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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.
PAPER REVISED: 2018-10-24
PAPER ACCEPTED: 2019-01-10
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THERMAL SCIENCE YEAR 2019, VOLUME 23, ISSUE Supplement 1, PAGES [S203 - S214]
  1. Miller, K. S., Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, USA, 1993
  2. Oldham, K. B., Spanier, J., The Fractional Calculus, Academic Press, New York, USA, 1974
  3. Jafari, H., et al., Application of Legendre Wavelets for Solving Fractional Differential Equations, Comput-ers and Mathematics with Applications,62 (2011), 3, pp. 1038-1045
  4. Balaji, S., Legendre Wavelet Operational Matrix Method for Solution of Fractional Order Riccati Differ-ential Equation, Journal of the Eqyptian Mathematical Society,23 (2015), 2, pp. 263-270
  5. Chen, Y.-M., et al., Numerical Solution for a Class of Nonlinear Variable Order Fractional Differential Equations with Legendre Wavelets, Applied Mathematics Letters, 46 (2015), Aug., pp. 83-88
  6. Rehman, M., Khan, R. A., The Legendre Wavelet Method for Solving Fractional Differential Equations, Commun Nonlinear Sci Numer Simulat,16 (2011), 11, pp. 4163-4173
  7. Saadatmandi, A., Dehghan, M., A New Operational Matrix for Solving Fractional-Order Differential Equations, Computers and Mathematics with Applications,59 (2010), 3, pp. 1326-1336
  8. Mohammadi, F., et al., 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
  9. Alshbool, M. H. T., et al., Solution of Fractional-Order Differential Equations Based on the Operational Matrices of New Fractional Bernstein Functions, Journal of King Saud University-Science, 29 (2017), 1, pp. 1-18
  10. Mohammadi, F., Numerical Solution of Bagley-Torvik Equation Using Chebyshev Wavelet Operational Matrix of Fractional Derivative, International Journal of Advances in Applied Mathematics and Mechan-ics,2, (2014), 1, pp. 83-91
  11. Isah, A., Phang, C., New Operational Matrix of Derivative for Solving Non-Linear Fractional Differential Equations Via Genocchi Polynomials, Journal of King Saud University-Science, 31 (2017), 1, pp. 1-7
  12. Isah, A., Phang, C., Genocchi Wavelet-Like Operational Matrix and Its Application for Solving Non-Lin-ear Fractional Differential Equations, Open Phys., 14 (2016), 1, pp. 463-472
  13. Isah, A., Phang, C., Legendre Wavelet Operational Matrix of Fractional Derivative through Wavelet-Poly-nomial Transformation and Its Applications in Solving Fractional Order Differential Equations, Interna-tional Journal of Pure and Applied Mathematics, 105 (2015), 1, pp. 97-114
  14. Doha, E. H., et al., A New Jacobi Operational Matrix: An Application for Solving Fractional Differential Equations, Applied Mathematical Modelling, 36 (2012), 10, pp. 4931-4943
  15. Şenol, M., Dolapci, I. T., On the Perturbation-Iteration Algorithm for Fractional Differential Equations, Journal of King Saud University-Science,28 (2016), 1, pp. 69-74
  16. Secer, A., et al., Sinc-Galerkin Method for Approximate Solutions of Fractional Order Boundary Value Problems, Boundary Value Problems,2013 (2013), 281
  17. Khader, M. M., et al., A Computational Matrix Method for Solving Systems of High Order Fractional Differential Equations, Applied Mathematical Modelling,37 (2013), 6, pp. 4035-4050
  18. Kurulay, M., et al., A New Approximate Analytical Solution of Kuramoto-Sivashinsky Equation Using Homotopy Analysis Method, Appl. Math. Inf. Sci., 7, (2013), 1, pp. 267-271
  19. Akinlar, M., et al., Numerical Solution of Fractional Benney Equation, Appl. Math. Inf. Sci., 8, (2014), 4, pp. 1633-1637
  20. Song, L., Wang, W., A New Improved Adomian Decomposition Method and Its Application to Fractional Differential Equations, Applied Mathematical Modelling,37 (2013), 3, pp. 1590-1598
  21. Mohammadi, F., 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), 8, pp. 1787-1796
  22. Venkatesh, S. G., et al., The Legendre Wavelet Method for Solving Initial Value Problems of Bratu-Type, Computers and Mathematics with Applications,63 (2012), 8, pp. 1287-1295
  23. Mohammadi, F., et al., Legendre Wavelet Galerkin Method for Solving Ordinary Differential Equations with Non-Analytic Solution, International Journal of Systems Science 42 (2011), 4, pp. 579-585
  24. Mishra, V., Sabina, Wavelet Galerkin Solutions of Ordinary Differential Equations, International Journal of Math. Analysis, 5 (2011), 9, pp. 407-424
  25. Khellat, F., Yousefi, S. A., The Linear Legendre Mother Wavelets Operational Matrix of Integration and its Application, Journal of The Franklin Institute,343 (2006), 2, pp. 181-190
  26. Mohammadi, F., Hosseini, M. M., A Comparative Study of Numerical Methods for Solving Quadratic Riccati Differential Equations, Journal of The Franklin Institute, 348 (2011), 2, pp. 156-164
  27. Kumar, P., et al., A Mathematical Model for Hyperbolic Space-Fractional Bioheat Transfer during Ther-mal Therapy, Procedia Engineering, 127 (2015), Dec., pp. 56-62
  28. Kumar, P., et al., Numerical Study on Non-Fourier Bioheat Transfer during Thermal Ablation, Procedia Engineering, 127 (2015), Dec., pp. 1300-1307
  29. Secer, A., Altun, S., A New Operational Matrix of Fractional Derivatives to Solve Systems of Fractional Differential Equations Via Legendre Wavelets, Mathematics, 6, (2018), 11, 238

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