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Numerical Solution of Multi-Order Fractional Differential Equations Using Generalized Sine-Cosine Wavelets

Year 2018, , 215 - 225, 20.12.2018
https://doi.org/10.32323/ujma.427381

Abstract

In this work, we propose a numerical method based on the generalized sine-cosine wavelets for solving multi-order fractional differential equations. After introducing generalized sine-cosine wavelets, the operational matrix of Riemann-Liouville fractional integration is constructed using the properties of the block-pulse functions. The fractional derivative in the problem is considered in the Caputo sense. This method reduces the considered problem to the problem of solving a system of nonlinear algebraic equations. Finally, some examples are included to demonstrate the applicability of the new approach.

References

  • [1] B. Ross, The development of fractional calculus 1695–1900, Hist. Math., 4(1) (1977), 75–89.
  • [2] K. S. Miller, B. Ross, An Introduction to the fractional calculus and fractional differential equations, New York, Wiley, 1993.
  • [3] K. B. Oldham, J. Spanier, The fractional calculus, New York, Academic Press, 1974.
  • [4] R. L. Bagley, P. J. Torvik, Fractional calculus in the transient analysis of viscoelastically damped structures, AIAA J., 23(6) (1985), 918–925.
  • [5] R. T. Baillie, Long memory processes and fractional integration in econometrics, J. Econometr. 73(1) (1996), 5–59.
  • [6] F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics, fractals and fractional calculus in continuum mechanics, A. Carpinteri and F. Mainardi, Eds. Vienna, Springer-Verlag, (1997), 291–348.
  • [7] Y. A. Rossikhin, M. V. Shitikova, Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Appl. Mech. Rev., 50(1) (1997), 15–67.
  • [8] K. B. Oldham, Fractional differential equations in electrochemistry, Adv. Eng. Softw., 41(1) (2010), 9–12.
  • [9] V. S. Ert ürk, Z. M. Odibat, S. Momani, An approximate solution of a fractional order differential equation model of human T-cell lymphotropic virus I (HTLV-I) infection of CD4+ T-cells, Comput. Math. Appl., 62(3) (2011), 996–1002.
  • [10] S. A. El-Wakil, E. M. Abulwafa, E. K. El-Shewy, A. A. Mahmoud, Ion-acoustic waves in unmagnetized collisionless weakly relativistic plasma of warm-ion and isothermal-electron using time-fractional KdV equation, Adv. Space Res., 49(12) (2012), 1721–1727.
  • [11] S. Momani, K. Al-Khaled, Numerical solutions for systems of fractional differential equations by the decomposition method, Appl. Math. Comput., 162 (2005), 1351–1365.
  • [12] Z. M. Odibat, S. Momani, Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. Nonlinear Sci. Numer. Simul., 7(1) (2006), 27–34.
  • [13] A. Arikoglu, I. Ozkol, Solution of fractional integro-differential equations by using fractional differential transform method, Chaos Solitons Fractals, 40(2) (2009), 521–529.
  • [14] A. Saadatmandi, M. Dehghan, A new operational matrix for solving fractional order differential equations, Comput. Math. Appl., 59 (2010), 1326–1336.
  • [15] M. Dehghan, J. Manafian, A. Saadatmandi, Solving nonlinear fractional partial differential equations using the homotopy analysis method, Numer. Methods Partial Differential Equations, 26 (2010), 448–479.
  • [16] Z. Odibat, N. Shawagfeh, Generalized Taylor’s formula, Appl. Math. Comput., 186 (2007), 286–293.
  • [17] Z. Odibat, S. Momani, Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order, Chaos Solitons Fractals, 36(1) (2008), 167–174.
  • [18] I. Podlubny, The Laplace transform method for linear differential equations of the fractional order, 1997, eprint arXiv:funct-an/9710005.
  • [19] I. Daubechies, Ten Lectures on Wavelets. CBMS-NFS Series in Applied Mathematics, SIAM, Philadelphia, PA, 1992.
