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Fractional-order generalized Legendre wavelets and their applications to fractional Riccati differential equations

  • Boonrod Yuttanan , Mohsen Razzaghi and Thieu N. Vo EMAIL logo
Published/Copyright: May 20, 2021
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International Journal of Nonlinear Sciences and Numerical Simulation
From the journal International Journal of Nonlinear Sciences and Numerical Simulation Volume 24 Issue 1

Abstract

In the present paper, fractional-order generalized Legendre wavelets (FOGLWs) are introduced. We apply the FOGLWs for solving fractional Riccati differential equation. By using the hypergeometric function, we obtain an exact formula for the Riemann–Liouville fractional integral operator (RLFIO) of the FOGLWs. By using this exact formula and the properties of the FOGLWs, we reduce the solution of the fractional Riccati differential equation to the solution of an algebraic system. This algebraic system can be solved effectively. This method gives very accurate results. The given numerical examples support this claim.

Keywords: fractional Riccati differential equation; fractional-order Legendre wavelet; hypergeometric function
Corresponding author: Thieu N. Vo, Fractional Calculus, Optimization and Algebra Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam, E-mail:

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

References

[1] U. N. Katugampola, “A new approach to generalized fractional derivatives,” Bull. Math. Anal. Appl., vol. 6, no. 4, pp. 1–15, 2014.Search in Google Scholar

[2] P. Rahimkhani and Y. Ordokhani, “A numerical scheme based on Bernoulli wavelets and collocation method for solving fractional partial differential equations with Dirichlet boundary conditions,” Numer. Methods Part. Differ. Equ., vol. 35, no. 1, pp. 34–59, 2019. https://doi.org/10.1002/num.22279.Search in Google Scholar

[3] P. Rahimkhani and Y. Ordokhani, “Numerical investigation of distributed-order fractional optimal control problems via Bernstein wavelets,” Optim. Contr. Appl. Methods, vol. 42, no. 1, pp. 355–373, 2020. https://doi.org/10.1002/oca.2679.Search in Google Scholar

[4] Y. Ordokhani, P. Rahimkhani, and E. Babolian, “Application of fractional-order Bernoulli functions for solving fractional Riccati differential equation,” Int. J. Nonlinear Anal. Appl., vol. 8, no. 2, pp. 277–292, 2017.Search in Google Scholar

[5] M. M. Khader, “Numerical solutions for the problem of the boundary layer flow of a Powell–Eyring fluid over an exponential sheet using the spectral relaxation method,” Indian J. Phys., vol. 94, pp. 1369–1374, 2020. https://doi.org/10.1007/s12648-019-01583-8.Search in Google Scholar

[6] M. M. Khader, “The numerical solution for BVP of the liquid film flow over an unsteady stretching sheet with thermal radiation and magnetic field using the finite element method,” Int. J. Mod. Phys. C, vol. 30, no. 11, p. 1950080, 2019. https://doi.org/10.1142/s0129183119500803.Search in Google Scholar

[7] M. M. Khader and R. P. Sharma, “Evaluating the unsteady MHD micropolar fluid flow past stretching/shirking sheet with heat source and thermal radiation: implementing fourth order predictor–corrector FDM,” Math. Comput. Simulat., vol. 181, pp. 333–350, 2021. https://doi.org/10.1016/j.matcom.2020.09.014.Search in Google Scholar

[8] M. M. Khader, “Using the generalized Adams-Bashforth-Moulton method for obtaining the numerical solution of some variable-order fractional dynamical models,” Int. J. Nonlinear Sci. Numer. Stimul., vol. 22, no. 1, pp. 93–98, 2021. https://doi.org/10.1515/ijnsns-2019-0307.Search in Google Scholar

[9] I. Podlubny, Fractional Differential Equations, San Diego, Academic Press, 1999.Search in Google Scholar

[10] P. Rahimkhani, Y. Ordokhani, and E. Babolian, “Fractional-order Legendre wavelets and their applications for solving fractional-order differential equations with initial/boundary conditions,” Comput. Methods Differ. Equ., vol. 5, no. 2, pp. 117–140, 2017.Search in Google Scholar

[11] P. Rahimkhani and Y. Ordokhani, “Generalized fractional-order Bernoulli–Legendre functions: an effective tool for solving two-dimensional fractional optimal control problems,” IMA J. Math. Contr. Inf., vol. 36, no. 1, pp. 185–212, 2019. https://doi.org/10.1093/imamci/dnx041.Search in Google Scholar

