THERMAL SCIENCE

International Scientific Journal

Thermal Science - Online First

online first only

Numerical simulation of three-dimensional fractional-order convection-diffusion PDEs by a local meshless method

ABSTRACT
In this article, we present an efficient local meshless method for the numerical treatment of three-dimensional convection-diffusion PDEs. The demand of meshless techniques increment because of its meshless nature and simplicity of usage in higher dimensions. This technique approximates the solution on set of uniform and scattered nodes. The space derivatives of the models are discretized by the proposed meshless procedure though the time fractional part is discretized by Liouville-Caputo fractional derivative. Some test problems on regular and irregular computational domains are presented to verify the validity, efficiency and accuracy of the method.
KEYWORDS
PAPER SUBMITTED: 2020-02-25
PAPER REVISED: 2020-05-02
PAPER ACCEPTED: 2020-02-07
PUBLISHED ONLINE: 2020-07-11
DOI REFERENCE: https://doi.org/10.2298/TSCI200225210S
REFERENCES
  1. S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives, Translated from the 1987 Russian original, Gordon and Breach, Yverdon, 1993.
  2. A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Vol. 204 (North-Holland Mathematics Studies), Elsevier (North-Holland) Science Publishers, Amsterdam, London and New York, 2006.
  3. H. M. Srivastava, Fractional-order derivatives and integrals: Introductory overview and recent developments, Kyungpook Mathematical Journal 60 (1) (2020) 73-116.
  4. H. M. Srivastava, R. M. Jena, S. Chakraverty, S. K. Jena, Dynamic response analysis of fractionally-damped generalized Bagley-Torvik equation subject to external loads, Russian Journal of Mathematical Physics 27 (2020) 254-268.
  5. H. Ahmad, T. A. Khan, Variational iteration algorithm-i with an auxiliary parameter for wave-like vibration equations, Journal of Low Frequency Noise, Vibration and Active Control 38 (3-4) (2019) 1113-1124.
  6. L. Mahto, S. Abbas, M. Hafayed, H. M. Srivastava, Approximate controllability of subdiffusion equation with impulsive condition, Mathematics 7 (2) (2019) 190.
  7. X.-J. Yang, H. M. Srivastava, J. A. T. Machado, A new fractional derivative without singular kernel: Application to the modelling of the steady heat flow, Thermal Science 20 (2016) 753-756.
  8. H. Ahmad, A. R. Seadawy, T. A. Khan, P. Thounthong, Analytic approximate solutions for some nonlinear parabolic dynamical wave equations, Journal of Taibah University for Science 14 (1) (2020) 346-358.
  9. Y. El-Dib, Stability analysis of a strongly displacement time-delayed during oscillator using multiple scales homotopy perturbation method, Journal of Applied and Computational Mechanics 4 (4) (2018) 260-274.
  10. R. M. Jena, S. Chakraverty, S. K. Jena, Dynamic response analysis of fractionally damped beams subjected to external loads using homotopy analysis method, Journal of Applied and Computational Mechanics 5 (2) (2019) 355-366.
  11. A. Yokus, H. Durur, H. Ahmad, Hyperbolic type solutions for the couple Boiti-Leon-Pempinelli system, Facta Universitatis, Series: Mathematics and Informatics 35 (2) (2020) 523-531.
  12. A. Yokus, H. Durur, H. Ahmad, S.-W. Yao, Construction of different types analytic solutions for the Zhiber-Shabat equation, Mathematics 8 (6) (2020) 908.
  13. H. M. Srivastava, K. M. Saad, Some new models of the time-fractional gas dynamics equation, Advanced Mathematical Models and Applications 3 (1) (2018) 5-17. 11
  14. M. Bisheh-Niasar, M. A. Ameri, Moving mesh non-standard finite difference method for nonlinear heat transfer in a thin finite rod, Journal of Applied and Computational Mechanics 4 (3) (2018) 161-166.
  15. K. V. Zhukovsky, H. M. Srivastava, Analytical solutions for heat diffusion beyond Fourier law, Applied Mathematics and Computation 293 (2017) 423-437.
  16. H. K. Jassim, D. Baleanu, A novel approach for Korteweg-de Vries equation of fractional order, Journal of Applied and Computational Mechanics 5 (2) (2019) 192-198.
  17. H. Ahmad, Variational iteration method with an auxiliary parameter for solving differential equations of the fifth order, Nonlinear Science Letters A 9 (1) (2018) 27-35.
  18. M. Inc, H. Khan, D. Baleanu, A. Khan, Modified variational iteration method for straight fins with temperature dependent thermal conductivity, Thermal Science 22 (Suppl. 1) (2018) 229-236.
  19. A. Kilicman, Y. Khan, A. Akgul, N. Faraz, E. K. Akgul, M. Inc, Analytic approximate solutions for fluid ow in the presence of heat and mass transfer, Thermal Science 22 (Suppl. 1) (2018) 259-264.
  20. H. M. Srivastava, H. I. Abdel-Gawad, K. M. Saad, Stability of traveling waves based upon the Evans function and Legendre polynomials, Applied Sciences 10 (3) (2020) 846.
  21. X.-J. Yang, H. M. Srivastava, D. F. M. Torres, A. Debbouche, General fractional-order anomalous diffusion with non-singular power-law kernel, Thermal Science 21 (1) (2017) 1-9.
  22. Z. Kucukarslan-Yuzbasi, E. Cavlak-Aslan, M. Inc, D. Baleanu, On exact solutions for new coupled nonlinear models getting evolution of curves in Galilean space, Thermal Science 23 (Suppl. 1) (2019) 227-233.
