problem,Journal of computational and Applied Mathematics, 233, pp.545-554.
  • Cant-Paz,E., (1995), A summary of research on parallel genetic algorithms, IlliGAL Report No 95007
  • University of Illinois. Pourgholi,R., Dana,H., and Tabasi,S.H., (2014), Solving an inverse heat conduction problem using ge- netic algorithm: Sequential and multi-core parallelization approach, Applied Mathematical Modelling, Volume 38, Issues 78.
  • Pourgholi,R., Molai,A.A., and Houlari,T., (2013), Resolution of an inverse parabolic problem using
  • Sinc-Galerkin method, TWMS J. App. Eng. Math.,V2(3), pp.160-181. Lund,J. and Bowers,K., (1991), Sinc Methods for Quadrature and Differential Equations, Siam, Philadelphia, PA.
  • Stenger,F., (1979), A Sinc-Galerkin method of solution of boundary-value problems, Math. Comp., , pp.85-109.
  • Whittaker,E.T., (1915), On the functions which are represented by the expansions of the interpolation theory, Proc. Roy. Soc. Edinburg, 35, pp.181-194.
  • Whittaker,J.M., (1935), Interpolation Function Theory, in: Cambridge Tracts in Mathematics and Mathematical Physics, Vol.33, Cambridge University Press, London.
  • Koonprasert,S. and Bowers,K., (2004), The Fully Sinc-Galerkin Method for Time-Dependent Bound- ary Conditions, Numerical Method for Partial Differential Equations, 20(4), pp.494-526.
  • Hansen,P.C., (1992), Analysis of discrete ill-posed problems by means of the L-curve, SIAM Rev., 34, pp.561-80.
  • Lawson,C.L. and Hanson,R.J., (1995), Solving Least Squares Problems, Philadelphia, PA:SIAM.
  • Tikhonov,A.N. and Arsenin,V.Y., (1977), On the solution of Ill-posed problems, New York, Wiley.
  • Tikhonov,A.N., and Arsenin,V.Y., (1977), Solution of Ill-Posed Problems, V.H.Winston and Sons, Washington, DC.
  • Martin,L., Elliott,L., Heggs,P.J., Ingham,D.B., Lesnic,D., and Wen,X., (2006), Dual Reciprocity
  • Boundary Element Method Solution of the Cauchy Problem for Helmholtz-type Equations with Vari- able Coefficients, Journal of sound and vibration, 297, pp.89-105. Elden,L., (1984), A Note on the Computation of the Generalized Cross-validation Function for Ill- conditioned Least Squares Problems, BIT, 24, pp.467-472
  • Golub,G.H., Heath,M., and Wahba,G., (1979), Generalized Cross-validation as a Method for Choosing a Good Ridge Parameter, Technometrics, 21(2), pp.215-223.
  • Wahba,G., (1990), Spline Models for Observational Data, CBMS-NSF Regional Conference Series in
  • Applied Mathematics, Vol.59, SIAM, Philadelphia. Holland,J.H., (1975), Adaptation in Natural and Artificial System, University of Michigan Press, Ann Arbor.
  • Liu,F-B., (2008), A modified genetic algorithm for solving the inverse heat transfer problem of esti- mating plan heat source, International Journal of Heat and Mass Transfer, 51, pp.3745-3752.
  • Reza Pourgholi, for the photograph and short biography, see TWMS J. Appl. and Eng. Math., V.2, No.2, 2012.
Toplam 37 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Bölüm Research Article
Yazarlar

Hassan Dana Mazraeh Bu kişi benim

R. Pourgholi Bu kişi benim

- T.houlari Bu kişi benim

Yayımlanma Tarihi 1 Haziran 2017
Yayımlandığı Sayı Yıl 2017 Cilt: 7 Sayı: 1

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