## THERMAL SCIENCE

International Scientific Journal

## Authors of this Paper

,

### Neslihan Ozdemir

online first only

### Modified Laguerre wavelet based Galerkin method for fractional and fractional-order delay differential equations

ABSTRACT
The application of modified Laguerre wavelet with respect to the given conditions by Galerkin method to an approximate solution of fractional and fractional-order delay differential equations is studied in this paper. For the concept of fractional derivative is used Caputo sense by using Riemann-Liouville fractional integral operator. The presented method here is tested on several problems. The approximate solutions obtained by presented method are compared with the exact solutions and is shown to be a very efficient and powerful tool for obtaining approximate solutions of fractional and fractional-order delay differential equations . Some tables and figures are presented to reveal the performance of the presented method.
KEYWORDS
PAPER SUBMITTED: 2018-09-12
PAPER REVISED: 2018-10-03
PAPER ACCEPTED: 2018-10-26
PUBLISHED ONLINE: 2018-12-16
DOI REFERENCE: https://doi.org/10.2298/TSCI180912326S
REFERENCES
1. Iqbal, M. A., et al., Modified Laguerre wavelets method for delay differential equations of fractional- order, Egypt. J. Basic Appl. Sci, (2015),2, 50, pp. 50-54
2. Wang, Z., A numerical method for delayed fractional-order differential equations, Journal of Applied Mathematics, (2013)
3. Momani, S.,Shawagfeh, N., Decomposition method for solving fractional Riccati differential equations, Applied Mathematics and Computation, 182(2) (2006), pp. 1083-1092
4. Shiralashetti, S. C., Kumbinarasaiah, S. , Theoretical study on continuous polynomial wavelet bases through wavelet series collocation method for nonlinear Lane-Emden type equations, Applied Mathematics and Computation, 315(2006), pp. 591-602
5. Cicelia, J. E., Solution of Weighted Residual Problems by Using Galerkin's Method, Indian Journal of Science and Technology, 7(3S) (2014), pp. 52-54
6. Secer, A., et al., Sinc-Galerkin method for approximate solutions of fractional order boundary value problems, Boundary Value Problems, 2013, pp 281
7. Yang, C., Hau, J., Chebyshev wavelets method for solving Bratu's problem, Boundary Value Problems, 2013, pp 1-9
8. Razzaghi, M., Yousefi, S., Legendre wavelets operational matrix of integration, Int. J. Syst. Sci., vol.32 (2001),pp. 495-502
9. Ray, S. S., Gupta, A.K., A numerical investigation of time- fractional modified Fornberg-Whitham equation for analyzing the behavior of water waves, Appl. Math. Comput, vol.266 (2015), pp 135-148
10. Zhou ,F., Xu, X., Numerical solutions for the linear and nonlinear singular boundary value problems using Laguerre wavelets, Adv. Differ. Equ., vol.2016, p.17
11. Celik, I., Haar wavelet approximation for magne to hydrodynamic flow equations ,Appl. Math. Model ,vol37(2013), pp. 3894-390
12. Celik, I., Chebyshev Wavelet cOllocation method for solving generalized Burgers'Huxley equation, Mathematical methods in the applied sciences, 39.3 (2016) ,pp. 366-377
13. Podlubny, I., Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of The Applications, Academic Press, New York,1998.
14. Rehman , M., Khan, R.A., The Legendre wavelet method for solving fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 16(2011), pp 4163-4173
15. Saeed , U., Rehman, M., Hermite wavelet method for fractional delay differential equations, J. Diff. Equ.,(2014)
16. Chen,M. Q., et al., The computation of wavelet-Galerkin approximation on a bounded interval, International journal for numerical methods in engineering, 39(17)(1996), pp 2921-2944
17. Rehman, M., Saeed, U., Gegenbauer wavelets Operational matrix method for fractional differential equations, J. Korean Math. Soc. 52 (2015), No. 5, pp. 1069-1096
18. Daftardar-Gejji, V., et. al. , Solving Fractional Delay Differential Equations: A New Approach, Fractional Calculus and Applied Analysis, (2015), pp.400-418
19. Kulish, V. V., Lage, J. L., Application of fractional calculus to fluid mechanics, Journal of Fluids Engineering, 124(3) (2002), pp. 803-806
20. He, J. H. , Some applications of nonlinear fractional differential equations and their approximations, Bull. Sci. Technol, 15(2) (1999), pp. 86-90
21. Panda, R., Dash, M., Fractional generalized splines and signal processing, Signal Process, (2006), pp. 2340-2350
22. Bohannan, G. W., Analog fractional order controller in temperature and motor control applications, J Vib Control (2008),14:1487-98