THERMAL SCIENCE

International Scientific Journal

Thermal Science - Online First

Authors of this Paper

External Links

online first only

Legendre wavelet operational matrix method for solving fractional differential equations in some special conditions

ABSTRACT
This paper proposes a new technique which rests upon Legendre wavelets for solving linear and non-linear forms of fractional order initial and boundary value problems. In some particular circumstances, a new operational matrix of fractional derivative is generated by utilizing some significant properties of wavelets and orthogonal polynomials. We approached the solution in a finite series with respect to Legendre wavelets and then by using these operational matrices, we reduced the FDEs into a system of algebraic equations. Finally, the introduced tecnique is tested on several illustrative examples. The obtained results demonstrate that this technique is a very impressive and applicable mathematical tool for solving FDEs.
KEYWORDS
PAPER SUBMITTED: 2018-09-20
PAPER REVISED: 2018-10-24
PAPER ACCEPTED: 2019-01-10
PUBLISHED ONLINE: 2019-03-09
DOI REFERENCE: https://doi.org/10.2298/TSCI180920034S
REFERENCES
  1. Mohammadi, F. and Hosseini, M. M., A new Legendre wavelet operational matrix of derivative and its applications in solving the singular ordinary differential equations, Journal of The Franklin Institute, 348 (2011), pp. 1787-1796
  2. Venkatesh, S. G., Ayyaswamy, S. K. and Balachandar, S.R., The Legendre wavelet method for solving initial value problems of Bratu-type, Computers and Mathematics with Applications, 63 (2012), pp. 1287-1295
  3. Mohammadi, F., Hosseini, M. M. and Mohyud-Din, S.T., Legendre wavelet Galerkin method for solving ordinary differential equations with non-analytic solution, International Journal of Systems Science 42 (2011) , 4, pp. 579-585
  4. Mishra, V. and Sabina, Wavelet Galerkin solutions of ordinary differential equations, International Journal of Math. Analysis, 5 (2011), pp. 407-424
  5. Khellat, F. and Yousefi, S.A., The linear Legendre mother wavelets operational matrix of integration and its application, Journal of The Franklin Institute, 343 (2006), pp. 181-190
  6. Mohammadi, F. and Hosseini, M.M., A comparative study of numerical methods for solving quadratic Riccati differential equations, Journal of The Franklin Institute, 348 (2011), pp. 156-164
  7. Miller, K.S. and Ross, B., An introduction to the fractional calculus and fractional differential equations, Wiley, New York, 1993
  8. Oldham, K.B. and Spanier, J., The fractional calculus, Academic Press, New York, 1974
  9. Jafari, H., Yousefi, S.A., Firoozjaee, M.A., Momani, S. and Khalique, C.M., Application of Legendre wavelets for solving fractional differential equations, Computers and Mathematics with Applications, 62 (2011), pp. 1038-1045.
  10. Balaji, S., Legendre wavelet operational matrix method for solution of fractional order Riccati differential equation, Journal of the Eqyptian Mathematical Society, 23 (2015), pp. 263-270
  11. Yi-Ming Chen, Yan-Qiao Wei and Da-Yan Liu, Hao Yu, Numerical solution for a class of nonlinear variable order fractional differential equations with Legendre wavelets, Applied Mathematics Letters, 46 (2015), pp. 83-88
  12. Rehman, M. and Khan, R.A., The Legendre wavelet method for solving fractional differential equations, Commun Nonlinear Sci Numer Simulat, 16 (2011), pp. 4163-4173
  13. Saadatmandi, A. and Dehghan, M., A new operational matrix for solving fractional-order differential equations, Computers and Mathematics with Applications, 59 (2010), pp.1326-1336
  14. Mohammadi, F., Hosseini, M.M. and Mohyud-Din, S.T., A new operational matrix for Legendre wavelets and its applications for solving fractional order boundary value problems, International Journal of Systems Science, 6, (2011), 32, pp. 7371-7378
  15. Khader, M.M., Danaf, T.S. and Hendy, A.S., A computational matrix method for solving systems of high order fractional differential equations, Applied Mathematical Modelling, 37 (2013), pp. 4035-4050
  16. Şenol, M. and Dolapçı, İ.T., On the perturbation-iteration algorithm for fractional differential equations, Journal of King Saud University-Science, 28 (2016), pp. 69-74
  17. Alshbool, M.H.T., Bataineh, A.S., Hashim, I. and Işık, O.R., Solution of fractional-order differential equations based on the operational matrices of new fractional Bernstein functions, Journal of King Saud University-Science, 29 (2017), pp. 1-18
  18. Mohammadi, F., Numerical solution of Bagley-Torvik equation using Chebyshev wavelet operational matrix of fractional derivative, International Journal of Advances in Applied Mathematics and Mechanics, 2, (2014), 1, pp. 83-91
  19. Isah, A. and Phang, C., New operational matrix of derivative for solving non-linear fractional differential equations via Genocchi polynomials, Journal of King Saud University-Science, (2017)
  20. Isah, A. and Phang, C., Genocchi wavelet-like operational matrix and its application for solving non-linear fractional differential equations, Open Phys., 14 (2016), pp. 463-472
  21. Isah, A. and Phang, C., Legendre wavelet operational matrix of fractional derivative through wavelet-polynomial transformation and its applications in solving fractional order differential equations, International Journal of Pure and Applied Mathematics, 105 (2015), pp. 97-114
  22. Doha, E.H., Bhrawy, A.H. and Ezz-Eldien, S.S., A new Jacobi operational matrix: An application for solving fractional differential equations, Applied Mathematical Modelling, 36 (2012), pp. 4931-4943.
  23. Demirci, E. and Ozalp, N., A method for solving differential equations of fractional order, Journal of Computational and Applied Mathematics, 236 (2012), pp. 2754-2762
  24. Song, L. and Wang, W., A new improved Adomian decomposition method and its application to fractional differential equations, Applied Mathematical Modelling, 37 (2013), pp. 1590-1598
  25. Feng, M., Zhang, X. and Ge, W., New existence results for high-order non-linear fractional differential equation with integral boundary conditions, Boundary Value Problems, (2011), doi:10.1155/2011/720702
  26. Kumar, P., Kumar, D. and Rai, K.N., A mathematical model for hyperbolic space-fractional bioheat transfer during thermal therapy, Procedia Engineering, 127 (2015), pp. 56 - 62
  27. Kumar, P., Kumar, D. and Rai, K.N., Numerical study on non-Fourier bioheat transfer during thermal ablation, Procedia Engineering, 127 (2015), pp. 1300 - 1307
  28. Secer, A. and Altun, S., A new operational matrix of fractional derivatives to solve systems of fractional differential equations via Legendre wavelets, Mathematics, 6, (2018), 11, doi:10.3390/math6110238
  29. Secer, A., Alkan, S., Akinlar, M.A. and Bayram, M., Sinc-Galerkin method for approximate solutions of fractional order boundary value problems, Boundary Value Problems, 281 (2013)
  30. Akinlar, M., Secer, A. and Bayram, M., Numerical solution of fractional Benney equation, Appl. Math. Inf. Sci., 8, (2014), 4, pp. 1633-1637
  31. Kurulay, M., Secer, A. and Akinlar, M. A., A new approximate analytical solution of Kuramoto -Sivashinsky equation using homotopy analysis method, Appl. Math. Inf. Sci., 7, (2013), 1, pp. 267-271