NUMERICAL SOLUTION OF FRACTIONAL ORDER DELAY DIFFERENTIAL
EQUATION USING SHIFTED CHEBYSHEV POLYNOMIALS OF SECOND KIND

AJMAL ALI

doi:https://doi.org/10.14445/22315373/IJMTT-V43P508

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 43 | Number 1 | Year 2017 | Article Id. IJMTT-V43P508 | DOI : https://doi.org/10.14445/22315373/IJMTT-V43P508

NUMERICAL SOLUTION OF FRACTIONAL ORDER DELAY DIFFERENTIAL EQUATION USING SHIFTED CHEBYSHEV POLYNOMIALS OF SECOND KIND

AJMAL ALI

Abstract

The main aim of this article is to present an effcient numerical method to solve the Delay Differential Equation of fractional order.We use the fractional derivative in Caputo's sense.The properties of Chebyshev polynomials of second kind are utilized to reduce Delay Fractional Differential Equation (DFDE) to a linear or non-linear easily solvable system of algebraic equations.Numerically illustrative solved examples are present.The results shows that proposed method is very effective and simple.Thats reveals the validity and applicability of method.

Keywords

The results shows that proposed method is very effective and simple.Thats reveals the validity and applicability of method.

References

[1] O. P. Agrawal, O. Defterli, and D. Baleanu,Fractional optimal control problems with several state and control vari- ables,Journal of Vibration and Control, vol. 16, no. 13, pp. 19671976, 2010
[2] Ali Mehmet Akinlar,Aydin Secer and Mustafa Bayram,Numerical Solution of fractional Benney equa- tion,Appl.Math.Inf.Sci.8,No.4,2014 pp 1633-1637
[3] R.L.Bagley,P.J. Torvik,Theoretical basis for the application of fractional calculus to viscoelasticity,Journal of Rheol- ogy,Vol.27,No.3, pp201-210,2013
[4] D. A. Benson, S. W. Wheatcraft, and M. M. Meerschaert,Application of a fractional advection-dispersion equation, Water Resources Research, vol. 36, no. 6, pp. 14031412, 2000
[5] Chang-Ming Chen,F. Liu,V.Anh and I.Turner, Numerical methods for solving a two dimensional variable-order anoma- lous subdiffusion equations, Mathematics of Computation Vol 81,No.277,Jan 2012, pp 345-366
[6] Concepcion A.Monje, Yangquan Chen,M.Vinangre,Dingyu Xue,Vicente Feliu,Fractional order systems and con- trols:fundamentals and applications Adva. Indus.Con.Springer-Verlag.London,2010
[7] Eid H Doha, Ali H Bhrawy, Dumitru Baleanu and Samer S Ezz-Eldien,The operational matrix formulation of the Jacobi tau approximation for space fractional diffusion equation,Springer open journal,Doha et al.Advances in difference equations,2014, 2014:231
[8] He, J.H.Approximate analytical solution for fractional derivatives in porous media. Computer Methods in Applied Mechanics and Engineering, 167 (1-2),1998: pp 57-68.
[9] I.I. Gorial,Numerical Solution for fractional Partial Differential euations of two sided,Journal of Al-Nahrain University Vol.12 (2), June,2009,pp 128-131
[10] A. A. Hemeda,Homotopy Perturbation Method for Solving Partial Differential Equations of Fractional Order,Int. Journal of Math. Analysis, Vol. 6, 2012, no. 49, 2431 - 2448
[11] Ibrahim Karatay,Nurdane Kale,Serife R.Bayramoglu,A New Difference Scheme for time fractional Heat equations based on the crank-nicholson method, An Int J for Theory and Applications, Vol 16,No.4 (2013)
[12] Ibrahim Karatay and Nurdane Kale,Finite difference method of fractional parabolic partial differential equations with variable coeffcients, Int J. of Contemporary Mathematics Sciences,Vol. 9,2014.No.16,pp 767-776
