Solution to fractional-order Riccati differential equations using Euler wavelet method

Document Type : Article


Department of Mathematical Engineering, Davutpasa Campus, Yildiz Technical University, 34220, Esenler, Istanbul, Turkey.


The fractional-order differential equations (FDEs) have the ability to model the real-life phenomena better in a variety of applied mathematics, engineering disciplines including diffusive transport, electrical networks, electromagnetic theory, probability and so forth. In most cases, there are no analytical solutions therefore a variety of numerical methods have been developed for the solution of the FDEs. In this paper, we derive the numerical solutions of the various fractional-order Riccati type differential equations using the Euler Wavelet Method (EWM). The Euler wavelet operational matrix method converts the fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and efficiency of the technique.


Main Subjects

1. Colinas-Armijo, N., Di Paola, M., and Pinnola, F.P. Fractional characteristic times and dissipated energy in fractional linear viscoelasticity", Commun. Nonlinear Sci. Numer. Simul., 37, pp. 14-30 (2016). 2. Rossikhin, Y.A. and Shitikova M.V. Application of fractional calculus for dynamic problems of solid mechanics: novel trends and recent results", Appl. Mech. Rev., 63(1), pp. 1-52 (2009). 3. Magin, R.L. and Ovadia M. Modeling the cardiac tissue electrode interface using fractional calculus", J. Vib. Control, 14(9-10), pp. 1431-1442 (2008). 4. Sommacal, L., Melchior, P., Oustaloup, A., Cabelguen, J.M., and Ijspeert, A.J. Fractional multi-models of the frog gastrocnemius muscle", J. Vib. Control, 14(9- 10), pp. 1415-1430 (2008). 5. Baillie, R.T. Long memory processes and fractional integration in econometrics", J. Econom., 73(1), pp. 5-59 (1996). 6. Carpinteri, A. and Mainardi, F., Fractals and Fractional Calculus in Continuum Mechanics, Springer- Verlag, Vien, New York (1997). 7. Lima, M.F.M., Machado, J.A.T., and Cris_ostomo, M. Experimental signal analysis of robot impacts in a fractional calculus perspective", J. Adv. Comput. Intell. Intell. Inform., 11, pp. 1079-1085 (2007). 8. Chen, C. and Hsiao, C. Haar wavelet method for solving lumped and distributed-parameter systems", IEE P-Contr. Theor. Appl., 144(1), pp. 87-94 (1997). 9. Karimi, H., Moshiri, B., Lohmann, B., and Maralani, P. Haar wavelet-based approach for optimal control of second-order linear systems in time domain", J. Dyn. Control Syst., 11(2), pp. 237-252 (2005). 10. Sadek, I., Abualrub, T., and Abukhaled, M. A computational method for solving optimal control of a system of parallel beams using Legendre wavelets", Math. Comput. Model, 45(9-10), pp. 1253-1264 (2007). 11. Babolian, E., Masouri, Z., and Hatamzadeh- Varmazyar, S. Numerical solution of nonlinear Volterra-Fredholm integro-di_erential equations via direct method using triangular functions", Comput. Math. Appl., 58(2), pp. 239-247 (2009). 12. Kajani, M. and Vencheh, A. The Chebyshev wavelets operational matrix of integration and product operation matrix", Int J. Comput. Math., 86(7), pp. 1118- 1125 (2008). 13. Razzaghi, M. and Youse_, S. The Legendre wavelets operational matrix of integration", Int. J. Syst. Sci., 32(4), pp. 495-502 (2001). 14. El-Wakil, S.A., Elhanbaly, A., and Abdou, M.A. Adomian decomposition method for solving fractional nonlinear di_erential equations", Appl. Math. Comput., 182(1), pp. 313-324 (2006). 15. Momani, S. and Odibat, Z. Numerical approach to di_erential equations of fractional order", J. Comput. Appl. Math., 207(1), pp. 96-110 (2007). 16. Das, S. Analytical solution of a fractional di_usion equation by variational iteration method", Comput. Math. Appl., 57(3), pp. 483-487 (2009). 17. Gaul, L., Klein, P., and Kemple, S. Damping description involving fractional operators", Mech. Syst. Signal Pr., 5(2), pp. 81-88 (1991). 18. Podlubny, I., Fractional Di_erential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of their Applications, New York, Academic Press (1999). 19. Suarez, L. and Shokooh, A. An eigenvector expansion method for the solution of motion containing fractional derivatives", J. Appl. Mech., 64(3), pp. 629-635 (1997). 