  23. A. Saravanan, N. Magesh, A comparison between the reduced differential transform method and the Adomian decomposition method for the Newell-Whitehead-Segel equation, Journal of the Egyptian Mathematical Society 21 (3) (2013) 259-265.
  24. A. Akgul, M. S. Hashemi, M. Inc, D. Baleanu, H. Khan, New method for investigating the density-dependent diffusion Nagumo equation, Thermal Science 22 (Suppl. 1) (2018) 143-152.
  25. X.-J. Yang, J. A. T. Machado, H. M. Srivastava, A new numerical technique for solving the local fractional diffusion equation: two-dimensional extended differential transform approach, Applied Mathematics and Computation 274 (2016) 143-151.
  26. X.-J. Yang, F. Gao, H. M. Srivastava, Exact travelling wave solutions for the local fractional two-dimensional Burgers-type equations, Computers & Mathematics with Applications 73 (2) (2017) 203-210.
  27. I. Ahmad, M. Riaz, M. Ayaz, M. Arif, S. Islam, P. Kumam, Numerical simulation of partial differential equations via local meshless method, Symmetry 11 (2019) 257.
  28. M. N. Khan, Sirj-ul-Islam, I. Hussain, I. Ahmad, H. Ahmad, A local meshless method for the numerical solution of space-dependent inverse heat problems, Mathematical Methods in the Applied Sciences (2020). doi: doi.org/10.1002/mma.6439
  29. G. Oguntala, R. Abd-Alhameed, Thermal analysis of convective-radiative fin with temperature-dependent thermal conductivity using Chebychev spectral collocation method, Journal of Applied and Computational Mechanics 4 (2) (2018) 87-94.
  30. H.Wendland, Approximation Scattered Data, Cambridge University Press, Cambridge, London and New York, 2005.
  31. Q. Shen, Local RBF-based differential quadrature collocation method for the boundary layer problems, Engineering Analysis with Boundary Elements 34 (2010) 213-228.
  32. X. Yan, Y. Pan, Performance analysis of heat accumulation of solar thermal generator units by computer numerical simulation, Thermal Science 24 (5B) (2020) 3279-3287.
  33. C. Shu, Differential Quadrature and Its Application in Engineering, Springer-Verlag, Berlin, Heidelberg and New York, 2000.
  34. I. Ahmad, Siraj-ul-Islam, A. Q. M. Khaliq, Local RBF method for multi-dimensional partial differential equations, Computers & Mathematics with Applications 74 (2017) 292-324.
  35. Siraj-ul-Islam, I. Ahmad, A comparative analysis of local meshless formulation for multiasset option models, Engineering Analysis with Boundary Elements 65 (2016) 159-176.
  36. M. N. Khan, I. Ahmad, H. Ahmad, A radial basis function collocation method for spacedependent inverse heat problems, Journal of Applied and Computational Mechanics (2020). doi: doi.org/10.22055/JACM.2020.32999.2123
  37. A. Mohebbi, M. Abbaszadeh, M. Dehghan, The use of a meshless technique based on collocation and radial basis functions for solving the time fractional nonlinear Schrodinger equation arising in quantum mechanics, Engineering Analysis with Boundary Elements 37 (2) (2013) 475-485.
  38. C. C. Piret, E. Hanert, A radial basis functions method for fractional diffusion equations, Journal of Computational Physics 238 (2013) 71-81.
  39. A. Mohebbi, M. Abbaszadeh, M. Dehghan, Solution of two-dimensional modified anomalous fractional sub-diffusion equation via radial basis functions (RBF) meshless method, Engineering Analysis with Boundary Elements 38 (2014) 72-82.
  40. V. R. Hosseini, W. Chen, Z. Avazzadeh, Numerical solution of fractional telegraph equation by using radial basis functions, Engineering Analysis with Boundary Elements 38 (2014) 31-39.
  41. S. Wei, W. Chen, Y.-C. Hon, Implicit local radial basis function method for solving twodimensional time fractional diffusion equations, Thermal Science 19 (1) (2015) S59-S67.
  42. H. R. Ghehsareh, S. H. Bateni, A. Zaghian, A meshfree method based on the radial basis functions for solution of two-dimensional fractional evolution equation, Engineering Analysis with Boundary Elements 61 (2015) 52-60.
  43. M. Aslefallah, E. Shivanian, Nonlinear fractional integro-differential reaction-diffusion equation via radial basis functions, The European Physical Journal Plus 130 (3) (2015) 47.
  44. A. Kumar, A. Bhardwaj, B. R. Kumar, A meshless local collocation method for time fractional diffusion wave equation, Computers & Mathematics with Applications 78 (6) (2019) 1851-1861.
  45. S. Wei, W. Chen, Y. Zhang, H. Wei, R. M. Garrard, A local radial basis function collocation method to solve the variable-order time fractional diffusion equation in a two-dimensional irregular domain, Numerical Methods for Partial Differential Equations 34 (4) (2018) 1209-1223.
  46. M. Dehghan, M. Abbaszadeh, A. Mohebbi, An implicit RBF meshless approach for solving the time fractional nonlinear Sine-Gordon and Klein-Gordon equations, Engineering Analysis with Boundary Elements 50 (2015) 412-434.
  47. Z. Avazzadeh, V. R. Hosseini, W. Chen, Radial basis functions and FDM for solving fractional diffusion-wave equation, Iranian Journal of Science and Technology Transactions A: Science 38 (3) (2014) 205-212.
  48. M. Caputo, Linear models of dissipation whose Q is almost frequency independent. II, Geophysical Journal International 13 (5) (1967) 529-539.