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The Ritz–Galerkin procedure for an inverse space-dependent heat source problem

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In this paper, an inverse problem of recovering the unknown space-dependent source term in a parabolic equation under a final overdetermination condition is considered. To keep matters simple, in our analysis the main problem has been considered in the one-dimensional case, however the proposed method can be applied for higher dimensional cases. The approximate solution of the inverse problem is implemented by the Ritz–Galerkin method. The shifted Legendre polynomials basis together with the Galerkin approach are employed to reduce the main problem to the solution of linear algebraic equations. To overcome the difficulties arising from solving the resultant ill-conditioned linear system, a type of regularization technique is utilized to obtain a stable solution. The convergence analysis of the suggested method using Gronwall’s inequality is studied. Finally, some numerical examples are provided to demonstrate the efficiency and applicability of the proposed algorithm in the presence of noise in input measured data.

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  1. Alifanov, O.M.: Inverse Heat Transfer Problems. Springer, Berlin (2012)

    Google Scholar 

  2. Askey, R.: Orthogonal Polynomials and Special Functions, vol. 21. SIAM, Philadelphia (1975)

    Book  Google Scholar 

  3. Berres, S.: Identification of piecewise linear diffusion function in convection-diffusion equation with overspecified boundary. In: AIP. Conf. Proc., vol. 1048, pp. 76–79. AIP (2008)

  4. Cheng, W., Zhao, L.L., Fu, C.L.: Source term identification for an axisymmetric inverse heat conduction problem. Comput. Math. Appl. 59(1), 142–148 (2010)

    Article  MathSciNet  Google Scholar 

  5. Dehghan, M.: An inverse problem of finding a source parameter in a semilinear parabolic equation. Appl. Math. Model. 25(9), 743–754 (2001)

    Article  Google Scholar 

  6. Ebel, A., Davitashvili, T.: Air, Water and Soil Quality Modelling for Risk and Impact Assessment. Springer, Heidelberg (2007)

    Book  Google Scholar 

  7. Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems, vol. 375. Springer, Berlin (1996)

    Book  Google Scholar 

  8. Fatullayev, A.G.: Numerical solution of the inverse problem of determining an unknown source term in a heat equation. Math. Comput. Simul. 58(3), 247–253 (2002)

    Article  MathSciNet  Google Scholar 

  9. Hansen, P.C.: Regularization tools: a MATLAB package for analysis and solution of discrete ill-posed problems. Numer. Algorithms 6(1), 1–35 (1994)

    Article  MathSciNet  Google Scholar 

  10. Hansen, P.C., O’Leary, D.P.: The use of the L-curve in the regularization of discrete ill-posed problems. SIAM J. Sci. Comput. 14(6), 1487–1503 (1993)

    Article  MathSciNet  Google Scholar 

  11. Hasanov, A., Pektaş, B.: Identification of an unknown time-dependent heat source term from overspecified Dirichlet boundary data by conjugate gradient method. Comput. Math. Appl. 65(1), 42–57 (2013)

    Article  MathSciNet  Google Scholar 

  12. Hazanee, A., Ismailov, M., Lesnic, D., Kerimov, N.: An inverse time-dependent source problem for the heat equation. Appl. Numer. Math. 69, 13–33 (2013)

    Article  MathSciNet  Google Scholar 

  13. Johansson, B.T., Lesnic, D.: A variational method for identifying a spacewise-dependent heat source. IMA. J. Appl. Math. 72(6), 748–760 (2007)

    Article  MathSciNet  Google Scholar 

  14. Johansson, T., Lesnic, D.: Determination of a spacewise dependent heat source. J. Comput. Appl. Math. 209(1), 66–80 (2007)

    Article  MathSciNet  Google Scholar 

  15. Kamynin, V.L.: On the unique solvability of an inverse problem for parabolic equations under a final overdetermination condition. Math. Notes 73(1–2), 202–211 (2003)

    Article  MathSciNet  Google Scholar 

  16. Kantorovich, L., Krylov, V.: Approximate Methods of Higher Analysis. Interscience Publishers, New York (1958)

    MATH  Google Scholar 

  17. Ladyzhenskaia, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasi-Linear Equations of Parabolic Type, Translated from the Russian by S. Smith, Translations of Mathematical Monographs 23. American Mathematical Society, Providence (1968)

  18. Lesnic, D., Elliott, L., Ingham, D.: Application of the boundary element method to inverse heat conduction problems. Int. J. Heat Mass Transf. 39(7), 1503–1517 (1996)

