메뉴 건너뛰기




Volumn 37, Issue 7, 2013, Pages 5498-5510

Fractional-order Legendre functions for solving fractional-order differential equations

Author keywords

Fractional order differential equations; Fractional order Legendre functions; Operational matrix; Tau method

Indexed keywords

ALGEBRAIC EQUATIONS; ENGINEERING PROCESS; FRACTIONAL CALCULUS; FRACTIONAL DERIVATIVES; FRACTIONAL DIFFERENTIAL EQUATIONS; FRACTIONAL INTEGRALS; ILLUSTRATIVE EXAMPLES; LEGENDRE FUNCTION; LEGENDRE POLYNOMIALS; OPERATIONAL MATRICES; TAU METHOD;

EID: 84872602206     PISSN: 0307904X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.apm.2012.10.026     Document Type: Article
Times cited : (298)

References (34)
  • 2
    • 0242354999 scopus 로고    scopus 로고
    • Geometric and physical interpretation of fractional integration and fractional differentiation
    • Podlubny I. Geometric and physical interpretation of fractional integration and fractional differentiation. Fract. Calc. Appl. Anal. 2002, 5:367-386.
    • (2002) Fract. Calc. Appl. Anal. , vol.5 , pp. 367-386
    • Podlubny, I.1
  • 3
    • 0007042083 scopus 로고    scopus 로고
    • Nonlinear oscillation with fractional derivative and its applications
    • in: International Conference on Vibrating Engineering'98, Dalian, China
    • J. He, Nonlinear oscillation with fractional derivative and its applications, in: International Conference on Vibrating Engineering'98, Dalian, China, 1998, pp. 288-291.
    • (1998) , pp. 288-291
    • He, J.1
  • 4
    • 79651469164 scopus 로고    scopus 로고
    • The non-standard finite difference scheme for linear fractional PDEs in fluid mechanics
    • Moaddy K., Momani S., Hashim I. The non-standard finite difference scheme for linear fractional PDEs in fluid mechanics. Comput. Math. Appl. 2011, 61:1209-1216.
    • (2011) Comput. Math. Appl. , vol.61 , pp. 1209-1216
    • Moaddy, K.1    Momani, S.2    Hashim, I.3
  • 5
    • 0347763940 scopus 로고    scopus 로고
    • Some applications of nonlinear fractional differential equations and their approximations
    • He J. Some applications of nonlinear fractional differential equations and their approximations. Bull. Sci. Technol. 1999, 15:86-90.
    • (1999) Bull. Sci. Technol. , vol.15 , pp. 86-90
    • He, J.1
  • 6
    • 0032307661 scopus 로고    scopus 로고
    • Approximate analytical solution for seepage flow with fractional derivatives in porous media
    • He J. Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput. Methods. Appl. Mech. Eng. 1998, 167:57-68.
    • (1998) Comput. Methods. Appl. Mech. Eng. , vol.167 , pp. 57-68
    • He, J.1
  • 7
    • 0041384356 scopus 로고    scopus 로고
    • Chaotic dynamics of the fractional lorenz system
    • Grigorenko I., Grigorenko E. Chaotic dynamics of the fractional lorenz system. Phys. Rev. Lett. 2003, 91:34101-34104.
    • (2003) Phys. Rev. Lett. , vol.91 , pp. 34101-34104
    • Grigorenko, I.1    Grigorenko, E.2
  • 8
    • 0001983732 scopus 로고    scopus 로고
    • Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics
    • Springer-Verlag, New York
    • Mainardi F. Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics. Fractals and Fractional Calculus in Continuum Mechanics 1997, Springer-Verlag, New York, pp. 291-348.
    • (1997) Fractals and Fractional Calculus in Continuum Mechanics , pp. 291-348
    • Mainardi, F.1
  • 10
    • 33751512215 scopus 로고    scopus 로고
    • Decomposition method for solving fractional Riccati differential equations
    • Momani S., Shawagfeh N.T. Decomposition method for solving fractional Riccati differential equations. Appl. Math. Comput. 2006, 182:1083-1092.
    • (2006) Appl. Math. Comput. , vol.182 , pp. 1083-1092
    • Momani, S.1    Shawagfeh, N.T.2
  • 11
    • 33750841854 scopus 로고    scopus 로고
    • Numerical methods for fourth-order fractional integro-differential equations
    • Momani S., Noor M.A. Numerical methods for fourth-order fractional integro-differential equations. Appl. Math. Comput. 2006, 182:754-760.
    • (2006) Appl. Math. Comput. , vol.182 , pp. 754-760
    • Momani, S.1    Noor, M.A.2
  • 12
    • 34248373867 scopus 로고    scopus 로고
