International Scientific Journal


In this work, we consider variable order diffusion-wave equations. We choose variable order derivative in the Caputo sense. First, we approximate the unknown functions and its derivatives using Bernstein basis. Then, we obtain operational matrices based on Bernstein polynomials. Finally, with the help of these operational matrices and collocation method, we can convert variable order diffusion-wave equations to an algebraic system. Few examples are given to demonstrate the accuracy and the competence of the presented technique.
PAPER REVISED: 2019-08-30
PAPER ACCEPTED: 2019-09-10
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THERMAL SCIENCE YEAR 2019, VOLUME 23, ISSUE Supplement 6, PAGES [S2063 - S2071]
  1. Bhrawy, A.H., et al., A review of operational matrices and spectral techniques for fractional calculus, Nonlinear Dyn, 81 (2016), pp. 1023-1052
  2. Caputo, M., Linear models of dissipation whose Q is almost frequency, Part II. Geophys. J. R. Astron. Soc., 13 (1967), pp. 529-539
  3. Chen, W., A speculative study of 2/3-order fractional Laplacian modeling of turbulence: some thoughts and conjectures, Chaos: An Interdisciplinary Journal of Nonlinear Science, 16 (2006), 2, pp. 120-126
  4. Chen, W., Anomalous diffusion modeling by fractal and fractional derivatives, Computers, Mathematics with Applications, 59 (2010), 5, pp. 1754-1758
  5. Cooper, G.R.J., Cowan, D.R., Filtering using variable order vertical derivatives, Computers and Geosciences, 30 (2004), pp. 455-459
  6. Rossikhin, Y.A., Shitikova, M.V., Application of fractional derivatives to the analysis of damped vibrations of viscoelastic single mass systems, Acta Mechanica, 120 (1997), 1, pp. 109-125
  7. Vazquez, L., Mendes, R.V., Fractionally coupled solutions of the diffusion equation, Appl. Math. Comput., 141 (2003), pp. 125-130
  8. Das, S., et al., Numerical solution of fractional order advection-reaction diffusion equation, Thermal Science, 22 (2018), 1, pp. 34-34
  9. Firoozjaee, M.A., Yousefi, S.A., A numerical approach for fractional partial differential equations by using Ritz approximation, Applied Mathematics and Computation, 338 (2018), pp. 711-721
  10. Heydari, M.H., et al., Legendre wavelets method for solving fractional partial differential equations with Dirichlet boundary conditions, Applied Mathematics and Computation, 234 (2014), pp. 267-276
  11. Hosseini, V.R., et al., Numerical solution of fractional telegraph equation by using radial basis functions, Engineering Analysis with Boundary Elements, 38 (2014), pp. 31-39
  12. Jafari, H., et al., Homotopy analysis method for solving Abel differential equation of fractional order, Central European Journal of Physics, 11 (2013), 10, pp. 1523-1527
  13. Körpinar, Z., On numerical solutions for the Caputo-Fabrizio fractional heat-like equation, Thermal Science, 22 (2017), 1, pp. 274-274
  14. Rahimkhani, P., et al., Fractional-order Bernoulli functions and their applications in solving fractional Fredholem-Volterra integro-differential equations, Applied Numerical Mathematics, 122 (2017), pp. 66-81
  15. Yang, A.M., On steady heat flow problem involving Yang-Srivastava-Machado fractional derivative without singular kernel, Thermal Science, 20 (2016), 3, pp. 717-721
  16. Yang, X.J., Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat-transfer problems, Thermal Science, 21 (2016), 12, pp. 1161-1171
  17. Samko, S.G., Ross, B., Integration and differentiation to a variable fractional order, Integral Transforms and Special Functions, 1 (1993), 4, pp. 277-300
  18. Xu, Y., Suat Ertürk, V., A Finite Difference Technique For Solving Variable-Order Fractional Integro-Differential Equations, Bulletin of the Iranian Mathematical Society, 40 (2014), 3, pp. 699- 712
  19. Jafari, H., et al., A numerical approach for solving variable order differential equations based on Bernstein polynomials, Computational and Mathematical Methods, (2019),
  20. Li, X., Wu, B., A numerical technique for variable fractional functional boundary value problems, Appl. Math. Lett., (2014), pp. 1-11
  21. Yang, J., H. Yao, B. Wu, An efficient numerical method for variable order fractional functional differential equation, Applied Mathematics Letters, 76 (2018), pp. 221-226
  22. Ganji, R.M., Jafari, H., A numerical approach for multi-variable orders differential equations using Jacobi polynomials, International Journal of Applied and Computational Mathematics, 5 (2019), 4, doi="10.1007/s40819-019-0610-6"
  23. Hassani, H., Naraghirad, E., A new computational method based on optimization scheme for solving variable-order time fractional Burgers equation, Mathematics and Computers in Simulation, (2019), pp. 1-20
  24. Heydari, M.H., et al., A wavelet approach for solving multi-term variable-order time fractional diffusion-wave equation, Applied Mathematics and Computation, 341 (2019), pp. 215-228
  25. Jiang, W., Guo, B., A new numerical method for solving two-dimensional variable-order anomalous sub-diffusion equation, Thermal Science, 20 (2016), pp. 701-710
  26. Doha, E., et al., Spectral technique for solving variable-order fractional Volterra integrodifferential equations, Numerical Methods for Partial Differential Equations, (2018), "10.1002/num.22233"
  27. Ganji, R.M., Jafari, H., Numerical solution of variable order integro-differential equations, Advanced Mathematical Models & Applications, 4 (2019), 1, pp. 64-69
  28. Bhrawya, A.H., Zakyc, M.A., A method based on the jacobi tau approximation for solving multi-term time-space fractional partial differential equations, J. Comput. Phys., 281 (2015), pp. 876- 895
  29. Gasca, M., Sauer, T., On the history of multivariate polynomial interpolation, J. Comput. Appl. Math., 122 (2000), pp. 23-35
  30. Villiers, J. de., Mathematics of Approximation, Atlantis Press, 2012.

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