[12] M. H. Heydari, M. R. Hooshmandasl, and F. Mohammadi, “Legendre wavelets method for solving fractional partial differential equations with Dirichlet boundary conditions,” Appl. Math. Comput., vol. 276, pp. 267–276, 2014. https://doi.org/10.1016/j.amc.2014.02.047.Search in Google Scholar

[13] Y. Li and W. Zhao, “Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations,” Appl. Math. Comput., vol. 216, pp. 2276–2285, 2010. https://doi.org/10.1016/j.amc.2010.03.063.Search in Google Scholar

[14] L. Zhu and Q. Fan, “Solving fractional nonlinear Fredholm integro-differential equations by the second kind Chebyshev wavelet,” Commun. Nonlinear Sci. Numer. Simulat., vol. 17, pp. 2333–2341, 2012. https://doi.org/10.1016/j.cnsns.2011.10.014.Search in Google Scholar

[15] S. Mashayekhi and M. Razzaghi, “Numerical solution of distributed order fractional differential equations by hybrid functions,” J. Comput. Phys., vol. 315, pp. 169–181, 2016. https://doi.org/10.1016/j.jcp.2016.01.041.Search in Google Scholar

[16] S. Mashayekhi and M. Razzaghi, “Numerical solution of the fractional Bagley–Torvik equation by using hybrid functions approximation,” Math. Methods Appl. Sci., vol. 39, no. 3, pp. 353–365, 2016. https://doi.org/10.1002/mma.3486.Search in Google Scholar

[17] F. Mohammadi and C. Cattani, “Fractional-order Legendre wavelet Tau method for solving fractional differential equations,” J. Comput. Appl. Math., vol. 339, pp. 306–316, 2018. https://doi.org/10.1016/j.cam.2017.09.031.Search in Google Scholar

[18] S. Kazem, S. Abbasbandy, and S. Kumar, “Fractional-order Legendre functions for solving fractional-order differential equations,” Appl. Math. Model., vol. 37, no. 7, pp. 5498–5510, 2013. https://doi.org/10.1016/j.apm.2012.10.026.Search in Google Scholar

[19] A. H. Bhrawy, Y. A. Alhamed, and D. Baleanu, “New spectral techniques for systems of fractional differential equations using fractional-order generalized Laguerre orthogonal functions,” Fract. Calc. Appl. Anal., vol. 17, pp. 1138–1157, 2014. https://doi.org/10.2478/s13540-014-0218-9.Search in Google Scholar

[20] P. Rahimkhani, Y. Ordokhani, and E. Babolian, “A new operational matrix based on Bernoulli wavelets for solving fractional delay differential equations,” Numer. Algorithm., vol. 74, pp. 223–245, 2017. https://doi.org/10.1007/s11075-016-0146-3.Search in Google Scholar

[21] P. Rahimkhani, Y. Ordokhani, and E. Babolian, “Numerical solution of fractional pantograph differential equations by using generalized fractional-order Bernoulli wavelet,” J. Comput. Appl. Math., vol. 309, pp. 493–510, 2017. https://doi.org/10.1016/j.cam.2016.06.005.Search in Google Scholar

[22] H. Singh and H. M. Srivastava, “Jacobi collocation method for the approximate solution of some fractional-order Riccati differential equations with variable coefficients,” Phys. A, vol. 523, pp. 1130–1149, 2019. https://doi.org/10.1016/j.physa.2019.04.120.Search in Google Scholar

[23] F. Ghomanjani and E. Khorram, “Approximate solution for quadratic Riccati differential equation,” J. Taibah Univ. Sci., vol. 11, no. 2, pp. 246–250, 2017. https://doi.org/10.1016/j.jtusci.2015.04.001.Search in Google Scholar

[24] M. G. Sakar, “Iterative reproducing kernel Hilbert spaces method for Riccati differential equations,” J. Comput. Appl. Math., vol. 309, pp. 163–174, 2017. https://doi.org/10.1016/j.cam.2016.06.029.Search in Google Scholar

[25] B. S. H. Kashkari and M. I. Syam, “Fractional-order Legendre operational matrix of fractional integration for solving the Riccati equation with fractional-order,” Appl. Math. Comput., vol. 290, pp. 281–291, 2016. https://doi.org/10.1016/j.amc.2016.06.003.Search in Google Scholar