    • Solving a multi-order fractional differential equation
    • Gejji V.D., Jafari H. Solving a multi-order fractional differential equation. Appl. Math. Comput. 2007, 189:541-548.
    • (2007) Appl. Math. Comput. , vol.189 , pp. 541-548
    • Gejji, V.D.1    Jafari, H.2
  • 13
    • 33751200976 scopus 로고    scopus 로고
    • Analytical approximate solution of nonlinear dynamic system containing fractional derivative by modified decomposition method
    • Ray S.S., Chaudhuri K.S., Bera R.K. Analytical approximate solution of nonlinear dynamic system containing fractional derivative by modified decomposition method. Appl. Math. Comput. 2006, 182:544-552.
    • (2006) Appl. Math. Comput. , vol.182 , pp. 544-552
    • Ray, S.S.1    Chaudhuri, K.S.2    Bera, R.K.3
  • 14
    • 33751513961 scopus 로고    scopus 로고
    • Numerical solutions for fractional KdV-Burgers equation by Adomian decomposition method
    • Wang Q. Numerical solutions for fractional KdV-Burgers equation by Adomian decomposition method. Appl. Math. Comput. 2006, 182:1048-1055.
    • (2006) Appl. Math. Comput. , vol.182 , pp. 1048-1055
    • Wang, Q.1
  • 16
    • 74249121146 scopus 로고    scopus 로고
    • Homotopy analysis method for solving multi-term linear and nonlinear diffusion-wave equations of fractional order
    • Jafari H., Golbabai A., Seifi S., Sayevand K. Homotopy analysis method for solving multi-term linear and nonlinear diffusion-wave equations of fractional order. Comput. Math. Appl. 2010, 59:1337-1344.
    • (2010) Comput. Math. Appl. , vol.59 , pp. 1337-1344
    • Jafari, H.1    Golbabai, A.2    Seifi, S.3    Sayevand, K.4
  • 17
    • 74149089078 scopus 로고    scopus 로고
    • Analytical approximate solutions of systems of fractional algebraic-differential equations by homotopy analysis method
    • Zurigat M., Momani S., Alawneh A. Analytical approximate solutions of systems of fractional algebraic-differential equations by homotopy analysis method. Comput. Math. Appl. 2010, 59:1227-1235.
    • (2010) Comput. Math. Appl. , vol.59 , pp. 1227-1235
    • Zurigat, M.1    Momani, S.2    Alawneh, A.3
  • 18
    • 80053341743 scopus 로고    scopus 로고
    • Homotopy analysis method for higher-order fractional integro-differential equations
    • Zhang Y.H.X., Tang B. Homotopy analysis method for higher-order fractional integro-differential equations. Comput. Math. Appl. 2011, 62:3194-3203.
    • (2011) Comput. Math. Appl. , vol.62 , pp. 3194-3203
    • Zhang, Y.H.X.1    Tang, B.2
  • 19
    • 33745712076 scopus 로고    scopus 로고
    • An approximate method for numerical solution of fractional differential equations
    • Kumar P., Agrawal O.P. An approximate method for numerical solution of fractional differential equations. Signal Process. 2006, 86:2602-2610.
    • (2006) Signal Process. , vol.86 , pp. 2602-2610
    • Kumar, P.1    Agrawal, O.P.2
  • 20
    • 1542425102 scopus 로고    scopus 로고
    • Numerical solution of the space fractional Fokker-Planck equation
    • Liu I.T.F., Anh V. Numerical solution of the space fractional Fokker-Planck equation. J. Comput. Appl. Math. 2004, 166:209-219.
    • (2004) J. Comput. Appl. Math. , vol.166 , pp. 209-219
    • Liu, I.T.F.1    Anh, V.2
  • 21
    • 33646128485 scopus 로고    scopus 로고
    • Weighted average finite difference methods for fractional diffusion equations
    • Yuste S.B. Weighted average finite difference methods for fractional diffusion equations. J. Comput. Phys. 2006, 216:264-274.
    • (2006) J. Comput. Phys. , vol.216 , pp. 264-274
    • Yuste, S.B.1
  • 22
    • 76449113714 scopus 로고    scopus 로고
    • Fractional diffusion equations by the Kansa method
    • Chen W., Ye L., Su H. Fractional diffusion equations by the Kansa method. Comput. Math. Appl. 2010, 59:1614-1620.
    • (2010) Comput. Math. Appl. , vol.59 , pp. 1614-1620
    • Chen, W.1    Ye, L.2    Su, H.3
  • 23
    • 84872612601 scopus 로고    scopus 로고
    • Analysis of a local discontinuous Galerkin method for time-fractional advection-diffusion equations
    • Int. J. Numer. Method H.
