THERMAL SCIENCE

International Scientific Journal

Thermal Science - Online First

online first only

Conformable heat equation on a radial symmetric plate

ABSTRACT
The conformable heat equation is defined in terms of a local and limit-based definition called conformable derivative which provides some basic properties of integer order derivative such that conventional fractional derivatives lose some of them due to their nonlocal structure. In this paper, we aim to find the fundamental solution of a conformable heat equation acting on a radial symmetric plate. Moreover, we give a comparison between the new conformable and the existing Grünwald-Letnikov solutions of heat equation. The computational results show that conformable formulation is quite successful to show the sub-behaviors of heat process. In addition, conformable solution can be obtained by a direct analytical method without need of any numerical scheme and any restrictions on the problem formulation. This is surely a significant advantageous compared to the Grünwald-Letnikov solution.
KEYWORDS
PAPER SUBMITTED: 2016-04-27
PAPER REVISED: 2016-05-30
PAPER ACCEPTED: 2016-06-27
PUBLISHED ONLINE: 2016-12-03
DOI REFERENCE: https://doi.org/10.2298/TSCI160427302A
REFERENCES
  1. Povstenko, Y., Fractional Heat Conduction Equation and Associated Thermal Stresses, J. Thermal Stresses, 28 (2005), pp. 83-102
  2. Povstenko, Y., Thermoelasticity which Uses Fractional Heat Conduction Equation, J. Math. Sci., 162 (2009), pp. 296-305
  3. Povstenko, Y., Fractional Thermoelasticity, Encyclopedia of Thermal Stresses (Ed. R.B. Hernarski), Springer, New York, USA, 2014, pp. 1778-1787
  4. Povstenko, Y., Linear Fractional Diffusion-Wave Equation for Scientists and Engineers, Birkhäuser, New York, USA, 2015
  5. Povstenko, Y., Fractional Thermoelasticity, Springer, New York, USA, 2015
  6. Kilbas, A. A., Srivastava, H. M., Trujillo, J. J., Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, Netherlands, 2006
  7. Podlubny, I., Fractional Differential Equations, Academic Press, San Diego, USA, 1999
  8. Samko, S. G., Kilbas, A. A., Marichev, O. I., Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Amsterdam, Netherlands, 1993
  9. Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J. J., Fractional Calculus: Models and Numerical Methods. Series on Complexity, Nonlinearity and Chaos- Vol.3, World Scientific Publishing Co. Pte. Ltd, Singapore, 2012
  10. Atanackovic, T. M., Pipilovic, S., Stankovic, B., Zorica, D., Fractional Calculus with Applications in Mechanics: Wave Propagation, Impact and Variational Principles, John Wiley & Sons, London, UK, 2014
  11. Li, C., Zeng, F., Numerical Methods for Fractional Calculus, CRC Press, Taylor & Francis, New York, USA, 2015
  12. Guo, B., Pu, X., Huang, F., Fractional Partial Differential Equations and Their Numerical Solutions, World Scientific Publishing, Singapore, 2015
  13. Hristov, J., Heat-Balance Integral to Fractional (Half-Time) Heat Diffusion Sub-Model, Thermal Science, 14 (2010), 2, pp. 291-316
  14. Hristov, J., An Approximate Analytical (integral-balance) Solution to A Nonlinear Heat Diffusion Equation, Thermal Science, 2 (2015), pp.723-733
  15. Hristov, J., Diffusion Models with Weakly Singular Kernels in The Fading Memories: How The Integral-Balance Method Can Be Applied? Thermal Science, 19 (2015), 3, pp. 947-957
  16. Hristov, J., An Alternative Integral-Balance Solutions to Transient Diffusion of Heat (Mass) by Time-Fractional Semi-Derivatives and Semi-Integrals, Thermal Science, (2016), doi:10.2298/TSCI150917010H
  17. Hristov, J., An Approximate Solution To The Transient Space-Fractional Diffusion Equation: Integral Balance Approach, Optimization Problems and Analyzes, Thermal Science, (2016), 10.2298/TSCI160113075H
  18. Yang, X. J., Baleanu, D., Srivastava, H. M., Local Fractional Integral Transforms and Their Applications, Elsevier, London, UK, 2015
  19. Atangana, A., Derivative with A New Parameter: Theory, Methods and Applications, Elsevier, London, UK, 2015
  20. Khalil, R., et al., A New Definition of Fractional Derivative, J. Comput. Appl. Math., 264 (2014), pp. 65-70
  21. Abdeljawad, T., On Conformable Fractional Calculus, J. Comput. Appl. Math., 279 (2015), pp. 57-66
  22. Abu Hammad, I., Khalil, R., Conformable Fractional Heat Differential Equation, Int. J. Pure Appl. Math., 94(2) (2014), pp. 215-221
  23. Abu Hammad, I., Khalil, R., Fractional Fourier Series with Applications, Am. J. Comput. Appl. Math., 4(6) (2014), pp. 187-191
  24. Khalil, R., Abu-Shaab, H., Solution of Some Conformable Fractional Differential Equations, Int. J. Pure Appl. Math., 103 (2015), 4, pp. 667-673
  25. Atangana, A., et al., New Properties of Conformable Derivative, Open Math., 13 (2015), pp. 889-898
  26. Çenesiz, Y., Kurt, A., The New Solution of Time Fractional Wave Equation with Conformable Fractional Derivative Definition, Journal of New Theory, 7 (2015), pp. 79-85
  27. Çenesiz, Y., Kurt, A., The Solution of Time and Space Conformable Fractional Heat Equations with Conformable Fourier Transform, Acta Univ. Sapientia, Mathematica, 7 (2015), 2, pp. 130-140
  28. Avcı, D., Eroğlu, B.B.İ, Özdemir, N., Conformable Fractional Wave-like Equation on A Radial Symmetric Plate, 8th Conference on Non-integer Order Calculus and its Applications, Zakopane, Poland, 2016 (Accepted)
  29. Neamaty, A., et al., On The Determination of The Eigenvalues for Airy Fractional Differential Equation with Turning Point, TJMM, 7 (2015), 2, pp. 149-153
  30. Avcı, D., Eroğlu, B. B. İ, Özdemir, N., Conformable Heat Problem in A Cylinder, Proceedings, International Conference on Fractional Differentiation and its Applications, Novi Sad, Serbia, 2016, pp. 572-58
  31. Iyiola, O. S., Nwaeze, E. R., Some New Results on The New Conformable Fractional Calculus with Application using D'Alembert Approach, Progr. Fract. Differ. Appl., 2 (2016), 2, pp.1-7
  32. Ghanbarl, K., Gholami, Y., Lyapunov Type Inequalities for Fractional Sturm-Liouville Problems and Fractional Hamiltonian Systems and Applications, J Fract. Calc. Appl., 7 (2016), 1, pp. 176-188
  33. Özdemir, N., et al., Analysis of An Axis-Symmetric Fractional Diffusion-Wave Problem, J. Phys. A-Math. Theor., 42 (2009), 35, 355208 (10pp)