[26] M. M. Khader and M. Adel, “Numerical approach for solving the Riccati and logistic equations via QLM-rational Legendre collocation method,” Comput. Appl. Math., vol. 39, p. 166, 2020. https://doi.org/10.1007/s40314-020-01207-6.Search in Google Scholar

[27] M. M. Khader, “Numerical treatment for solving fractional Riccati differential equation,” J. Egyptian Math. Soc., vol. 21, no. 1, pp. 32–37, 2013. https://doi.org/10.1016/j.joems.2012.09.005.Search in Google Scholar

[28] W. T. Reid, Riccati Differential Equations, New York, Academic Press, 1972.Search in Google Scholar

[29] C. S. Sin, “Well-posedness of general Caputo-type fractional differential equations,” Fract. Calc. Appl. Anal., vol. 21, no. 3, pp. 819–832, 2018. https://doi.org/10.1515/fca-2018-0043.Search in Google Scholar

[30] S. Esmaeili, M. Shamsi, and Y. Luchko, “Numerical solution of fractional differential equations with a collocation method based on Muntz polynomials,” Comput. Math. Appl., vol. 62, pp. 918–929, 2011. https://doi.org/10.1016/j.camwa.2011.04.023.Search in Google Scholar

[31] M. I. Syam, H. Siyyam, and I. Al-Subaihi, “Tau-path following method for solving the Riccati equation with fractional order,” J. Comput. Methods Phys., vol. 2014, nos 1–7, p. 207916, 2014. https://doi.org/10.1155/2014/207916.Search in Google Scholar

[32] N. Khan, A. Ara, and M. Jamil, “An efficient approach for solving the Riccati equation with fractional orders,” Comput. Math. Appl., vol. 61, pp. 2683–2689, 2011. https://doi.org/10.1016/j.camwa.2011.03.017.Search in Google Scholar

[33] H. Jafari, N. Kadkhoda, H. Tajadodi, and S. A. Hosseini Matikolai, “Homotopy perturbation Padé technique for solving fractional Riccati differential equations,” Int. J. Nonlinear Sci. Numer. Stimul., vol. 11, pp. 271–275, 2010. https://doi.org/10.1515/ijnsns.2010.11.s1.271.Search in Google Scholar

[34] M. Merdan, “On the solutions fractional Riccati differential equation with modified Riemann–Liouville derivative,” Int. J. Differ. Equ., vol. 2012, pp. 1–17, 2012. https://doi.org/10.1155/2012/346089.Search in Google Scholar

[35] P. Rahimkhani and Y. Ordokhani, “Numerical solution a class of 2D fractional optimal control problems by using 2D Münzt-Legendre wavelets,” Optim. Contr. Appl. Methods, vol. 39, no. 6, pp. 1916–1934, 2018. https://doi.org/10.1002/oca.2456.Search in Google Scholar

[36] P. Rahimkhani, Y. Ordokhani, and E. Babolian, “Fractional-order Bernoulli wavelets and their applications,” Appl. Math. Model., vol. 40, pp. 8087–8107, 2016. https://doi.org/10.1016/j.apm.2016.04.026.Search in Google Scholar

[37] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, New York, Wiley, 1993.Search in Google Scholar

[38] J. S. Gu and W. S. Jiang, “The Haar wavelets operational matrix of integration,” Int. J. Syst. Sci., vol. 27, pp. 623–628, 1996.10.1080/00207729608929258Search in Google Scholar

[39] Z. M. Odibat and N. T. Shawagfeh, “Generalized Taylors formula,” Appl. Math. Comput., vol. 186, no. 1, pp. 286–293, 2007. https://doi.org/10.1016/j.amc.2006.07.102.Search in Google Scholar

[40] S. S. Ezz-Eldien, J. A. T. Machado, Y. Wang, and A. A. Aldraiweesh, “An algorithm for the approximate solution of the fractional Riccati differential equation,” Int. J. Nonlinear Sci. Numer. Stimul., vol. 20, pp. 661–674, 2019. https://doi.org/10.1515/ijnsns-2018-0146.Search in Google Scholar

[41] C. Bota and B. Caruntu, “Analytical approximate solutions for quadratic Riccati differential equation of fractional order using the polynomial least squares method,” Chaos, Solit. Fractals, vol. 102, pp. 339–345, 2017. https://doi.org/10.1016/j.chaos.2017.05.002.Search in Google Scholar

