A numerical Algorithm Based on Chebyshev Polynomials for Solving some Inverse Source Problems

Authors
Abstract
In this paper‎, two inverse problems of determining an unknown source term in a parabolic‎ equation are considered‎. ‎First‎, ‎the unknown source term is ‎estimated in the form of a combination of Chebyshev functions‎. ‎Then‎, ‎a numerical algorithm based on Chebyshev polynomials is presented for obtaining the solution of the problem‎. ‎For solving the problem‎, ‎the operational matrices of integration and derivation are introduced and utilized to reduce the mentioned problem into the matrix equations which correspond to a system of linear algebraic equations with unknown Chebyshev coefficients‎. Due‎ to ill-posedness of these inverse problems‎, ‎the Tikhonov regularization method with generalized cross validation (GCV) criterion is applied to find stable‎ solutions. ‎Finally‎, some examples are presented to illustrate the efficiency of this numerical method‎. The numerical results show that the proposed method is a reliable method and can give high accuracy approximate solutions.
Keywords

1. ‎Isakov V‎.‎‎, "Inverse source problems‎, ‎Mathematical surveys and monographs", American mathematical society, ‎Providence‎, ‎Rhode Island‎ ‎(1990)‎.‎

‎2. Isakov V‎.‎‎‎, "Inverse Problems for Partial Differential Equations", Springer‎, ‎New York‎, ‎(2005)‎.‎

‎3. Ladyzhenskaya‎ O. A.‎‎, ‎ Solonnikov‎‎ V. A‎.‎, ‎ Ural′ceva N. N‎.‎, "Linear and Quasilinear Equations of Parabolic Type", American mathematical society‎, ‎Providence‎, ‎Rhode Island (1968).

‎4. ‎Prilepko‎‎ A‎. ‎I., ‎‎Orlovsky‎‎ D‎. ‎G‎., ‎Vasin I‎. ‎A‎.‎, "Methods for Solving Inverse Problems in Mathematical Physics", Marcel Dekker‎, ‎Inc.‎, ‎New York‎‎‎ ‎(2000)‎.

‎5. ‎Samarskii‎‎ A‎.‎ A‎., ‎Vabishchevich P‎. ‎N‎.‎‎, "Methods for Solving Inverse Problems of Mathematical Physics", Walter de Gruyter GmbH & Company‎, ‎Berlin‎, ‎Germany (2007)‎.‎

‎6. ‎Cannon‎‎‎ J‎.‎ R‎., "The One Dimensional Heat Equation", Addison-Wesley‎, ‎Reading‎, ‎MA‎.‎, ‎(1984)‎.‎‎

7. ‎Friedman A‎.‎‎‎‎, "Partial Differential Equations of Parabolic Type", ‎American mathematical society‎, ‎Prentice-Hall Inc‎.‎‎, ‎(1964)‎.‎‎‎

8. Tuan N. H., Trong D. D., Thong T. H., Minh N. D., "Identification of the pollution source of a parabolic equation with the time-dependent heat conduction", Journal of Inequalities and Applications (2014) 161.

‎9. ‎Ebel‎‎ A‎.,‎ Davitashvili‎‎ T., "Air‎, ‎water and soil quality modeling for risk and impact assessment", ‎Springer‎, ‎Dordrecht ‎(2007)‎.‎‎‎‎

10. ‎Kamynin‎‎ V‎. ‎L., "The unique solvability of an inverse problem for parabolic equations under a final overdetermination condition", Mathematical Notes‎, 73 (2003) 202-211‎.‎‎‎

11. Beznoshchenko N.Ya., Prilepko A.I., "Inverse problems for parabolic equations, in: Problems of Mathematical Physics and Computational Mathematics", Nauka, Moscow, (1977) 51–63 [in Russian].

12. Gol'dman‎‎ N. L‎.‎‎, "Determination of the right-hand side in a quasilinear parabolic equation with a terminal observation", Differential Equations‎, ‎‎41(3) (2005)‎ 384-392‎.‎

13. Gol'dman N. L‎.‎‎‎, "Properties of solutions of parabolic equations with unknown right-hand side and adjoint problems", Differential Equations‎, 77(3) (2008) 350-355‎.‎‎

14. ‎‎Prilepko‎‎ A‎. ‎I‎., ‎Soloviev V‎. ‎V‎.‎‎‎, "On the solvability of an inverse boundary value problem of determining a coefficient at a lower order term in a parabolic equation‎, Mathematical Notes‎", ‎23(1) (1987)‎ 136-143‎.‎‎‎‎

15. ‎Prilepko‎‎ A‎. ‎I‎., ‎Soloviev‎ V‎. ‎V‎., "Solvability theorems and Rothe’s method for inverse problems for a parabolic ‎equation‎.‎I", Differential Equations‎, 23 (1) (1988) 1230-1237‎.‎‎

‎16. Rundell‎‎ W‎., "Determination of an unknown nonhomogeneous term in a linear partial differential equation from overspecified boundary data", Applicable Analysis‎, 10 (1980)‎ 231-242‎.‎‎‎