  • [20] I. Daubechies, J. C. Lagarias, Two-scale difference equations II. local regularity, infinite products of matrices and fractals, SIAM J. Math. Anal., 23(4) (1992), 1031–1079.
  • [21] S. Mallat, A theory of multiresolution signal decomposition: the wavelet representation, IEEE Trans. Pattern Anal. Mach. Intell., 11(7) (1989), 674–693.
  • [22] I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math., 41(7) (1988), 909–996.
  • [23] X. Li, Numerical solution of fractional differential equations using cubic B-spline wavelet collocation method, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 3934–3946.
  • [24] Y. M. Chen, M. X. Yi, C. X. Yu, Error analysis for numerical solution of fractional differential equation by Haar wavelets method, J. Comput. Sci., 3(5) (2012), 367–373.
  • [25] Y. L. Li, Solving a nonlinear fractional differential equation using Chebyshev wavelets, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 2284–2292.
  • [26] H. Jafari, S. A. Yousefi, M. A. Firoozjaee, S. Momani, C.M. Khalique, Application of Legendre wavelets for solving fractional differential equations, Comput. Appl. Math., 62 (2011), 1038–1045.
  • [27] P. Rahimkhani, Y. Ordokhani, E. Babolian, Numerical solution of fractional pantograph differential equations by using generalized fractional-order Bernoulli wavelet, J. Comput. Appl. Math., 309 (2017), 493–510.
  • [28] M. Razzaghi, S. Yousefi, Sine-cosine wavelets operational matrix of integration and its applications in the calculus of variations, Int. J. Syst. Sci., 33 (2002), 805–810.
  • [29] M. Tavassoli Kajani, M. Ghasemi, E.Babolian, Numerical solution of linear integro-differential equation by using sine-cosine wavelets, Appl. Math. Comput., 180 (2006), 569–574.
  • [30] M. Ghasemi, E. Babolian, M. Tavassoli Kajani, Numerical solution of linear Fredholm integral equations using sine-cosine wavelets, Int. J. Comput. Math., 84 (2007), 979–987.
  • [31] N. Irfan, A. H. Siddiqi, Sine-cosine wavelets approach in numerical evaluation of Hankel transform for seismology, Appl. Math. Model., 40 (2016), 4900–4907.
  • [32] I. Podlubny, Fractional Differential Equations, Math. Sci. Eng., 198, Academic Press, 1999.
  • [33] O. Christensen, K. L. Christensen, Approximation theory: from Taylor polynomial to wavelets, Birkhauser, Boston, 2004.
  • [34] D. Gottlieb, S.A. Orszag, Numerical analysis of spectral methods, SIAM, Philadelphia, PA, 1997.
  • [35] Y. Wang, T. Yin, L. Zhu, Sine-cosine wavelet operational matrix of fractional order integration and its applications in solving the fractional order Riccati differential equations, Adv. Differ. Equ., (2017), 2017: 222.
  • [36] A. Kilicman, Z. A. A. Al Zhour, Kronecker operational matrices for fractional calculus and some applications, Appl. Math. Comput., 187 (2007), 250–265.
  • [37] S. K. Damarla, M. Kundu, Numerical solution of multi-order fractional differential equations using generalized triangular function operational matrices, Appl. Math. Comput., 263 (2015), 189 – 203.
  • [38] A. E. M. El-Mesiry, A. M. A. El-Sayed, H. A. A. El-Saka, Numerical methods for multi-term fractional (arbitrary) orders differential equations, Appl. Math. Comput., 160 (2005), 683–699.
  • [39] M. Seifollahi, A. S. Shamloo, Numerical solution of nonlinear multi-order fractional differential equations by operational matrix of Chebyshev polynomials, World Appl. Program., 3 (2013), 85–92.
  • [40] A. H. Bhrawy, M. M. Tharwat, M.A. Alghamdi, A new operational matrix of fractional integration for shifted Jacobi polynomials, Bull. Malays. Math. Sci. Soc., 37 (2014), 983–995.
Year 2018, , 215 - 225, 20.12.2018
https://doi.org/10.32323/ujma.427381

