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Volumn 281, Issue , 2015, Pages 876-895

A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations

Author keywords

Advection diffusion equation; Multi term time fractional wave diffusion equations; Operational matrix; Power law wave equation; Spectral method; Telegraph equation

Indexed keywords

ADVECTION; APPROXIMATION ALGORITHMS; DIFFERENTIATION (CALCULUS); JACOBIAN MATRICES; WAVE EQUATIONS;

EID: 84919935961     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2014.10.060     Document Type: Article
Times cited : (302)

References (74)
  • 1
    • 3042776917 scopus 로고    scopus 로고
    • Fractional calculus in bioengineering
    • Magin R.L. Fractional calculus in bioengineering. Crit. Rev. Biomed. Eng. 2004, 32(1):1-104.
    • (2004) Crit. Rev. Biomed. Eng. , vol.32 , Issue.1 , pp. 1-104
    • Magin, R.L.1
  • 2
    • 33750340240 scopus 로고    scopus 로고
    • Fractional calculus in biomechanics: a 3D viscoelastic model using regularized fractional derivative kernels with application to the human calcaneal fat pad
    • Freed A.D., Diethelm K. Fractional calculus in biomechanics: a 3D viscoelastic model using regularized fractional derivative kernels with application to the human calcaneal fat pad. Biomech. Model. Mechanobiol. 2006, 5:203-215.
    • (2006) Biomech. Model. Mechanobiol. , vol.5 , pp. 203-215
    • Freed, A.D.1    Diethelm, K.2
  • 3
    • 84875915899 scopus 로고    scopus 로고
    • Fractional calculus model of electrical impedance applied to human skin
    • Vosika Z.B., Lazovic G.M., Misevic G.N., Simic-Krstic J.B. Fractional calculus model of electrical impedance applied to human skin. PLoS ONE 2013, 8(4):1-12.
    • (2013) PLoS ONE , vol.8 , Issue.4 , pp. 1-12
    • Vosika, Z.B.1    Lazovic, G.M.2    Misevic, G.N.3    Simic-Krstic, J.B.4
  • 5
    • 77958009389 scopus 로고    scopus 로고
    • A class of fractional evolution equations and optimal controls
    • Wang J.R., Zhou Y. A class of fractional evolution equations and optimal controls. Nonlinear Anal., Real World Appl. 2011, 12:262-272.
    • (2011) Nonlinear Anal., Real World Appl. , vol.12 , pp. 262-272
    • Wang, J.R.1    Zhou, Y.2
  • 8
    • 84899637563 scopus 로고    scopus 로고
    • Numerical simulation of a new two-dimensional variable-order fractional percolation equation in non-homogeneous porous media
    • Chen S., Liu F., Burrage K. Numerical simulation of a new two-dimensional variable-order fractional percolation equation in non-homogeneous porous media. Comput. Math. Appl. 2014, 67:1673-1681.
    • (2014) Comput. Math. Appl. , vol.67 , pp. 1673-1681
    • Chen, S.1    Liu, F.2    Burrage, K.3
  • 9
    • 84885954007 scopus 로고    scopus 로고
    • Fractional-order modeling of neutron transport in a nuclear reactor
    • Vyawahare V.A., Nataraj P.S.V. Fractional-order modeling of neutron transport in a nuclear reactor. Appl. Math. Model. 2013, 37:9747-9767.
    • (2013) Appl. Math. Model. , vol.37 , pp. 9747-9767
    • Vyawahare, V.A.1    Nataraj, P.S.V.2
  • 10
    • 84897870748 scopus 로고    scopus 로고
    • The use of a Riesz fractional differential-based approach for texture enhancement in image processing
    • Yu Q., Liu F., Turner I., Burrage K., Vegh V. The use of a Riesz fractional differential-based approach for texture enhancement in image processing. ANZIAM J. 2013, 54:C590-C607.
    • (2013) ANZIAM J. , vol.54 , pp. C590-C607
    • Yu, Q.1    Liu, F.2    Turner, I.3    Burrage, K.4    Vegh, V.5
  • 11
    • 76649094637 scopus 로고    scopus 로고
    • Solving nonlinear fractional partial differential equations using the homotopy analysis method
    • Dehghan M., Manafian J., Saadatmandi A. Solving nonlinear fractional partial differential equations using the homotopy analysis method. Numer. Methods Partial Differ. Equ. 2010, 26:448-479.
    • (2010) Numer. Methods Partial Differ. Equ. , vol.26 , pp. 448-479