    Article  Google Scholar 

  19. Luke, Y.L.: Special Functions and Their Approximations, vol. 2. Academic Press, New York (1969)

    MATH  Google Scholar 

  20. Mierzwiczak, M., Kołodziej, J.A.: Application of the method of fundamental solutions and radial basis functions for inverse transient heat source problem. Comput. Phys. Commun. 181(12), 2035–2043 (2010)

    Article  MathSciNet  Google Scholar 

  21. Mikhlin, S.G.: Variational Methods in Mathematical Physics. Pergamon Press, Oxford (1964)

    MATH  Google Scholar 

  22. Pourgholi, R., Dana, H., Tabasi, S.H.: Solving an inverse heat conduction problem using genetic algorithm: sequential and multi-core parallelization approach. Appl. Math. Model. 38(7–8), 1948–1958 (2014)

    Article  MathSciNet  Google Scholar 

  23. Prilepko, A., Tkachenko, D.: An inverse problem for a parabolic equation with final overdetermination. J. Inverse Ill-Posed Probl. pp. 345–381 (2002)

  24. Prilepko, A.I., Kostin, A.B.: On certain inverse problems for parabolic equations with final and integral observation. Mat. Sb. 183(4), 49–68 (1992)

    MATH  Google Scholar 

  25. Prilepko, A.I., Solov’ev, V.V.: Solvability theorems and the Rothe’s method in inverse problems for an equation of parabolic type. i. Differ. Uravn 23(11), 1971–1980 (1987)

    MathSciNet  Google Scholar 

  26. Rashedi, K., Adibi, H., Dehghan, M.: Determination of space-time-dependent heat source in a parabolic inverse problem via the Ritz–Galerkin technique. Inverse Probl. Sci. Eng. 22(7), 1077–1108 (2014)

    Article  MathSciNet  Google Scholar 

  27. Samarskii, A.A., Vabishchevich, P.N.: Numerical Methods for Solving Inverse Problems of Mathematical Physics. De Gruyter, Berlin (2008)

    MATH  Google Scholar 

  28. Shidfar, A., Karamali, G., Damirchi, J.: An inverse heat conduction problem with a nonlinear source term. Nonlinear Anal. 65(3), 615–621 (2006)

    Article  MathSciNet  Google Scholar 

  29. Shidfar, A., Pourgholi, R., Ebrahimi, M.: A numerical method for solving of a nonlinear inverse diffusion problem. Comput. Math. Appl. 52(6–7), 1021–1030 (2006)

    Article  MathSciNet  Google Scholar 

  30. Shidfar, A., Zolfaghari, R., Damirchi, J.: Application of Sinc-collocation method for solving an inverse problem. J. Comput. Appl. Math. 233(2), 545–554 (2009)

    Article  MathSciNet  Google Scholar 

  31. Slodička, M., Lesnic, D., Onyango, T.: Determination of a time-dependent heat transfer coefficient in a nonlinear inverse heat conduction problem. Inverse Probl. Sci. Eng. 18(1), 65–81 (2010)

    Article  MathSciNet  Google Scholar 

  32. Tarantola, A.: Inverse Problem Theory and Methods for Model Parameter Estimation. SIAM, Philadelphia (2005)

    Book  Google Scholar 

  33. Tikhonov, A., Arsenin, V.Y.: Solutions of Ill-Posed Problems. Wiley, New York (1977)

    MATH  Google Scholar 

  34. Yan, L., Yang, F.L., Fu, C.L.: A meshless method for solving an inverse spacewise-dependent heat source problem. J. Comput. Phys. 228(1), 123–136 (2009)

    Article  MathSciNet  Google Scholar 

  35. Yang, L., Dehghan, M., Yu, J.N., Luo, G.W.: Inverse problem of time-dependent heat sources numerical reconstruction. Math. Comput. Simul. 81(8), 1656–1672 (2011)

    Article  MathSciNet  Google Scholar 

  36. Yousefi, S., Barikbin, Z., Dehghan, M.: Ritz–Galerkin method with Bernstein polynomial basis for finding the product solution form of heat equation with non-classic boundary conditions. Int. J. Numer. Methods Heat Fluid Flow 22(1), 39–48 (2012)

    Article  MathSciNet  Google Scholar 

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Shahsahebi, F.S., Damirchi, J. & Janmohammadi, A. The Ritz–Galerkin procedure for an inverse space-dependent heat source problem. Japan J. Indust. Appl. Math. 38, 625–643 (2021).

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