International Scientific Journal

Authors of this Paper

External Links


This paper gives a literature review on various analytical methods and numerical methods for heat problems. Fractal models and fractional models are emphasized. Beginning at the classic heat equation, fractional Fourier law and fractional conservation of energy are considered for 1-D heat equation in fractal media, its solution properties are discussed using the fractional complex transform. The emphasis of this literature review is put upon recent publications in Thermal Science, and the references are not exhaustive.
PAPER ACCEPTED: 2016-06-06
CITATION EXPORT: view in browser or download as text file
  1. Chen, R. X., et al., Bubble Rupture in Bubble Electrospinning, Thermal Science, 19 (2015), 4, pp. 1141-1149
  2. He, C. H., et al., Bubbfil Spinning for Fabrication of PVA Nanofibers, Thermal Science, 19 (2015), 2, pp. 743-746
  3. Gao, S. W., et al., Near-Infraed Scattering Method for Fabric Thermal Comfort, Thermal Science, 18 (2014), 5, pp. 1469-1472
  4. Ma, J., et al., Architectural Design of Passive Solar Residential Building, Thermal Science, 19 (2015), 4, pp. 1415-1418
  5. He, J.-H., A New Fractal Derivation, Thermal Science, 15 (2011), Suppl. 1, pp. S145-S147
  6. Fan, J., He, J. H., Fractal Derivative Model for Air Permeability in Hierarchic Porous Media, Abstract and Applied Analysis, 2012, 354701
  7. Fan, J., et al., Model of Moisture Diffusion in Fractal Media, Thermal Science, 19 (2015), 4, pp. 1161-1166
  8. Hu, Y., et al., On Fractal Space-Time and Fractional Calculus, Thermal Science, 20 (2016), 3, pp. 773-777
  9. Fan, J. Shang, X. M., Water Permeation in the Branching Channelnet of Wool Fiber, Heat Transfer Research, 44 (2013), 5, pp. 465-472
  10. Fan, J., Shang, X. M., Fractal Heat Trans fer in Wool Fiber Hierarchy, Heat Transfer Research, 44 (2013), 5, pp. 399-407
  11. Fei, D. D., et al., Fractal Approach to Heat Transfer in Silkworm Cocoon Hierarchy, Thermal Science, 17 (2013), 5, pp. 1546-1548
  12. Lu, T., et al., Analysis of Fractional Flow for Transient Two-Phase Flow in Fractal Porous Medium, Fractals-Complex Geometry Patters and Scaling in Nature and Society, 24 (2016), 1, 1650013
  13. He, J. H., A Tutorial Review on Fractal Spacetime and Fractional Calculus, International Journal of Theoretical Physics, 53 (2014), 11, pp. 3698-3718
  14. Liu, F. J., et al., He's Fractional Derivative for Heat Conduction in a Fractal Medium Arising in Silkworm Cocoon Hierarchy, Thermal Science, 19 (2015), 4, pp. 1155-1159
  15. Liu, F. J., He, J. H., A Fractional Model for Heat-Insulating Coating with Cocoon-Like Hierarchy on Surface of Buildings, Computers & Mathematics with Applications, doi:10.1016/j.camwa.2016.04.018
  16. Sayevand, K., Pichaghchi, K., Analysis of Nonlinear Fractional KdV Equation Based on He's Fractional Derivative, Nonlinear Science Letters A, 7 (2016), 3, pp. 77-85
  17. Wang, K., Liu, S., A New Solution Procedure for Nonlinear Fractional Porous Media Equation Based on a New Fractional Derivative, Nonlinear Science Letters A, 7 (2016), 4, pp. 135-140
  18. Abbas, I. A., Youssef, H. M., Two-Dimensional Fractional Order Generalized Thermoelastic Porous Material, Latin American Journal of Solids and Structures, 12 (2015), 7, pp. 1415-1431
  19. Fou rier, J. B. J., The Analytical Theory of Heat (Translated by A. Freeman), Cambridge University Press, Cambridge, UK, 1878
  20. He, J. H., Lee, E. W. M., A Constrained Variational Principle for Heat Conduction, Physics Letters A, 373 (2009), 31, pp. 2614-2615
  21. Fei, D. D., et al., A Short Remark on He-Lee Variational Principle for Heat Conduction, Thermal Science, 17 (2013), 5, pp. 1561-1563