    • Dehghan, M.1    Manafian, J.2    Saadatmandi, A.3
  • 12
    • 0002641421 scopus 로고    scopus 로고
    • The random walks guide to anomalous diffusion: a fractional dynamics approach
    • Metzler R., Klafter J. The random walks guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 2000, 339:177.
    • (2000) Phys. Rep. , vol.339 , pp. 177
    • Metzler, R.1    Klafter, J.2
  • 13
    • 0036887936 scopus 로고    scopus 로고
    • Chaos, fractional kinetics, and anomalous transport
    • Zaslavsky G.M. Chaos, fractional kinetics, and anomalous transport. Phys. Rep. 2002, 371(6):461-580.
    • (2002) Phys. Rep. , vol.371 , Issue.6 , pp. 461-580
    • Zaslavsky, G.M.1
  • 14
    • 1542425102 scopus 로고    scopus 로고
    • Numerical solution of the space fractional Fokker-Planck equation
    • Liu F., Anh V., Turner I. 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, F.1    Anh, V.2    Turner, I.3
  • 15
    • 4444368867 scopus 로고    scopus 로고
    • Finite difference approximations for fractional advection-dispersion flow equations
    • Meerschaert M.M., Tadjeran C. Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 2004, 172:65-77.
    • (2004) J. Comput. Appl. Math. , vol.172 , pp. 65-77
    • Meerschaert, M.M.1    Tadjeran, C.2
  • 16
    • 0034032484 scopus 로고    scopus 로고
    • Application of a fractional advection-dispersion equation
    • Benson D.A., Wheatcraft S.W., Meerschaert M.M. Application of a fractional advection-dispersion equation. Water Resour. Res. 2000, 36(6):1403-1412.
    • (2000) Water Resour. Res. , vol.36 , Issue.6 , pp. 1403-1412
    • Benson, D.A.1    Wheatcraft, S.W.2    Meerschaert, M.M.3
  • 17
    • 77955927812 scopus 로고    scopus 로고
    • 2N) finite difference method for fractional diffusion equations
    • 2N) finite difference method for fractional diffusion equations. J. Comput. Phys. 2010, 229:8095-8104.
    • (2010) J. Comput. Phys. , vol.229 , pp. 8095-8104
    • Wang, H.1    Wang, K.2    Sircar, T.3
  • 18
    • 84868214613 scopus 로고    scopus 로고
    • Numerical methods and analysis for aclass of fractional advection-dispersion models
    • Liu F., Zhuang P., Burrage K. Numerical methods and analysis for aclass of fractional advection-dispersion models. Comput. Math. Appl. 2012, 64:2990-3007.
    • (2012) Comput. Math. Appl. , vol.64 , pp. 2990-3007
    • Liu, F.1    Zhuang, P.2    Burrage, K.3
  • 19
    • 28044468843 scopus 로고    scopus 로고
    • Finite difference approximations for two-sided space fractional partial differential equations
    • Meerschaert M.M., Tadjeran C. Finite difference approximations for two-sided space fractional partial differential equations. Appl. Numer. Math. 2006, 56:80-90.
    • (2006) Appl. Numer. Math. , vol.56 , pp. 80-90
    • Meerschaert, M.M.1    Tadjeran, C.2
  • 20
    • 84868209623 scopus 로고    scopus 로고
    • A second order explicit finite difference method for the fractional advection-diffusion equation
    • Sousa E. A second order explicit finite difference method for the fractional advection-diffusion equation. Comput. Math. Appl. 2012, 64:3141-3152.
    • (2012) Comput. Math. Appl. , vol.64 , pp. 3141-3152
    • Sousa, E.1
  • 21
    • 23944449029 scopus 로고    scopus 로고
    • A mass balance based numerical method for the fractional advection dispersion equation: theory and application
    • Zhang X.X., Crawford J.W., Deeks L.K., Stutter I.M., Bengough A.G., Young I.M. A mass balance based numerical method for the fractional advection dispersion equation: theory and application. Water Resour. Res. 2005, 41(7).
    • (2005) Water Resour. Res. , vol.41 , Issue.7
    • Zhang, X.X.1    Crawford, J.W.2    Deeks, L.K.3    Stutter, I.M.4    Bengough, A.G.5    Young, I.M.6
  • 22
    • 34249805393 scopus 로고    scopus 로고
    • Numerical approximation of Levy-Feller diffusion equation and its probability interpretation