[13] Ibrahim Karatay,Nurdane Kale,Serife R.Bayramoglu,A characheristics difference scheme for time fractional heat equa- tions based on the crank-nicholson difference scheme,Hindawi publishing corporation Abstract and Applied Analysis Vol. 2012,Article ID 548292,11pages doi:10.1155/2012/548292
[14] M.A Iqbal, Ayyaz Ali and S.T. Mohyud-Din,Chebyshev Wavelets Method for Fractional Delay Differential Equa- tions,International Journal of Modern Applied Physics, 2013, 4(1): 49-61
[15] H.Jafari, M. Soleymanivaraki, M. A.Firoozjaee,Legendre Wavelets for Solving Fractional Differential Equations,Journal of Applied Mathematics, Vol.7, No.4(27), Winter 2011, pp 65-70
[16] M.M Khader,A.S Hendy,The approximate and exact solutions of the fractional-order delay differential equations using legendre seudospectral method, Int J. Pure and Applied Mathematics,Vol 74 No.3 2012,pp 287-297
[17] M.M Khader,Numerical solution of Nonlinear Multi-order fractional differential equations by implementation of the operational derivative, Studies in Nonlinear Sciences 2(1):05-12,2011
[18] M.M Khadar,N.H Sweilam,T.A Assiri, On numerical solution for the fractional wave equation using legendre psedospec- tral method, Int J. Pure and Applied Mathematics,Vol 84 No.4 2013,pp 307-319
[19] M.M Khadar,N.H Sweilam,A.M.S. Mahdy,An Effcient method for solving the fractional diffusion equation, Int J. Pure and Applied Mathematics,Vol 1, No.2 2011,pp 1-12
[20] M.M. Khader,On the numerical solutions for the fractional diffusion equation,Communications in Nonlinear Science and Numerical Simulation,doi: 10.1016/j.cnsns.2010.09.007
[21] Mahmoud N. Sherif,Ibrahim Abouelfarag,T.S. Amer,Numerical solution of fractional Delay dufferential euations using spline functions, International Journal of Pure and Applied Mathematics,Volume 90 No. 1 2014, 73-83
[22] M. M. Meerschaert, H.PSchefer,C.Tadjeran,Finite difference methods for two-dimensional fractional dispersion equa- tion,Journal of Computational Physics 211 (2006) 249261
[23] Osama H. Mohammed and Abbas I. Khlaif,Adomian Decomposition Method for Solving Delay Differential Equations of Fractional Order,IOSR Journal of Mathematics,Volume 10, Issue 6 Ver. I (Nov - Dec. 2014), PP 01-05
[24] E. Scalas, R. Goren o, and F. Mainardi,Fractional calculus and continuous-time nance, Physica A,vol. 284, no. 14, pp. 376384, 2000
[25] M.Seifollahi,A.S Shamloo,Numerical solution of non linear muti-order fractional differential equations by operational matrix of chebyshev polynomials,World Applied Programming Vol(3),March 2013,85-92
[26] Sweilam, N.H., M.M. Khader and R.F. Al-Bar,Homotopy perturbation method for multidimensional nonlinear coupled system of parabolic and hyperbolic equations,Topological Methods in Nonlinear Analysis, 31: 295-304.
[27] Wanhai Geng,Yiming Chen,Wavelet Method for Nonlinear Partial Differential Equations of Fractional Order,Computer and Information Science, Vol. 4, No. 5; September 2011,doi:10.5539/cis.v4n5p28

Citation :

AJMAL ALI, "NUMERICAL SOLUTION OF FRACTIONAL ORDER DELAY DIFFERENTIAL EQUATION USING SHIFTED CHEBYSHEV POLYNOMIALS OF SECOND KIND," International Journal of Mathematics Trends and Technology (IJMTT), vol. 43, no. 1, pp. 45-54, 2017. Crossref, https://doi.org/10.14445/22315373/IJMTT-V43P508