THERMAL SCIENCE

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V. D. Beibalaev

,

M. R. Shabanova

A FINITE-DIFFERENCE SCHEME FOR SOLUTION OF A FRACTIONAL HEAT DIFFUSION-WAVE EQUATION WITHOUT INITIAL CONDITIONS

ABSTRACT
Efficient finite-difference scheme to solve fractional diffusion-wave equations without initial conditions has been developed. The efficient approximation of the Riesz fractional derivatives is demonstrated and efficiently exemplified by two simple problems with/without source terms.
KEYWORDS
fractional diffusion-wave equations, Riesz derivative, finite-difference scheme
PAPER SUBMITTED: 2012-04-18
PAPER REVISED: 2012-05-02
PAPER ACCEPTED: 2012-08-25
DOI REFERENCE: https://doi.org/10.2298/TSCI120418148B
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2015, VOLUME 19, ISSUE Issue 2, PAGES [531 - 536]
REFERENCES
  1. Oldham , K.B., J. Spanier ,J., The Fractional calculus , Academic Press, New York, 1974.
  2. I. Podlubny, I., Fractional Differential Equations, Academic Press, San Diego, 1999.
  3. Siddique, I., Vieru, D., Stokes flows of a Newtonian fluid with fractional derivatives and slip at the wall, Int. Rev. Chem. Eng., 3 (2011),6, pp. 822- 826.
  4. Qi , H., Xu, M., Some unsteady unidirectional flows of a generalized Oldroyd-B fluid with fractional derivative, Appl. Math. Model., 33 (2009) ,11, pp. 4184-4191.
  5. Hristov, J., Transient Flow of A Generalized Second Grade Fluid Due to a Constant Surface Shear Stress: an approximate Integral-Balance Solution , Int. Rev. Chem. Eng., 3 (2011), 6, pp. 802-809.
  6. Agrawal , O.P., Application of Fractional Derivatives in Thermal Analysis of Disk Brakes , Nonlinear Dynamics , 38( 2004),1, pp. 191-206.
  7. Hristov, J., Heat-Balance Integral to Fractional (Half-Time) Heat Diffusion Sub-Model, Thermal Science, 14 (2) (2010), 2, pp. 291-316. DOI: 10.2298/TSCI1002291H
  8. Kulish , V.V., Lage, J.L., Fractional-diffusion solutions for transient local temperature and heat flux, J. Heat Transfer, 122 (2000),2,pp.372-376.
  9. dos Santos, M.C., Lenzi, E., Gomes, E.M., Lenzi,, M.K., Lenzi, E.K., Development of Heavy Metal sorption Isotherm Using Fractional Calculus, Int. Rev. Chem. Eng., 3 (2011),6,pp. 814-817.
  10. Hristov, J., Starting radial subdiffusion from a central point through a diverging medium (a sphere): Heat-balance Integral Method, Thermal Science, 15 (2011), S1, pp. S5-S20 .
  11. Voller, V.R., An exact solution of a limit case Stefan problem governed by a fractional diffusion equation, Int. J. Heat Mass Transfer, 53 (2010),23-24, pp. 5622-5625.
  12. Pfaffenzeller, R.A., Lenzi,M.K., Lenzi, E.K., Modeling of Granular Material Mixing Using Fractional Calculus, Int. Rev. Chem. Eng., 3 (2011),6,pp. 818-821.
  13. Meilanov , R.P., Shabanova, M.R., Akhmedov, E.N., A Research Note on a Solution of Stefan Problem with Fractional Time and Space Derivatives, Int. Rev. Chem. Eng., 3 (2011),6, pp. 810-813.
  14. Liu, J., Xu, M., Some exact solutions to Stefan problems with fractional differential equations, J. Math. Anal. Appl., 351(2010),2, pp.536-542
  15. Jafari,H. Kadkhoda,N., Tajadodi, H.et al. , Homotopy Perturbation Pade Technique for Solving Fractional Riccati Differential Equations, Int. J. Nonlinear Sci. Num., 11(2010),s, pp. 271-275. DOI: 10.1515/IJNSNS.2010.11.S1.271
  16. Golbabai A., Sayevand,K., The Homotopy Perturbation Method for Multi-order Time Fractional Differential Equations, Nonlinear Science Letters A, 1(2010), 2, pp.147-154
  17. 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
  18. Hristov, J., Approximate solutions to fractional subdiffusion equations, European Physical Journal, 193(2011), 1, pp. 229-243
  19. Hristov, J., A Short-Distance Integral-Balance Solution to a Strong Subdiffusion Equation: A Weak Power-Law Profile, Int. Rev. Chem. Eng..,5(2) (2010),5, pp. 555-563
  20. Zhang, S., Zhang, H.Q., Fractional sub-equation method and its applications to nonlinear fractional PDEs, Physics Letters A, 375(2011), 7, pp. 1069-1073
  21. He J.H., Analytical methods for thermal science-An elementary introduction, Thermal Science, 15(2011), s, pp. S1-S3
  22. He, J.H., A New Fractal Derivation, Thermal Science, 15(2011), s, pp. S145-S147
  23. Meerschaert,M.M., Tadjeran, C., Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math. 172(2004), 1, pp.65-77.
  24. Lynch V.E., Carreras B.A., del-Castill-Negrete D., Ferreira-Mejias K.M., Hicks H.R. Numerical methods for the solution of partial differential equations of fractional order. J. Comp. Phys. , 192 (2003),2, pp. 406-421.
  25. Beibalaev, B.D.Б., Mathematical model of transport on media with fractal structures, Mathematical Modelling, 21(2009),5, pp.55-62 (in Russian)
  26. Garg M., Manohar P. , Numerical solution of fractional diffusion-wave equation with two space variables by matrix method, Fractional Calculus and Applied Analysis , 13 (2005), 2, pp. 191-207.
  27. Katsikadelis J.T., The BEM for numerical solution of partial fractional differential equations, Computers & Mathematics with Applications, 62 (2011),3, pp.891-901. doi:10.1016/j.camwa.2011.04.001.
  28. Liu, J., Hou, G., Numerical solutions of the space- and time-fractional coupled Burgers equations by generalized differential transform method, Applied Mathematics and Computation, 217 (2011), 16,pp. 7001-7008, doi:10.1016/j.amc.2011.01.111
  29. Li C., Zhao Z, Chen Y-Q., Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion, Computers & Mathematics with Applications, 62 (2011),3, pp.855-875. doi:10.1016/j.camwa.2011.02.045.
  30. Odibat, Z.M., Rectangular decomposition method for fractional diffusion-wave equations, Applied Mathematics and Computation, 179 (2006),1, pp. 92-97. dx.doi.org/10.1016/j.amc.2005.11.088
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A finite-difference scheme for solution of a fractional heat diffusion-wave equation without initial conditions

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