Abstract

References

  • [1] B. Ross, The development of fractional calculus 1695–1900, Hist. Math., 4(1) (1977), 75–89.
  • [2] K. S. Miller, B. Ross, An Introduction to the fractional calculus and fractional differential equations, New York, Wiley, 1993.
  • [3] K. B. Oldham, J. Spanier, The fractional calculus, New York, Academic Press, 1974.
  • [4] R. L. Bagley, P. J. Torvik, Fractional calculus in the transient analysis of viscoelastically damped structures, AIAA J., 23(6) (1985), 918–925.
  • [5] R. T. Baillie, Long memory processes and fractional integration in econometrics, J. Econometr. 73(1) (1996), 5–59.
  • [6] F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics, fractals and fractional calculus in continuum mechanics, A. Carpinteri and F. Mainardi, Eds. Vienna, Springer-Verlag, (1997), 291–348.
  • [7] Y. A. Rossikhin, M. V. Shitikova, Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Appl. Mech. Rev., 50(1) (1997), 15–67.
  • [8] K. B. Oldham, Fractional differential equations in electrochemistry, Adv. Eng. Softw., 41(1) (2010), 9–12.
  • [9] V. S. Ert ürk, Z. M. Odibat, S. Momani, An approximate solution of a fractional order differential equation model of human T-cell lymphotropic virus I (HTLV-I) infection of CD4+ T-cells, Comput. Math. Appl., 62(3) (2011), 996–1002.
  • [10] S. A. El-Wakil, E. M. Abulwafa, E. K. El-Shewy, A. A. Mahmoud, Ion-acoustic waves in unmagnetized collisionless weakly relativistic plasma of warm-ion and isothermal-electron using time-fractional KdV equation, Adv. Space Res., 49(12) (2012), 1721–1727.
  • [11] S. Momani, K. Al-Khaled, Numerical solutions for systems of fractional differential equations by the decomposition method, Appl. Math. Comput., 162 (2005), 1351–1365.
  • [12] Z. M. Odibat, S. Momani, Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. Nonlinear Sci. Numer. Simul., 7(1) (2006), 27–34.
  • [13] A. Arikoglu, I. Ozkol, Solution of fractional integro-differential equations by using fractional differential transform method, Chaos Solitons Fractals, 40(2) (2009), 521–529.
  • [14] A. Saadatmandi, M. Dehghan, A new operational matrix for solving fractional order differential equations, Comput. Math. Appl., 59 (2010), 1326–1336.
  • [15] M. Dehghan, J. Manafian, A. Saadatmandi, Solving nonlinear fractional partial differential equations using the homotopy analysis method, Numer. Methods Partial Differential Equations, 26 (2010), 448–479.
  • [16] Z. Odibat, N. Shawagfeh, Generalized Taylor’s formula, Appl. Math. Comput., 186 (2007), 286–293.
  • [17] Z. Odibat, S. Momani, Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order, Chaos Solitons Fractals, 36(1) (2008), 167–174.
  • [18] I. Podlubny, The Laplace transform method for linear differential equations of the fractional order, 1997, eprint arXiv:funct-an/9710005.
  • [19] I. Daubechies, Ten Lectures on Wavelets. CBMS-NFS Series in Applied Mathematics, SIAM, Philadelphia, PA, 1992.
  • [20] I. Daubechies, J. C. Lagarias, Two-scale difference equations II. local regularity, infinite products of matrices and fractals, SIAM J. Math. Anal., 23(4) (1992), 1031–1079.
  • [21] S. Mallat, A theory of multiresolution signal decomposition: the wavelet representation, IEEE Trans. Pattern Anal. Mach. Intell., 11(7) (1989), 674–693.
  • [22] I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math., 41(7) (1988), 909–996.
  • [23] X. Li, Numerical solution of fractional differential equations using cubic B-spline wavelet collocation method, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 3934–3946.
  • [24] Y. M. Chen, M. X. Yi, C. X. Yu, Error analysis for numerical solution of fractional differential equation by Haar wavelets method, J. Comput. Sci., 3(5) (2012), 367–373.
  • [25] Y. L. Li, Solving a nonlinear fractional differential equation using Chebyshev wavelets, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 2284–2292.
  • [26] H. Jafari, S. A. Yousefi, M. A. Firoozjaee, S. Momani, C.M. Khalique, Application of Legendre wavelets for solving fractional differential equations, Comput. Appl. Math., 62 (2011), 1038–1045.
  • [27] P. Rahimkhani, Y. Ordokhani, E. Babolian, Numerical solution of fractional pantograph differential equations by using generalized fractional-order Bernoulli wavelet, J. Comput. Appl. Math., 309 (2017), 493–510.
  • [28] M. Razzaghi, S. Yousefi, Sine-cosine wavelets operational matrix of integration and its applications in the calculus of variations, Int. J. Syst. Sci., 33 (2002), 805–810.
  • [29] M. Tavassoli Kajani, M. Ghasemi, E.Babolian, Numerical solution of linear integro-differential equation by using sine-cosine wavelets, Appl. Math. Comput., 180 (2006), 569–574.
  • [30] M. Ghasemi, E. Babolian, M. Tavassoli Kajani, Numerical solution of linear Fredholm integral equations using sine-cosine wavelets, Int. J. Comput. Math., 84 (2007), 979–987.
  • [31] N. Irfan, A. H. Siddiqi, Sine-cosine wavelets approach in numerical evaluation of Hankel transform for seismology, Appl. Math. Model., 40 (2016), 4900–4907.
  • [32] I. Podlubny, Fractional Differential Equations, Math. Sci. Eng., 198, Academic Press, 1999.
  • [33] O. Christensen, K. L. Christensen, Approximation theory: from Taylor polynomial to wavelets, Birkhauser, Boston, 2004.
  • [34] D. Gottlieb, S.A. Orszag, Numerical analysis of spectral methods, SIAM, Philadelphia, PA, 1997.
  • [35] Y. Wang, T. Yin, L. Zhu, Sine-cosine wavelet operational matrix of fractional order integration and its applications in solving the fractional order Riccati differential equations, Adv. Differ. Equ., (2017), 2017: 222.
  • [36] A. Kilicman, Z. A. A. Al Zhour, Kronecker operational matrices for fractional calculus and some applications, Appl. Math. Comput., 187 (2007), 250–265.
  • [37] S. K. Damarla, M. Kundu, Numerical solution of multi-order fractional differential equations using generalized triangular function operational matrices, Appl. Math. Comput., 263 (2015), 189 – 203.
  • [38] A. E. M. El-Mesiry, A. M. A. El-Sayed, H. A. A. El-Saka, Numerical methods for multi-term fractional (arbitrary) orders differential equations, Appl. Math. Comput., 160 (2005), 683–699.
  • [39] M. Seifollahi, A. S. Shamloo, Numerical solution of nonlinear multi-order fractional differential equations by operational matrix of Chebyshev polynomials, World Appl. Program., 3 (2013), 85–92.
  • [40] A. H. Bhrawy, M. M. Tharwat, M.A. Alghamdi, A new operational matrix of fractional integration for shifted Jacobi polynomials, Bull. Malays. Math. Sci. Soc., 37 (2014), 983–995.
There are 40 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Somayeh Nemati