    • Y. He, L. Wei, X. Zhang, Analysis of a local discontinuous Galerkin method for time-fractional advection-diffusion equations, Int. J. Numer. Method H. (2012).
    • (2012)
    • He, Y.1    Wei, L.2    Zhang, X.3
  • 24
    • 84870242888 scopus 로고    scopus 로고
    • An integral operational matrix based on jacobi polynomials for solving fractional-order differential equations
    • Kazem S. An integral operational matrix based on jacobi polynomials for solving fractional-order differential equations. Appl. Math. Model. 2012, 10.1016/j.apm.2012.03.033.
    • (2012) Appl. Math. Model.
    • Kazem, S.1
  • 25
    • 74149085984 scopus 로고    scopus 로고
    • An algorithm for the numerical solution of differential equations of fractional order
    • Odibat Z., Momani S. An algorithm for the numerical solution of differential equations of fractional order. J. Appl. Math. Inform. 2008, 26:15-27.
    • (2008) J. Appl. Math. Inform. , vol.26 , pp. 15-27
    • Odibat, Z.1    Momani, S.2
  • 27
    • 84872596071 scopus 로고    scopus 로고
    • On the fractional order Rodrigues formula for the Legendre polynomials
    • Rida S.Z., Yousef A.M. On the fractional order Rodrigues formula for the Legendre polynomials. Adv. Appl. Math. Sci. 2011, 10:509-518.
    • (2011) Adv. Appl. Math. Sci. , vol.10 , pp. 509-518
    • Rida, S.Z.1    Yousef, A.M.2
  • 28
    • 74249095517 scopus 로고    scopus 로고
    • A new operational matrix for solving fractional-order differential equations
    • Saadatmandi A., Dehghan M. A new operational matrix for solving fractional-order differential equations. Comput. Math. Appl. 2010, 59:1326-1336.
    • (2010) Comput. Math. Appl. , vol.59 , pp. 1326-1336
    • Saadatmandi, A.1    Dehghan, M.2
  • 29
    • 0012899160 scopus 로고    scopus 로고
    • Numerical solution of the Bagley-Torvik equation
    • Diethelm K., Ford N.J. Numerical solution of the Bagley-Torvik equation. BIT 2002, 42:490-507.
    • (2002) BIT , vol.42 , pp. 490-507
    • Diethelm, K.1    Ford, N.J.2
  • 30
    • 33846930099 scopus 로고    scopus 로고
    • A Tau method for the one-dimensional parabolic inverse problem subject to temperature overspecification
    • Dehghan M., Saadatmandi A. A Tau method for the one-dimensional parabolic inverse problem subject to temperature overspecification. Comput. Math. Appl. 2006, 52:933-940.
    • (2006) Comput. Math. Appl. , vol.52 , pp. 933-940
    • Dehghan, M.1    Saadatmandi, A.2
  • 31
    • 57749203983 scopus 로고    scopus 로고
    • Numerical solution of a mathematical model for capillary formation in tumor angiogenesis via the Tau method
    • Saadatmandi A., Dehghan M. Numerical solution of a mathematical model for capillary formation in tumor angiogenesis via the Tau method. Commun. Numer. Methods Eng. 2008, 24:1467-1474.
    • (2008) Commun. Numer. Methods Eng. , vol.24 , pp. 1467-1474
    • Saadatmandi, A.1    Dehghan, M.2
  • 32
    • 54349113107 scopus 로고    scopus 로고
    • Analytical comparison between the homotopy perturbation method and variational iteration method for differential equations of fractional order
    • Odibat Z., Momani S. Analytical comparison between the homotopy perturbation method and variational iteration method for differential equations of fractional order. Int. J. Mod. Phys. B 2008, 22:4041-4058.
    • (2008) Int. J. Mod. Phys. B , vol.22 , pp. 4041-4058
    • Odibat, Z.1    Momani, S.2
  • 33
    • 35349007940 scopus 로고    scopus 로고
    • Numerical studies for a multi-order fractional differential equation
    • Sweilam N.H., Khader M.M., Al-Bar R.F. Numerical studies for a multi-order fractional differential equation. Phys. Lett. A 2007, 371:26-33.
    • (2007) Phys. Lett. A , vol.371 , pp. 26-33
    • Sweilam, N.H.1    Khader, M.M.2    Al-Bar, R.F.3
  • 34
    • 35348869861 scopus 로고    scopus 로고
    • Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order
    • Odibat Z., Momani S. Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order. Chaos Solitons Fract. 2008, 36:167-174.
    • (2008) Chaos Solitons Fract. , vol.36 , pp. 167-174
    • Odibat, Z.1    Momani, S.2


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.