## THERMAL SCIENCE

International Scientific Journal

### 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

**THERMAL SCIENCE** YEAR

**2019**, VOLUME

**23**, ISSUE

**Supplement 1**, PAGES [S203 - S214]

- Miller, K. S., Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, USA, 1993
- Oldham, K. B., Spanier, J., The Fractional Calculus, Academic Press, New York, USA, 1974
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- Secer, A., et al., Sinc-Galerkin Method for Approximate Solutions of Fractional Order Boundary Value Problems, Boundary Value Problems,2013 (2013), 281
- 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
- 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
- Akinlar, M., et al., Numerical Solution of Fractional Benney Equation, Appl. Math. Inf. Sci., 8, (2014), 4, pp. 1633-1637
- 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
- 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
- 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
- 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
- Mishra, V., Sabina, Wavelet Galerkin Solutions of Ordinary Differential Equations, International Journal of Math. Analysis, 5 (2011), 9, pp. 407-424
- 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
- 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
- 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
- Kumar, P., et al., Numerical Study on Non-Fourier Bioheat Transfer during Thermal Ablation, Procedia Engineering, 127 (2015), Dec., pp. 1300-1307
- 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