NUMERICAL SOLUTION OF FRACTIONAL RICCATI DIFFERENTIAL EQUATION VIA SHIFTED CHEBYSHEV POLYNOMIALS OF THE THIRD KIND

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