Anas Al-haboobi This is me

Publication Date December 20, 2018
Submission Date May 26, 2018
Acceptance Date October 26, 2018
Published in Issue Year 2018

Cite

APA Nemati, S., & Al-haboobi, A. (2018). Numerical Solution of Multi-Order Fractional Differential Equations Using Generalized Sine-Cosine Wavelets. Universal Journal of Mathematics and Applications, 1(4), 215-225. https://doi.org/10.32323/ujma.427381
AMA Nemati S, Al-haboobi A. Numerical Solution of Multi-Order Fractional Differential Equations Using Generalized Sine-Cosine Wavelets. Univ. J. Math. Appl. December 2018;1(4):215-225. doi:10.32323/ujma.427381
Chicago Nemati, Somayeh, and Anas Al-haboobi. “Numerical Solution of Multi-Order Fractional Differential Equations Using Generalized Sine-Cosine Wavelets”. Universal Journal of Mathematics and Applications 1, no. 4 (December 2018): 215-25. https://doi.org/10.32323/ujma.427381.
EndNote Nemati S, Al-haboobi A (December 1, 2018) Numerical Solution of Multi-Order Fractional Differential Equations Using Generalized Sine-Cosine Wavelets. Universal Journal of Mathematics and Applications 1 4 215–225.
IEEE S. Nemati and A. Al-haboobi, “Numerical Solution of Multi-Order Fractional Differential Equations Using Generalized Sine-Cosine Wavelets”, Univ. J. Math. Appl., vol. 1, no. 4, pp. 215–225, 2018, doi: 10.32323/ujma.427381.
ISNAD Nemati, Somayeh - Al-haboobi, Anas. “Numerical Solution of Multi-Order Fractional Differential Equations Using Generalized Sine-Cosine Wavelets”. Universal Journal of Mathematics and Applications 1/4 (December 2018), 215-225. https://doi.org/10.32323/ujma.427381.
JAMA Nemati S, Al-haboobi A. Numerical Solution of Multi-Order Fractional Differential Equations Using Generalized Sine-Cosine Wavelets. Univ. J. Math. Appl. 2018;1:215–225.
MLA Nemati, Somayeh and Anas Al-haboobi. “Numerical Solution of Multi-Order Fractional Differential Equations Using Generalized Sine-Cosine Wavelets”. Universal Journal of Mathematics and Applications, vol. 1, no. 4, 2018, pp. 215-2, doi:10.32323/ujma.427381.
Vancouver Nemati S, Al-haboobi A. Numerical Solution of Multi-Order Fractional Differential Equations Using Generalized Sine-Cosine Wavelets. Univ. J. Math. Appl. 2018;1(4):215-2.

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