Research Article
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Numerical Solution of Fractional Riccati Differential Equation via Shifted Chebyshev Polynomials of the Third Kind

Year 2017, Volume: 2 Issue: 1, 1 - 11, 30.04.2017
https://doi.org/10.30931/jetas.304377

Abstract

In this paper, Chebyshev collocation method is applied to fractional Riccati differential equation (FRDE) using the shifted Chebyshev polynomials of the third kind. Approximate analytical solution of FRDE is considered as Chebyshev series expansion. The fractional derivative is described in the Caputo sense. Using properties of Chebyshev polynomials FRDE with initial condition is reduced to a nonlinear system of algebraic equations which solved by the Newton iteration method. The accuracy and efficiency of the proposed method is illustrated by numerical examples.

References

  • [1] S. Balaji, Solution of nonlinear Riccati differential equation using Cheby- shev wavelets, WSEAS Trans. Math. 13 (2014) 441 - 451.
  • [2] M. Dalir, M. Bashour, Applications of fractional calculus, Appl. Math. Sci. 4 (2010) 1021 - 1032.
  • [3] G.A. Einicke, L.B. White, R.R. Bitmead, The use of fake algebraic Ric- cati equations for co-channel demodulation, IEEE Trans. Signal Process 51 (2003) 2288 - 2293.
  • [4] M. Gerber, N. Hasselblatt, D. Keesing, The Riccati equation: pinching of forcing and solutions, Experiment Math. 12 (2003) 129 - 134.
  • [5] I. Grigorenko I, E. Grigorenko, Chaotic dynamics of the fractional Lorenz system, Phys. Rev. Lett. 91 (2003) 034101 - 034104.
  • [6] J. H. He, Some applications of nonlinear fractional differential equations and their approximations, Bull. Sc. Technol. 15 (1999) 86 - 90.
  • [7] J. H. He, Approximate analytical solution for seepage flow with frac- tional derivatives in porous media, Comput. Methods. Appl. Mech Eng. 167 (1998) 57 - 68.
  • [8] H. Jafari, H. Tajadodi, He’s Variational Iteration Method for Solving Fractional Riccati Differential Equation, J. Differ. Equations 2010 (2010) 1 - 8 .
  • [9] M. M. Khader, Numerical treatment for solving fractional Riccati dif- ferential equation, J. Egyptian Math. Soc. 21 (2013) 32 - 37.
  • [10] M. M. Khader, A.M.S. Mahdy, E.S. Mohamed, On Approximate So- lutions For Fractional Riccati Differential Equation, J. Eng. Appl. Sci. (2014) 2683 - 2689.
  • [11] M.M. Khader, On the numerical solutions for the fractional diffusion equation, Commun. Nonlinear Sci. Nume. Simulat. 16 (2011) 2535 - 2542.
  • [12] N.A. Khan, A. Ara, J. Muhammad, An efficient approach for solving the Riccati equation with fractional orders, Comput. Math. Appl. 61 (2011) 2683 - 2689.
  • [13] Y. Li, Solving a nonlinear fractional differential equation using Cheby- shev wavelets, Commun. Nonlinear Sci. Numer. Simulat. 15 (2010) 2284 - 2292.
  • [14] F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics, In: A. Carpinteri, F. Mainardi, Editors, Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, New York, 1997, 291 - 348.
  • [15] J. C. Mason, D. C. Handscomb, Chebyshev Polynomials, Chapman and Hall, CRC, New York, NY, Boca Raton, 2003.
  • [16] M. Merdan, On the Solutions Fractional Riccati Differential Equation with Modified Riemann-Liouville Derivative, J. Differ. Equations 2012 (2012) 1 - 17.
  • [17] Z. Odibat, S. Momani, Modified homotopy perturbation method: ap- plication to quadratic Riccati differential equation of fractional order, Chaos Soliton. Fract. 36 (2008) 167 - 174.
  • [18] I. Podlubny, Geometric and physical interpretation of fractional integra- tion and fractional differentiation, Fract Calculus App. Anal. 5 (2002) 367 - 386.
  • [19] I. Podlubny, Fractional differential equations, Academic press, New York, 1999.
  • [20] M.A.Z. Raja, J. A. Khan, I.M. Qureshi, A new stochastic approach for solution of Riccati differential equation of fractional order, Ann. Math. Artif. Intell. 60 (2010) 229 - 250.
  • [21] E. Sousa, Numerical approximations for fractional diffusion equation via splines, Compu.t Math. App. 62 (2011) 983 - 944.
  • [22] N. H. Sweilam, A.M. Nagy, A. El-Sayed, On the numerical solution of space fractional order diffusion equation via shifted Chebyshev polyno- mials of the third kind, J. King Saud Univ. Sci. 28 (2016) 41-47.
  • [23] N.H. Sweilam, M.M. Khader,A.M.S. Mahdy, Numerical Studies for Solv- ing Fractional Riccati Differential Equation, Appl. Appl. Math. 7 (2012) 595 - 608.
  • [24] C. Tadjeran, M. Meerschaert, H.P. Scheffler, A second-order accurate numerical approximation for the fractional diffusion equation, J. Com- put. Phys. 21 (2006) 205 - 213.
Year 2017, Volume: 2 Issue: 1, 1 - 11, 30.04.2017
https://doi.org/10.30931/jetas.304377

