Solving an inverse problem for a parabolic equation with a nonlocal boundary condition in the reproducing kernel space

Document Type : Research Article

Authors

Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156- 83111, Iran.

Abstract

On the basis of a reproducing kernel space, an iterative algorithm for solving the inverse problem for heat equation with a nonlocal boundary condition is presented. The analytical solution in the reproducing kernel space is shown in a series form and the approximate solution vn is constructed by truncating the series to n terms. The convergence of vn to the analytical solution is also proved. Results obtained by the proposed method imply that it can be considered as a simple and accurate method for solving such inverse problems.

Keywords


1. Cannon J.R. and Van de Hoek J. The one phase stefan problem subject to energy, J. Math. Anal. Appl. 86 (1982) 281-292.
2. Dehghan M. An inverse problem of finding a source parameter in a semi linear parabolic equation, Appl. Math. Model. 25 (2001) 743-754.
3. Dehghan M. Numerical solution of one-dimensional parabolic inverse problem, Appl. Math. Comput. 136 (2003) 333-344.
4. Fatullayev A. and Can E. procedures for determining unknown source parameter in parabolic equations, Math. Comput. Simulat. 54 (2000) 159-167.
5. Cannon JR., Lin Y. and Wang S. Determination of a control parameter in a parabolic partial differential equation, J. Aust. Math. Soc. B. 33 (1991) 149-163.
6. Ivanchov MI. and Pabyrivska NV. Simultaneous determination of two coefficients of a parabolic equation in the case of nonlocal and integral conditions, Ukr. Math. J. 53 (2001) 674-684.
7. Ivanchov MI. Inverse Problems for Equations of Parabolic Type, VNTL Publishers: Lviv, Ukraine. 2003.
8. Namazov GK. Definition of the unknown coefficient of a parabolic equation with nonlocal boundary and complementary-conditions, Transactions of Academy of Sciences of Azerbaijan, Series of Physical-Technicaland Mathematical Sciences. 19 (1999) 113-117.
9. Sapagovas M. and Jakub˙elien˙e K. Alternating direction method for two dimensional parabolic equation with nonlocal integral condition, Nonlinear Anal. Modelling Control. 17(1) (2012) 91-98.
10. Sapagovas M. and Stikonien˙e O. ˇ Alternating-direction method for a mildly nonlinear elliptic equation with nonlocal integral conditions, Nonlinear Anal Modelling Control. 16(2) (2011) 220-230.
11. Aronszajn N. Theory of reproducing kernels, Trans. Amer. Math. Soc. 68(1950) 337-404.
12. Geng F. Solving singular second order three-point boundary value problems using reproducing kernel Hilbert space method, Appl. Math. Comput. 215 (2009) 2095-2102.
13. Mohammadi M. and Mokhtari R. Solving the generalized regularized long wave equation on the basis of a reproducing kernel space, J. Comput. Appl. Math. 235 (2011) 4003-4011.
14. Geng F.and Cui M. A reproducing kernel method for solving nonlocal fractional boundary value problems, Appl. Math. Lett. 25 (2012) 818-823.
15. Mokhtari R., Toutian Isfahani F. and Mohammadi M. Solving a class of nonlinear differential-difference equations in the reproducing kernel space,Abstr. Appl. Anal. 2012 (2012) Articel ID 514103.
16. Mohammadi M. and Mokhtari R. A new algorithm for solving nonlinear Shr¨odinger equation in the reproducing kernel space, to appear in IJS &T-Transaction A.
17. Mohammadi M. and Mokhtari R. A reproducing kernel method for solving a class of nonlinear systems of PDEs, to appear in Math. Model. Anal.
18. Mohammadi M., Mokhtari R. and Panahipour H. A Galerkin-reproducing kernel method: application to the 2D nonlinear coupled Burgers’ equations, Eng. Anal. Bound. Elem. 37 (2013) 1642-1652.
19. Cui M. and Lin Y. Nonlinear Numerical Analysis in the Reproducing Kernel Space. Nova Science Publisher: New York. 2008.
20. Ismailova M. and Kancab F. An inverse coefficient problem for a parabolic equation in the case of nonlocal boundary and overdetermination condi tions, Math. Meth. Appl. Sci. 34 (2011) 692-702.
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