[42] Z. Meng, M. Yi, J. Huang, and L. Song, “Numerical solutions of nonlinear fractional differential equations by alternative Legendre polynomials,” Appl. Math. Comput., vol. 336, pp. 454–464, 2019.10.1016/j.amc.2018.04.072Search in Google Scholar

[43] S. Momani and N. Shawagfeh, “Decomposition method for solving fractional Riccati differential equations,” Appl. Math. Comput., vol. 182, pp. 1083–1092, 2006. https://doi.org/10.1016/j.amc.2006.05.008.Search in Google Scholar

[44] S. H. Hosseinnna, A. Ranjbar, and S. Momani, “Using an enhanced homotopy perturbation method in fractional differential equations via deforming the linear part,” Comput. Math. Appl., vol. 56, no. 12, pp. 3138–3149, 2008.10.1016/j.camwa.2008.07.002Search in Google Scholar

[45] Z. M. Odibat and S. Momani, “Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order,” Chaos, Solit. Fractals, vol. 36, no. 1, pp. 167–174, 2008. https://doi.org/10.1016/j.chaos.2006.06.041.Search in Google Scholar

[46] S. Yuzbasi, “Numerical solutions of fractional Riccati type differential equations by means of the Bernstein polynomials,” App. Math. Comput., vol. 219, no. 11, pp. 6328–6343, 2013.10.1016/j.amc.2012.12.006Search in Google Scholar
Received: 2020-06-22
Revised: 2021-02-11
Accepted: 2021-04-12
Published Online: 2021-05-20
Published in Print: 2023-02-23

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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https://doi.org/10.1515/ijnsns-2020-0137
Keywords for this article
fractional Riccati differential equation; fractional-order Legendre wavelet; hypergeometric function

Articles in the same Issue

  1. Frontmatter
  2. Original Research Articles
  3. Modeling and assessment of the flow and air pollutants dispersion during chemical reactions from power plant activities
  4. Stochastic dynamics of dielectric elastomer balloon with viscoelasticity under pressure disturbance
  5. Unsteady MHD natural convection flow of a nanofluid inside an inclined square cavity containing a heated circular obstacle
  6. Fractional-order generalized Legendre wavelets and their applications to fractional Riccati differential equations
  7. Battery discharging model on fractal time sets
  8. Adaptive neural network control of second-order underactuated systems with prescribed performance constraints
  9. Optimal control for dengue eradication program under the media awareness effect
  10. Shifted Legendre spectral collocation technique for solving stochastic Volterra–Fredholm integral equations
  11. Modeling and simulations of a Zika virus as a mosquito-borne transmitted disease with environmental fluctuations
  12. Mathematical analysis of the impact of vaccination and poor sanitation on the dynamics of poliomyelitis
  13. Anti-sway method for reducing vibrations on a tower crane structure
  14. Stable soliton solutions to the time fractional evolution equations in mathematical physics via the new generalized G ′ / G -expansion method
  15. Convergence analysis of online learning algorithm with two-stage step size
  16. An estimative (warning) model for recognition of pandemic nature of virus infections
  17. Interaction among a lump, periodic waves, and kink solutions to the KP-BBM equation
  18. Global exponential stability of periodic solution of delayed discontinuous Cohen–Grossberg neural networks and its applications
  19. An efficient class of fourth-order derivative-free method for multiple-roots
  20. Numerical modeling of thermal influence to pollutant dispersion and dynamics of particles motion with various sizes in idealized street canyon
  21. Construction of breather solutions and N-soliton for the higher order dimensional Caudrey–Dodd–Gibbon–Sawada–Kotera equation arising from wave patterns
  22. Delay-dependent robust stability analysis of uncertain fractional-order neutral systems with distributed delays and nonlinear perturbations subject to input saturation
  23. Construction of complexiton-type solutions using bilinear form of Hirota-type
  24. Inverse estimation of time-varying heat transfer coefficients for a hollow cylinder by using self-learning particle swarm optimization
  25. Infinite line of equilibriums in a novel fractional map with coexisting infinitely many attractors and initial offset boosting
  26. Lump solutions to a generalized nonlinear PDE with four fourth-order terms
  27. Quantum motion control for packaging machines
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