    • Zhang H., Liu F., Anh V. Numerical approximation of Levy-Feller diffusion equation and its probability interpretation. J. Comput. Appl. Math. 2007, 206:1098-1115.
    • (2007) J. Comput. Appl. Math. , vol.206 , pp. 1098-1115
    • Zhang, H.1    Liu, F.2    Anh, V.3
  • 23
    • 33646494026 scopus 로고    scopus 로고
    • On using random walks to solve the space-fractional advection-dispersion equations
    • Zhang Y., Benson D.A., Meerschaert M.M., Scheffler H.P. On using random walks to solve the space-fractional advection-dispersion equations. J. Stat. Phys. 2006, 367(1):89-110.
    • (2006) J. Stat. Phys. , vol.367 , Issue.1 , pp. 89-110
    • Zhang, Y.1    Benson, D.A.2    Meerschaert, M.M.3    Scheffler, H.P.4
  • 24
    • 33747784932 scopus 로고    scopus 로고
    • Random walk approximation of fractional-order multiscaling anomalous diffusion
    • Zhang Y., Benson D.A., Meerschaert M.M., LaBolle E.M., Scheffler H.P. Random walk approximation of fractional-order multiscaling anomalous diffusion. Phys. Rev. E 2006, 74(2):026706.
    • (2006) Phys. Rev. E , vol.74 , Issue.2 , pp. 026706
    • Zhang, Y.1    Benson, D.A.2    Meerschaert, M.M.3    LaBolle, E.M.4    Scheffler, H.P.5
  • 25
    • 84922803426 scopus 로고    scopus 로고
    • New numerical approximations for space-time fractional Burgers' equations via a Legendre spectral-collocation method
    • Bhrawy A.H., Zaky M.A., Baleanu D. New numerical approximations for space-time fractional Burgers' equations via a Legendre spectral-collocation method. Rom. Rep. Phys. 2015, 2:1221-1451.
    • (2015) Rom. Rep. Phys. , vol.2 , pp. 1221-1451
    • Bhrawy, A.H.1    Zaky, M.A.2    Baleanu, D.3
  • 26
    • 84890864220 scopus 로고    scopus 로고
    • Numerical approximations for fractional diffusion equations via a Chebyshev spectral-tau method
    • Doha E.H., Bhrawy A.H., Ezz-Eldien S.S. Numerical approximations for fractional diffusion equations via a Chebyshev spectral-tau method. Cent. Eur. J. Phys. 2013, 11:1494-1503.
    • (2013) Cent. Eur. J. Phys. , vol.11 , pp. 1494-1503
    • Doha, E.H.1    Bhrawy, A.H.2    Ezz-Eldien, S.S.3
  • 27
    • 79960977916 scopus 로고    scopus 로고
    • A tau approach for solution of the space fractional diffusion equation
    • Saadatmandi A., Dehghan M. A tau approach for solution of the space fractional diffusion equation. Comput. Math. Appl. 2011, 62:1135-1142.
    • (2011) Comput. Math. Appl. , vol.62 , pp. 1135-1142
    • Saadatmandi, A.1    Dehghan, M.2
  • 28
    • 84894731513 scopus 로고    scopus 로고
    • Fully discrete local discontinuous Galerkin method for solving the fractional telegraph equation
    • Wei L., Dai H., Zhang D., Si Z. Fully discrete local discontinuous Galerkin method for solving the fractional telegraph equation. Calcolo 2014, 51(1):175-192.
    • (2014) Calcolo , vol.51 , Issue.1 , pp. 175-192
    • Wei, L.1    Dai, H.2    Zhang, D.3    Si, Z.4
  • 29
    • 84887273517 scopus 로고    scopus 로고
    • Numerical solution of fractional telegraph equation by using radial basis functions
    • Hosseini V.R., Chen W., Avazzadeh Z. Numerical solution of fractional telegraph equation by using radial basis functions. Eng. Anal. Bound. Elem. 2014, 38:31-39.
    • (2014) Eng. Anal. Bound. Elem. , vol.38 , pp. 31-39
    • Hosseini, V.R.1    Chen, W.2    Avazzadeh, Z.3
  • 30
    • 84871790575 scopus 로고    scopus 로고
    • Numerical methods for solving the multi-term time-fractional wave-diffusion equation
    • Liu F., Meerschaert M.M., McGough R.J., Zhuang P., Liu Q. Numerical methods for solving the multi-term time-fractional wave-diffusion equation. Fract. Calc. Appl. Anal. 2013, 16(1):9-25.
    • (2013) Fract. Calc. Appl. Anal. , vol.16 , Issue.1 , pp. 9-25
    • Liu, F.1    Meerschaert, M.M.2    McGough, R.J.3    Zhuang, P.4    Liu, Q.5
  • 31
    • 84861716404 scopus 로고    scopus 로고