Abstract

References

  • [1] S. Balaji, Solution of nonlinear Riccati differential equation using Cheby- shev wavelets, WSEAS Trans. Math. 13 (2014) 441 - 451.
  • [2] M. Dalir, M. Bashour, Applications of fractional calculus, Appl. Math. Sci. 4 (2010) 1021 - 1032.
  • [3] G.A. Einicke, L.B. White, R.R. Bitmead, The use of fake algebraic Ric- cati equations for co-channel demodulation, IEEE Trans. Signal Process 51 (2003) 2288 - 2293.
  • [4] M. Gerber, N. Hasselblatt, D. Keesing, The Riccati equation: pinching of forcing and solutions, Experiment Math. 12 (2003) 129 - 134.
  • [5] I. Grigorenko I, E. Grigorenko, Chaotic dynamics of the fractional Lorenz system, Phys. Rev. Lett. 91 (2003) 034101 - 034104.
  • [6] J. H. He, Some applications of nonlinear fractional differential equations and their approximations, Bull. Sc. Technol. 15 (1999) 86 - 90.
  • [7] J. H. He, Approximate analytical solution for seepage flow with frac- tional derivatives in porous media, Comput. Methods. Appl. Mech Eng. 167 (1998) 57 - 68.
  • [8] H. Jafari, H. Tajadodi, He’s Variational Iteration Method for Solving Fractional Riccati Differential Equation, J. Differ. Equations 2010 (2010) 1 - 8 .
  • [9] M. M. Khader, Numerical treatment for solving fractional Riccati dif- ferential equation, J. Egyptian Math. Soc. 21 (2013) 32 - 37.
  • [10] M. M. Khader, A.M.S. Mahdy, E.S. Mohamed, On Approximate So- lutions For Fractional Riccati Differential Equation, J. Eng. Appl. Sci. (2014) 2683 - 2689.
  • [11] M.M. Khader, On the numerical solutions for the fractional diffusion equation, Commun. Nonlinear Sci. Nume. Simulat. 16 (2011) 2535 - 2542.
  • [12] N.A. Khan, A. Ara, J. Muhammad, An efficient approach for solving the Riccati equation with fractional orders, Comput. Math. Appl. 61 (2011) 2683 - 2689.
  • [13] Y. Li, Solving a nonlinear fractional differential equation using Cheby- shev wavelets, Commun. Nonlinear Sci. Numer. Simulat. 15 (2010) 2284 - 2292.
  • [14] F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics, In: A. Carpinteri, F. Mainardi, Editors, Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, New York, 1997, 291 - 348.
  • [15] J. C. Mason, D. C. Handscomb, Chebyshev Polynomials, Chapman and Hall, CRC, New York, NY, Boca Raton, 2003.
  • [16] M. Merdan, On the Solutions Fractional Riccati Differential Equation with Modified Riemann-Liouville Derivative, J. Differ. Equations 2012 (2012) 1 - 17.
  • [17] Z. Odibat, S. Momani, Modified homotopy perturbation method: ap- plication to quadratic Riccati differential equation of fractional order, Chaos Soliton. Fract. 36 (2008) 167 - 174.
  • [18] I. Podlubny, Geometric and physical interpretation of fractional integra- tion and fractional differentiation, Fract Calculus App. Anal. 5 (2002) 367 - 386.
  • [19] I. Podlubny, Fractional differential equations, Academic press, New York, 1999.
  • [20] M.A.Z. Raja, J. A. Khan, I.M. Qureshi, A new stochastic approach for solution of Riccati differential equation of fractional order, Ann. Math. Artif. Intell. 60 (2010) 229 - 250.
  • [21] E. Sousa, Numerical approximations for fractional diffusion equation via splines, Compu.t Math. App. 62 (2011) 983 - 944.
  • [22] N. H. Sweilam, A.M. Nagy, A. El-Sayed, On the numerical solution of space fractional order diffusion equation via shifted Chebyshev polyno- mials of the third kind, J. King Saud Univ. Sci. 28 (2016) 41-47.
  • [23] N.H. Sweilam, M.M. Khader,A.M.S. Mahdy, Numerical Studies for Solv- ing Fractional Riccati Differential Equation, Appl. Appl. Math. 7 (2012) 595 - 608.
  • [24] C. Tadjeran, M. Meerschaert, H.P. Scheffler, A second-order accurate numerical approximation for the fractional diffusion equation, J. Com- put. Phys. 21 (2006) 205 - 213.
There are 24 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Handan Yaslan

