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The fractional (half-time) sub-model of the heat diffusion equation, known as Dirac-like evolution diffusion equation has been solved by the heat-balance integral method and a parabolic profile with unspecified exponent. The fractional heat-balance integral method has been tested with two classic examples: fixed temperature and fixed flux at the boundary. The heat-balance technique allows easily the convolution integral of the fractional half-time derivative to be solved as a convolution of the time-independent approximating function. The fractional sub-model provides an artificial boundary condition at the boundary that closes the set of the equations required to express all parameters of the approximating profile as function of the thermal layer depth. This allows the exponent of the parabolic profile to be defined by a straightforward manner. The elegant solution performed by the fractional heat-balance integral method has been analyzed and the main efforts have been oriented towards the evaluation of fractional (half-time) derivatives by use of approximate profile across the penetration layer.
PAPER REVISED: 2010-03-16
PAPER ACCEPTED: 2010-05-03
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THERMAL SCIENCE YEAR 2010, VOLUME 14, ISSUE Issue 2, PAGES [291 - 316]
  1. Samko, S. G., Kilbas, A. A., Marichev, O. I., Fractional Integrals and Derivatives - Theory and Applications, Gordon and Breach, Longhorne, Penn., USA, 1993
  2. Oldham, K. B., Spanier, J., The Fractional Calculus, Academic Press, New York, USA, 1974
  3. Miller, K. S., Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, New York, USA, 1993
  4. Podlubny, I., Fractional Differential Equations, Academic Press, New York, USA, 1999
  5. Babenko, Yu. I., Heat-Mass Transfer: Methods for Calculation of Diffusion Fluxes (in Russian), Khimia, Leningrad, USSR, 1986
  6. Narahari Achar, B. N., Hanneken, J. W., Fractional Radial Diffusion in a Cylinder, Journal of Molecular Liquids, 114 (2004), 1, pp. 147-151
  7. Djordjevic, V. D., Atanackovic, T. M., Similarity Solutions to Nonlinear Heat Conduction and Burgers/Korteweg-deVries Fractional Equations, Journal of Computational and Applied Mathematics, 222 (2008), 2, pp. 701-714
  8. Langlands, T. A. M., Solution of a Modified Fractional Diffusion Equation, Physica A, 367 (2006), July, pp. 136-144
  9. Pskhu, A. V., Solution of Boundary Value Problems for the Fractional Difusion Equation by the Green Function Method, Differential Equations (Russia), 39 (2003), 10, pp. 1509-1513
  10. Luchko, Yu. F., Srivastava , H. M., The Exact Solution of Certain Differential Equations of Fractional Order by Using Operational Calculus, Computers and Mathematical Application., 29 (1995), 8, pp. 73-85
  11. Diethelm, K., et al., Algorithms for the Fractional Calculus: a Selection of Numerical Methods, Comput. Methods Appl. Mech. Engrg., 194 (2005), 6-8, pp. 743-773
  12. He, J.-H. Approximate Analytical Solution for Seepage Flow with Fractional Derivatives in Porous Media, Computer Methods in Applied Mechanics and Engineering, 167 (1998), 1-2, pp. 57-68
  13. Ghorbani, A., Toward a New Analytical Method for Solving Nonlinear Fractional Differential Equations , Computer Methods in Applied Mechanics and Engineering, 197 (2008), 49-50, pp. 4173-4179
  14. Kurulay, M., Bayram, M., Approximate Analytical Solution for the Fractional Modified KdV by Differential Transform Method, Communications in Nonlinear Science and Numerical Simulation 15 (2010), 7, pp. 1777-1782
  15. Al-Rabtah, A., Ertürk, V. S., Momani, S., Solutions of a Fractional Oscillator by Using Differential Transform Method, Computers and Mathematics with Applications, 59 (2009), 3, pp. 1356-1362
  16. Goodman, T. R., The Heat Balance Integral and its Application to Problems Involving a Change of Phase, Transactions of ASME, 80 (1958), 1-2, pp. 335-342
  17. Goodman,T. R., Application of Integral Methods to Transient Nonlinear Heat Transfer, in: Advances in Heat Transfer, (Eds. T. F. Irvine, J. P. Hartnett), Vol. 1, 1964, Academic Press, San Diego, Cal., pp. 51-122
