یک الگوریتم عددی بر پایه چندجمله‌ای‌های چبیشف برای حل برخی مسائل منبع معکوس

نویسندگان
1 دانشگاه مازندران، گروه ریاضی، بابلسر
2 دانشگاه صنعتی بوئین زهرا
چکیده
در این مقاله، دو مسئله معکوس تعیین جمله منبع مجهول در یک معادله سهموی بررسی می­شوند. ابتدا جمله منبع مجهول به‌صورت ترکیبی از توابع چبیشف تخمین زده می­شود. آن‌گاه یک الگوریتم عددی بر پایه چندجمله­ای­های چبیشف برای تعیین جواب مسئله ارائه می­شود. برای حل مسئله، ماتریس­های عملیاتی انتگرال­گیری و مشتق­گیری معرفی شده و برای تبدیل مسئله اشاره شده به معادلات ماتریسی استفاده می­شوند که متناظر با دستگاه معادلات جبری خطی با ضرایب چبیشف مجهول است. با توجه به بدوضعی این مسائل معکوس، روش منظم‌سازی تیخونوف با معیار اعتبارسنجی متقابل کلی، برای تعیین جواب­های پایدار به‌کار برده می­شود. در نهایت، در بخش آخر، چند مثال برای مشخص کردن توانایی و کارایی این روش ارائه می­شوند. نتایج عددی نشان می­دهند که روش پیشنهادی یک روش قابل اعتماد بوده و جواب‌های عددی با دقت زیاد است.
کلیدواژه‌ها

عنوان مقاله English

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

نویسندگان English

S Nemati 1
A Babaei 1
S Sedaghat 2
چکیده English

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.

کلیدواژه‌ها English

Parabolic equation‎
Inverse problem‎
‎Unknown source term‎
‎Tikhonov regularization‎
‎Chebyshev polynomials‎
‎Operational matrix
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.