Publication Date April 30, 2017
Published in Issue Year 2017 Volume: 2 Issue: 1

Cite

APA Yaslan, H. (2017). Numerical Solution of Fractional Riccati Differential Equation via Shifted Chebyshev Polynomials of the Third Kind. Journal of Engineering Technology and Applied Sciences, 2(1), 1-11. https://doi.org/10.30931/jetas.304377
AMA Yaslan H. Numerical Solution of Fractional Riccati Differential Equation via Shifted Chebyshev Polynomials of the Third Kind. JETAS. April 2017;2(1):1-11. doi:10.30931/jetas.304377
Chicago Yaslan, Handan. “Numerical Solution of Fractional Riccati Differential Equation via Shifted Chebyshev Polynomials of the Third Kind”. Journal of Engineering Technology and Applied Sciences 2, no. 1 (April 2017): 1-11. https://doi.org/10.30931/jetas.304377.
EndNote Yaslan H (April 1, 2017) Numerical Solution of Fractional Riccati Differential Equation via Shifted Chebyshev Polynomials of the Third Kind. Journal of Engineering Technology and Applied Sciences 2 1 1–11.
IEEE H. Yaslan, “Numerical Solution of Fractional Riccati Differential Equation via Shifted Chebyshev Polynomials of the Third Kind”, JETAS, vol. 2, no. 1, pp. 1–11, 2017, doi: 10.30931/jetas.304377.
ISNAD Yaslan, Handan. “Numerical Solution of Fractional Riccati Differential Equation via Shifted Chebyshev Polynomials of the Third Kind”. Journal of Engineering Technology and Applied Sciences 2/1 (April 2017), 1-11. https://doi.org/10.30931/jetas.304377.
JAMA Yaslan H. Numerical Solution of Fractional Riccati Differential Equation via Shifted Chebyshev Polynomials of the Third Kind. JETAS. 2017;2:1–11.
MLA Yaslan, Handan. “Numerical Solution of Fractional Riccati Differential Equation via Shifted Chebyshev Polynomials of the Third Kind”. Journal of Engineering Technology and Applied Sciences, vol. 2, no. 1, 2017, pp. 1-11, doi:10.30931/jetas.304377.
Vancouver Yaslan H. Numerical Solution of Fractional Riccati Differential Equation via Shifted Chebyshev Polynomials of the Third Kind. JETAS. 2017;2(1):1-11.