‎17. Soloviev V‎.‎ V‎.‎‎‎, "Solvability of the inverse problem of finding a source‎, ‎using overdetermination on the upper base for a parabolic equation", Differential Equations‎, 25 (1990)‎ 1114-1119‎.‎‎‎‎‎‎‎

‎18. Ashyralyev A., ‎‎Erdogan A. S., ‎Demirdag O.‎, "On the determination of the right-hand side in a parabolic equation"‎, Applied Numerical Mathematics‎, 62 (2012)‎ 1672-1683‎.‎‎‎‎‎‎‎‎

19. Erdem A.‎, Lesnic D., Hasanov‎ A.,‎‎" Identification of a spacewise dependent heat source", Applied Mathematical Modelling‎, 37 (2013)‎ 10231-10244‎.‎‎‎‎‎‎‎‎

‎20. Johansson T‎., Lesnic‎ D.‎,‎‎‎‎ "Determination of a spacewise dependent heat source", Journal of Computational and Applied Mathematics‎, 209 (2007)‎ 66-80‎.‎‎‎‎‎‎‎‎

21. Meng J. H., "Ritz approximation method for inverse source problem of heat equation", www.paper.edu.cn/download/downPaper/200904-296.

22. Shidfar‎‎‎‎ A., Babaei‎ A., "The Sinc-Galerkin method for solving an inverse parabolic problem with unknown source term", Numerical Methods for Partial Differential Equations‎, 29 (1) (2013)‎‎ 64-78‎.‎‎‎‎‎‎‎‎‎

23. Farcas‎‎ A‎., ‎Lesnic D.‎, "The boundary-element method for the determination of a heat source dependent on one variable", Journal of Engineering Mathematics‎, 54 (2006)‎ 375-388‎.‎‎‎‎‎‎‎‎

24. Borukhov‎‎ V‎. ‎T‎., ‎Vabishchevich P‎. ‎N.‎,‎‎" Numerical solution of the inverse problem of reconstructing a distributed right-hand side of a parabolic equation", Computer Physics Communications‎, 126 (1) (2000)‎ 32-36‎.‎‎‎‎‎‎‎‎‎‎

25. Hasanov A‎., ‎‎Pektas B.‎‎‎,‎‎" Identification of an unknown time-dependent heat source term from ‎‎overspecified Dirichlet boundary data by conjugate gradient method‎",‎‎‎‎ Computers and Mathematics with Applications‎, 56 (2013)‎ 42-57‎.‎‎‎‎‎‎‎‎‎‎‎‎

‎26. Hazanee A.‎, Ismailov‎ M. I.‎‎, ‎Lesnic D., ‎Kerimovand‎‎ N‎.‎ B., "An inverse time-dependent source problem for the heat equation", Computers and Mathematics with Applications‎, 69 (2013)‎ 13-33‎.‎‎‎‎‎‎‎‎‎‎‎‎

27. Rashedi‎ K., Adibi‎‎ H.‎‎, Dehghan‎‎‎‎ M., "Determination of space-time-dependent heat source in a parabolic inverse problem via the Ritz–Galerkin technique", Inverse Problems in Science and Engineering, 22 (2014)‎ 1077-1108‎.‎‎‎‎‎‎‎‎‎‎‎‎

28. Wang W.‎‎‎, ‎Han ‎‎‎‎‎‎‎‎ B., Yamamoto M.‎‎‎‎, "Inverse heat problem of determining time-dependent source parameter in reproducing kernel space", Nonlinear Analysis‎: ‎Real World Applications‎, ‎14 (2014) 875-887‎.‎‎‎‎‎‎‎‎‎‎‎‎‎

‎29. ‎Mason‎‎ J. C‎., ‎Handscomb D‎.‎ C.‎‎, "Chebyshev ‎‎‎Polynomials", CRC Press LLC (2003)‎‎.‎‎‎‎‎‎‎‎‎‎‎‎‎‎

30. Laub A. J., "Matrix Analysis for Scientists & Engineers", Siam, Philadelphia (2005).

31. Tikhonov A. N.‎, ‎Arsenin‎‎‎‎‎‎‎‎‎‎‎‎‎‎V. Y‎.,‎‎‎ "Solutions of ill-posed problems,‎ Winston and Sons"‎, ‎Washington ‎‎‎D.C‎. (1977)‎‎.‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎

32. Hansen‎‎‎‎‎‎‎‎‎‎‎‎ P‎. ‎C.‎,‎‎‎" Rank-Defficient and Discrete Ill-Posed Problems", SIAM, Philadelphia, (1998).

‎‎33. Wahba G‎.‎ A.‎‎‎‎‎‎‎‎‎‎‎‎,‎‎‎ "Survey of some smoothing problems and the method of‎ generalized cross-validation for solving them", Applications of statistics ‎(Proceedings Symposium on Wright State University‎, ‎Dayton‎, ‎Ohio‎, ‎1976)‎. Amsterdam‎: ‎North-Holland (1977) 507-23‎.‎

34. ‎Hansen P.‎ C.‎‎‎‎‎‎‎‎‎‎‎‎,‎‎‎" Regularization tools‎: ‎a Matlab package for analysis and solution‎ of discrete ill-posed problems", Numer Algorithms ‎6 (1-2) ‎(1994) 1-35‎.