## THERMAL SCIENCE

International Scientific Journal

## Authors of this Paper

,

### RECONSTRUCTION OF THE BOUNDARY CONDITION FOR THE HEAT CONDUCTION EQUATION OF FRACTIONAL ORDER

ABSTRACT
This paper describes reconstruction of the heat transfer coefficient occurring in the boundary condition of the third kind for the time fractional heat conduction equation. Fractional derivative with respect to time, occurring in considered equation, is defined as the Caputo derivative. Additional information for the considered inverse problem is given by the temperature measurements at selected points of the domain. The direct problem is solved by using the implicit finite difference method. To minimize functional defining the error of approximate solution the Nelder-Mead algorithm is used. The paper presents results of computational examples to illustrate the accuracy and stability of the presented algorithm.
KEYWORDS
PAPER SUBMITTED: 2014-10-10
PAPER REVISED: 2015-02-13
PAPER ACCEPTED: 2015-02-24
PUBLISHED ONLINE: 2015-08-02
DOI REFERENCE: https://doi.org/10.2298/TSCI15S1S35B
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2015, VOLUME 19, ISSUE Supplement 1, PAGES [S35 - S42]
REFERENCES
1. Hu, M. S., et. al., One-Phase Problems for Discontinuous Heat Transfer in Fractal Media, Mathematical Problems in Engineering, 2013 (2013), ID 358473
2. Yang, X. J., Baleanu, D., Fractal Heat Conduction Problem Solved by Local Fractional Variation Iteration Method, Thermal Science, 17 (2013), 2, pp. 625-628
3. Hristov, J., An Inverse Stefan Problem Relevant to Boilover: Heat Balance Integral Solutions and Analysis, Thermal Science, 11 (2007), 2, pp. 141-160
4. Cialkowski, M. J., Grysa, K., Trefftz Method in Solving the Inverse Problems, Journal of Inverse and Ill-Posed Problems, 18 (2010), 6, pp. 595-616
5. Grysa, K., Lesniewska, R., Different Finite Element Approaches for Inverse Heat Conduction Problems, Inverse Problems in Science and Engineering, 18 (2010), 1, pp. 3-17
6. Słota, D., Reconstruction of the Boundary Condition in the Problem of the Binary Alloy Solidification, Archives of Metallurgy and Materials, 56 (2011), 2, pp. 279-285
7. Nowak, I., et. al., Application of Bezier Surfaces to the 3-D Inverse Geometry Problem in Continuous Casting, Inverse Problems in Science and Engineering, 19 (2011), 1, pp. 75-86
8. Johansson, B., et. al., A Method of Fundamental Solutions for the One Dimensional Inverse Stefan Problem, Applied Mathematical Modeling, 35 (2011), 9, pp. 4367-4378
9. Hetmaniok, E., et. al., Solution of the Inverse Heat Conduction Problem with Neumann Boundary Condition by Using the Homotopy Perturbation Method, Thermal Science, 17 (2013), 3, pp. 643-650
10. Hetmaniok, E., et. al., Experimental Verification of Immune Recruitment Mechanism and Clonal Selection Algorithm Applied for Solving the Inverse Problems of Pure Metal Solidification, Numerical Heat Transfer B, 66 (2014), 4, pp. 343-359
11. Murio, D. A., Stable Numerical Solution of a Fractional-Diffusion Inverse Heat Conduction Problem, Computers and Mathematics with Applications, 53 (2007), 10, pp. 1492-1501
12. Murio, D. A., Time Fractional IHCP with Caputo Fractional Derivatives, Computers and Mathematics with Applications, 56 (2008), 9, pp. 2371-2381
13. Murio, D. A., Mejia, C. E., Generalized Time Fractional IHCP with Caputo Fractional Derivatives, Journal of Physics: Conferece Series, 135 (2008), ID 012074
14. Murio, D. A., Stable Numerical Evaluation of Grünwald-Letnikov Fractional Derivatives Applied to a Fractional IHCP, Inverse Problems in Science and Engineering, 17 (2009), 2, pp. 229-243
15. Miller, L., Yamamoto, M., Coefficient Inverse Problem for a Fractional Diffusion Equation, Inverse Problems, 29 (2013), 7, ID 075013
16. Wei, T., Zhang Z. Q., Reconstructon of Time-Dependent Source Term in a Time-Fractional Diffusion Equation, Engineering Analysis with Boundary Elements, 37 (2013), 1, pp. 23-31
17. Jin, B., Rundell., W., An Inverse Problem for a One-Dimensional Time-Fractional Diffusion Problem, Inverse Problems, 28 (2012), 7, ID 075010
18. Mitkowski, W., Obrączka, A., Simple Identification of Fractional Differential Equation, Solid State Phenomena, 180 (2012), pp. 331-338
19. Obraczka, A., Mitkowski, W., The Comparison of Parameter Identification Methods for Fractional Partial Differential Equation, Solid State Phenomena, 210 (2014), pp. 265-270
20. Brociek, R., et. al, Reconstruction of the Thermal Conductivity Coefficient in the Time Fractional Diffusion Equation, in: Advances in Modeling and Control of Non-Integer-Order Systems (Eds. K. J. Latawiec, et. al.) Lecture Notes in Electrical Engineering, 320, (2015), pp. 239-247
21. Bondarenko, A. N., Ivaschenko, D. S., Numerical Methods for Solving Inverse Problems for Time Fractional Diffusion Equation with Variable Coefficient, Journal of Inverse and Ill-Posed Problems, 17 (2009), 5, pp. 419-440
22. Murio, D. A., Implicit Finite Difference Approximation for Time Fractional Diffusion Equations, Computers and Mathematics with Applications, 56 (2008), 4, pp. 1138-1145
23. Brociek, R., Implicite Finite Difference Method for Time Fractional Diffusion Equation with Mixed Boundary Conditions, Zeszyty Naukowe Politechniki Śląskiej, Matematyka Stosowana, 4 (2014), pp. 73-87
24. Podlubny, I., Fractional Differential Equations, Academic Press, San Diego, Cal., USA, 1999
25. Diethelm, K., The Analysis of Fractional Differential Equations, Springer, Berlin, 2010