20. Kumar, S. A new fractional analytical approach for treatment of a system of physical models using Laplace transform", Sci. Iran. B, 21(5), pp.1693-1699 (2014). 21. Khader, M.M. Application of homotopy perturbation method for solving nonlinear fractional heat-like equations using Sumudu transform", Sci. Iran. B, 24(2), pp. 648-655 (2017). 22. Xu, X. and Xu, D. Legendre wavelets method for approximate solution of fractional-order di_erential equations under multi-point boundary conditions", Int. J. Comput. Math., 95(5), pp. 998-1014 (2018). 23. Wang, Y.X. and Fan, Q.B. The second kind Chebyshev wavelet method for solving fractional di_erential equation", Appl. Math. Comput., 218(17), pp. 8592- 8601 (2012). 1616 A.T. Dincel/Scientia Iranica, Transactions D: Computer Science & ... 26 (2019) 1608{1616 24. Shah, F.A. and Abass, R. Haar wavelet operational matrix method for the numerical solution of fractional order di_erential equations", Nonlinear Engin., 4(4), pp. 203-213 (2015). 25. Rahimkhani, P., Ordokhani, Y., and Babolian, E. Numerical solution of fractional pantograph di_erential equations by using generalized fractional-order Bernoulli wavelet", J. Comput. Appl. Math., 309(1), pp. 493-510 (2017). 26. El-Tawil, M.A., Bahnasawi, A.A., and Abdel-Naby, A. Solving Riccati di_erential equation using Adomian's decomposition method", Appl. Math. Comput., 157(2), pp. 503-514 (2004). 27. Sakar M. Iterative reproducing kernel Hilbertspaces method for Riccati di_erential equations", J. Comput. Appl. Math., 309, pp. 163-174 (2017). 28. Batiha, B., Noorani, M.S.M., and Hashim, I. Application of variational iteration method to general Riccati equation", Int. Math. Forum, 2(56), pp. 2759-2770 (2007). 29. Geng, F., Lin, Y., and Cui, M. A piecewise variational iteration method for Riccati di_erential equations", Comput. Math. Appl., 58(11-12), pp. 2518-2522 (2009). 30. Yuzbas_, S. Numerical solutions of fractional Riccati type di_erential equations by means of the Bernstein polynomials", Appl. Math. Comput., 219(11), pp. 6328-6343 (2013). 31. Mabood, F., Ismail, A.I., and Hashim, I. Application of optimal homotopy asymptotic method for the approximate solution of Riccati equation", Sains Malays., 42(6), pp. 863-867 (2013b). 32. Li, X.Y., Wu, B.Y., and Wang, R.T. Reproducing kernel method for fractional Riccati di_erential equations", Abstr. Appl. Anal., Article ID 970967, 6 pages (2014). 33. Odibat, Z. and Momani, S. Modi_ed homotopy perturbation method: application to quadratic Riccati di_erential equation of fractional order", Chaos Solitons Fractals, 36(1), pp. 167-174 (2008). 34. Khader, M.M. Numerical treatment for solving fractional Riccati di_erential equation", J. Egyptian Math. Soc., 21(1), pp. 32-37 (2013). 35. Sakar, M.G., Akgul, A., and Baleanu, D. On solutions of fractional Riccati di_erential equations", Adv. Di_er. Equ., 39, pp. 1-10 (2017). 36. Beylkin, G., Coifman, R., and Rokhlin, V. Fast wavelet transforms and numerical algorithms", I. Commun. Pure Appl. Math., 44(2), pp. 141-183 (1991). 37. Wang, Y. and Zhu, L. Solving nonlinear Volterra integro-di_erential equations of fractional order by using Euler wavelet method", Adv. Di_er. Equ., 27, pp. 1-16 (2017). 38. He, Y. and Wang, C. Recurrence formulae for Apostol-Bernoulli and Apostol-Euler polynomials", Adv. Di_er. Equ., 209, pp. 1-16 (2012). 39. Kilicman, A. Kronecker operational matrices for fractional calculus and some applications", Appl. Math. Comput., 187(1), pp. 250-265 (2007). 40. Atkinson, K.E., An Introduction to Numerical Analysis, Wiley, New York (1978). 41. Majak, J., Shvartsman, B., Karjust, K., Mikola, M., Haavaj oe, A., and Pohlak, M. On the accuracy of the Haar wavelet discretization method", Compos. Part B, 80, pp. 321-327 (2015). 42. Majak, J., Pohlak, M., Karjust, K., Eerme, M., Kurnitski, J., and Shvartsman, B.S. New higher order Haar wavelet method: Application to FGM structures", Compos. Struct., 201, pp. 71-78 (2018). 43. Tural Polat S.N. The vector-matrix form numerical simulations for time-derivative cellular neural networks", Int. J. Numer. Model., 31(5), pp. 1-13 (2018).