  18. Wood, A. S., Kutluay, S., A Heat Balance Integral Model of the Thermistor, Int. J. Heat Mass Transfer, 38 (1995), 10, pp. 1831-1840
  19. Moghtaderi, B., et al., An Integral Model for Transient Pyrolysis of Solid Materials, Fire and Materials, 21 (1997), 1, pp. 7-16
  20. Hristov, J., An Inverse Stefan Problem Relevant to Boilover: Heat Balance Integral Solutions and Analysis, Thermal Science, 11 (2007), 2, pp. 141-160
  21. Theuns, E., et al., Critical Evaluation of an Integral Model for the Pyrolysis of Charring Materials, Fire Safety Journal, 40 (2005), 2, pp.121-140
  22. Myers, T., Optimizing the Exponent in the Heat Balance and Refined Integral Methods, Int. Comm. Heat Mass Transfer, 36 (2009), 2, pp. 143-147
  23. Sadoun, N., Si-Ahmed, E-K., Colinet, P., On the Refined Integral Method for the One-Phase Stefan Problem with Time-Dependent Boundary Conditions, Applied Mathematical Modelling, 30 (2006), 6, pp. 531-544
  24. Sadoun, N., et al., On the Goodman Heat-Balance Integral Method for Stefan Like-Problems, Thermal Science, 13 (2009), 2, pp. 91-96
  25. Volkov, V. N., Li-Orlov, V. K., Refinement of the Integral Method in Solving the Heat Conduction Equation, Heat Trans-Sov Res, 2 (1970), 2, pp. 41-47
  26. Myers, T. G., Optimal Exponent Heat Balance and Refined Integral Methods Applied to Stefan Problem, Int. J. Heat Mass Transfer, 53 (2010), 5-6, pp. 1119-1127
  27. Hristov, J., Research Note on a Parabolic Heat-Balance Integral Method with Unspecified Exponent: An Entropy Generation Approach in Optimal Profile Determination, Thermal Science, 13 (2009), 2, pp. 49-59
  28. Hristov, J., The Heat-Balance Integral Method by a Parabolic Profile with Unspecified Exponent: Analysis and Benchmark Exercises, Thermal Science, 13 (2009), 2, pp. 22-48
  29. Mitchell , S. L., Myers, T.,G., Application of Standard and Refined Heat Balance Integral Methods to One-Dimensional Stefan Problems, SIAM Review, 52 (2010), 1, pp. 57-86
  30. Agrawal, O. P., Response of Diffusion-Wave System Subjected to Deterministic and Stochastic Fields, ZAMM, 83 (2003), 4, pp. 265-274
  31. Agrawal, O. P., Application of Fractional Derivatives in Thermal Analysis of Disk Brakes, Nonlinear Dynamics, 38 (2004), 1-4, pp. 191-206
  32. Kilbas, A. A., et al., On General Fractional Evolution-Diffusion Equation, in: Fractional Derivatives and Their Applications: Mathematical Tools, Geometrical and Physical Aspect (Eds. A. Le Meauté, J. A. Tenreiro Machado, J. C. Trigeassou, J. Sabatier), pp. 19-44
  33. Pierantozzi, T., Vazques, L., A Numerical Study of Fractional Evolution-Diffusion Dirac-Like Equations, Proceedings (Eds. B. H. V. Topping, G. Montero, R. Montero), 5th International Conference on Engineering Computational Technology, Las Palmas de Gran Canaria, Spain, 2006, Civil-Comp Press, Stirlingshire, Scotland, paper 20, pp. 1-11
  34. Usero, T., Vasquez, L., Pierantozzi, T., From Fractional Mechanical Model to a Fractional Generat\Lizationof the Dirac Equation, Proceedings (Eds. B. H. V. Topping, G. Montero, R. Montero), 5th International Conference on Engineering Computational Technology, Las Palmas de Gran Canaria, Spain, 2006, Civil-Comp Press, Stirlingshire, Scotland, 2006, paper 21, pp, 1-11
  35. Kosztolowicz, T., Subdiffusion in a System with a Thick Membrane, Journal of Membrane Science, 320 (2008), 1-2, pp. 492-499
  36. Dworecki, K., et al., Evolution of Concentration Field in a Membrane System, J. Biochem. Biophys. Methods, 62 (2005), 2, pp. 153-162
  37. Carslaw, H. S., Jaeger, J. C., Conduction of Heat in Solids, 2nd ed. Oxford Science Publications, Oxford University Press, Oxford, UK, 1992
  38. Langford, D., The Heat Balance Integral Method, Int. J. Heat Mass Transfer, 16 (1973), 12, pp. 2424-2428
  39. Zien, T. F., Integral Solutions of Ablation Problems with Time-Dependent Heat Flux, AIAA Journal, 16 (1978), 12, pp. 1287-1295
  40. Mosally, F., Wood, A. S., Al-Fhaid, A., An Exponential Heat Balance Integral Method, Applied Mathematics and Computation, 130 (2002), 1, pp. 87-100

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