  22. Tao, Z. L., Chen, G. H., Remark on a Constrained Variational Principle for Heat Conduction, Thermal Science, 17 (2013), 3, pp. 951-952
  23. Liu, H. Y, et al., A Short Remark on Stewart 1962 Variational Principle for Laminar Flow in a Uniform Duct, Thermal Science, 20 (2106), 1, pp. 359-361
  24. Jia, Z., et al., Variational Principle for Unsteady Heat Conduction Equation, Thermal Science, 18 (2014), 3, pp. 1045-1047
  25. Li, Z. B., Liu, J., Vartional for mu lations for Solton Equations Arising in Water Transport in Porous Soils, Thermal Science, 17 (2013), 5, pp. 1483-1485
  26. Ghaneai, H., Hosseini, M. M., Variational Iteration Method with an Auxiliary Parameter for Solving Wave-Like and Heat-Like Equations in Large Domains, Computers and Mathematics with Applications, 69 (2015), 5, pp. 363-373
  27. Liu, J. F., Modified Variational Iteration Method for Varian Boussinesq Equation, Thermal Science, 19 (2015), 4, pp. 1195-1199
  28. Dehghan, M., et al., Convection-Radiation Heat Transfer in Solar Heat Exchangers Filled with a Porous Medium: Homotopy Perturbation Method versus Numerical Analysis, Renewable Energy, 74 (2015), Feb., pp. 448-455
  29. Rahim-Esbo, M., et al., Analytical and Numerical Investigation of Natural Convection in a Heated Cylinder Using Homotopy Perturbation Method, Acta Scientiarum-Technology, 36 (2014), 4, pp. 669-677
  30. Patel, T., Meher, R., Adomian Decomposition Sumudu Transform Method for Solving a Solid and Porous Fin with Temperature Dependent Internal Heat Generation, Springer Plus, 5 (2016), Article Number489
  31. He, J. H., An Elementary Introduction to Recently Developed Asymptotic Methods and Nanomechanics in Textile Engineering, Int. J. Mod. Phys. B, 22 (2008), 21, pp. 3487-3578
  32. Minea, A. A., A Review on Analytical Techniques for Natural Convection Investigation in a Heated Closed Enclosure, Case Study, Thermal Science, 19 (2015), 3, pp. 1077-1095
  33. Sahu, S. K., et al., Analytical and Semi-Analytical Models of Conduction Controlled Rewetting, A State-of-Art Review, Thermal Science, 19 (2015), 5, pp. 1479-1496
  34. He, J. H., Asymptotic Methods for Solitary Solutions and Compactons, Abstract and Applied Analysis, 2012, 916793
  35. He, J. H., Some Asymptotic Methods for Strongly Nonlinear Equations, Int. J. Mod. Phys. B, 20 (2006), 10, pp. 1141-1199
  36. Tarasov, V. E., On Chain Rule for Fractional Derivatives, Communications in Nonlinear Science and Numerical Simulation, 30 (2016), 1-3, pp. 1-4
  37. He, J. H., et al. Geometrical Explanation of the Fractional Complex Transform and Derivative Chain Rule for Fractional Calculus, Phys. Lett. A, 376 (2012), pp. 257-259
  38. He, J.-H., Li, Z.-B., Converting Fractional Differential Equations into Partial Differential Equations, Thermal Science, 16 (2012), 2, pp. 331-334
  39. Li, Z. B., He, J. H., Fractional Complex Transform for Fractional Differential Equations, Math. Comput. Appl, 15 (2010), 5, pp. 970-973
  40. Zhang, M. F., et al., Efficient Homotopy Perturbation Method for Fractional Nonlinear Equations Using Sumudu Transform, Thermal Science, 19 (2015), 4, pp. 1167-1171
  41. Ma, H. C., et al., Exact Solutions of Nonlinear Fractional Partial Differential Equations by Fractional Sub-Equation Method, Thermal Science, 19 (2015), 4, pp. 1239-1244
  42. Hristov, J., An Approximate Analytical (Integral-Balance) Solution to a Nonlinear Heat Diffusion Equatons, Thermal Science, 19 (2015), 2, pp. 723-733
  43. Hristov, J., Approximate Solutions to Fractional Subdiffusion Equations, European Physical Journal-Special Topics, 193 (2011), 1, pp. 229-243

© 2019 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence