Reconstruction Robin Boundary Condition in the Heat Conduction Inverse Problem of Fractional Order

This paper describes a parallel algorithm for reconstruction the boundary condition for the heat conduction equation with derivative of fractional order with respect to the time. The heat transfer coefficient, occurring in the boundary condition of the third kind, was reconstructed. Additional information for the considered inverse problem is given by the temperature measurements at selected points of the domain. The direct problem was solved by using the implicit finite difference method. To minimize functional defining the error of approximate solution parallel Ant Colony Optimization algorithm (ACO) was used. Calculations have been performed in parallel way (multi-threaded). The paper presents examples to illustrate the accuracy and stability of the presented algorithm.

[1]  Jacek Smolka,et al.  Application of Bezier surfaces to the 3-D inverse geometry problem in continuous casting , 2011 .

[2]  Damian Słota,et al.  RECONSTRUCTION OF THE BOUNDARY CONDITION FOR THE HEAT CONDUCTION EQUATION OF FRACTIONAL ORDER , 2015 .

[3]  Dervis Karaboga,et al.  A comparative study of Artificial Bee Colony algorithm , 2009, Appl. Math. Comput..

[4]  Diego A. Murio,et al.  Implicit finite difference approximation for time fractional diffusion equations , 2008, Comput. Math. Appl..

[5]  Marco Dorigo,et al.  Ant colony optimization for continuous domains , 2008, Eur. J. Oper. Res..

[6]  T. Wei,et al.  A new regularization method for solving a time-fractional inverse diffusion problem , 2011 .

[7]  Y. C. Hon,et al.  An inverse problem for fractional diffusion equation in 2-dimensional case: Stability analysis and regularization , 2012 .

[8]  M. Duran Toksari,et al.  Ant colony optimization for finding the global minimum , 2006, Appl. Math. Comput..

[9]  D. Karaboga,et al.  On the performance of artificial bee colony (ABC) algorithm , 2008, Appl. Soft Comput..

[10]  Diego A. Murio,et al.  Time fractional IHCP with Caputo fractional derivatives , 2008, Comput. Math. Appl..

[11]  Marcin Gabryel,et al.  Can We Process 2D Images Using Artificial Bee Colony? , 2015, ICAISC.

[12]  Mahmud Khodadad,et al.  Identification of thermal conductivity and the shape of an inclusion using the boundary elements method and the particle swarm optimization algorithm , 2009 .

[13]  Masahiro Yamamoto,et al.  Coefficient inverse problem for a fractional diffusion equation , 2013 .

[14]  T. Wei,et al.  A New Regularization Method for the Time Fractional Inverse Advection-Dispersion Problem , 2011, SIAM J. Numer. Anal..

[15]  Carlos E. Mejía,et al.  Generalized time fractional IHCP with Caputo fractional derivatives , 2008 .

[16]  A. F. Santos,et al.  Firefly optimization to determine the precipitation field on South America , 2013 .

[17]  Krzysztof Grysa,et al.  Different finite element approaches for inverse heat conduction problems , 2010 .

[18]  Ting Wei,et al.  Reconstruction of a time-dependent source term in a time-fractional diffusion equation , 2013 .

[19]  Damian Slota,et al.  Reconstruction of the Thermal Conductivity Coefficient in the Time Fractional Diffusion Equation , 2014, RRNR.

[20]  Diego A. Murio,et al.  Stable numerical evaluation of Grünwald–Letnikov fractional derivatives applied to a fractional IHCP , 2009 .

[21]  Thomas Stützle,et al.  Ant Colony Optimization , 2009, EMO.

[22]  Christian Napoli,et al.  Is the Colony of Ants Able to Recognize Graphic Objects? , 2015, ICIST.

[23]  Damian Słota,et al.  Experimental Verification of Selected Artificial Intelligence Algorithms Used for Solving the Inverse Stefan Problem , 2014 .

[24]  Fangfang Dou,et al.  Kernel-based approximation for Cauchy problem of the time-fractional diffusion equation , 2012 .

[25]  William Rundell,et al.  An inverse problem for a one-dimensional time-fractional diffusion problem , 2012 .

[26]  J. Hristov.  An inverse Stefan problem relevant to boilover: Heat balance integral solutions and analysis , 2007, 1012.2534.

[27]  B. Tomas Johansson,et al.  A Meshless Regularization Method for a Two-Dimensional Two-Phase Linear Inverse Stefan Problem , 2013 .

[28]  Diego A. Murio,et al.  Stable numerical solution of a fractional-diffusion inverse heat conduction problem , 2007, Comput. Math. Appl..