    • The Sinc-Legendre collocation method for a class of fractional convection-diffusion equations with variable coefficients
    • Saadatmandi A., Dehghan M., Azizi M.R. The Sinc-Legendre collocation method for a class of fractional convection-diffusion equations with variable coefficients. Commun. Nonlinear Sci. Numer. Simul. 2012, 17:4125-4136.
    • (2012) Commun. Nonlinear Sci. Numer. Simul. , vol.17 , pp. 4125-4136
    • Saadatmandi, A.1    Dehghan, M.2    Azizi, M.R.3
  • 32
    • 69049086472 scopus 로고    scopus 로고
    • Compact finite difference method for the fractional diffusion equation
    • Cui M.R. Compact finite difference method for the fractional diffusion equation. J. Comput. Phys. 2009, 228:7792-7804.
    • (2009) J. Comput. Phys. , vol.228 , pp. 7792-7804
    • Cui, M.R.1
  • 33
    • 84874394694 scopus 로고    scopus 로고
    • Convergence analysis of high-order compact alternating direction implicit schemes for the two-dimensional time fractional diffusion equation
    • Cui M.R. Convergence analysis of high-order compact alternating direction implicit schemes for the two-dimensional time fractional diffusion equation. Numer. Algorithms 2013, 62:383-409.
    • (2013) Numer. Algorithms , vol.62 , pp. 383-409
    • Cui, M.R.1
  • 34
    • 77957822720 scopus 로고    scopus 로고
    • Initial-boundary value problems for the generalized multi-term time-fractional diffusion equation
    • Luchko Y. Initial-boundary value problems for the generalized multi-term time-fractional diffusion equation. J. Math. Anal. Appl. 2011, 374:538-548.
    • (2011) J. Math. Anal. Appl. , vol.374 , pp. 538-548
    • Luchko, Y.1
  • 35
    • 84868214613 scopus 로고    scopus 로고
    • Numerical methods and analysis for a class of fractional advection-dispersion models
    • Liu F., Zhuang P., Burrage K. Numerical methods and analysis for a class of fractional advection-dispersion models. Comput. Math. Appl. 2012, 64:2990-3007.
    • (2012) Comput. Math. Appl. , vol.64 , pp. 2990-3007
    • Liu, F.1    Zhuang, P.2    Burrage, K.3
  • 36
    • 84882382834 scopus 로고    scopus 로고
    • A Jacobi-Gauss-Lobatto collocation method for solving generalized Fitzhugh-Nagumo equation with time-dependent coefficients
    • Bhrawy A.H. A Jacobi-Gauss-Lobatto collocation method for solving generalized Fitzhugh-Nagumo equation with time-dependent coefficients. Appl. Math. Comput. 2013, 222:255-264.
    • (2013) Appl. Math. Comput. , vol.222 , pp. 255-264
    • Bhrawy, A.H.1
  • 37
    • 40849129906 scopus 로고    scopus 로고
    • A direct solver for the least-squares spectral collocation system on rectangular elements for the incompressible Navier-Stokes equations
    • Heinrichs W., Kattelans T. A direct solver for the least-squares spectral collocation system on rectangular elements for the incompressible Navier-Stokes equations. J. Comput. Phys. 2008, 227:4776-4796.
    • (2008) J. Comput. Phys. , vol.227 , pp. 4776-4796
    • Heinrichs, W.1    Kattelans, T.2
  • 38
    • 84892836298 scopus 로고    scopus 로고
    • Jacobi-Gauss-Lobatto collocation method for the numerical solution of 1+1 nonlinear Schrödinger equations
    • Doha E.H., Bhrawy A.H., Abdelkawy M.A., Van Gorder R.A. Jacobi-Gauss-Lobatto collocation method for the numerical solution of 1+1 nonlinear Schrödinger equations. J. Comput. Phys. 2014, 261:244-255.
    • (2014) J. Comput. Phys. , vol.261 , pp. 244-255
    • Doha, E.H.1    Bhrawy, A.H.2    Abdelkawy, M.A.3    Van Gorder, R.A.4
  • 39
    • 84934292514 scopus 로고    scopus 로고
    • Numer. solution of fractional advection-diffusion equation with a nonlinear source term, Numer. Algorithms , in press.
    • M. Parvizi, M.R. Eslahchi, M. Dehghan, Numer. solution of fractional advection-diffusion equation with a nonlinear source term, Numer. Algorithms (2014), in press. doi:10.1007/s11075-014-9863-7.
    • (2014)
    • Parvizi, M.1    Eslahchi, M.R.2    Dehghan, M.3
  • 40
    • 84884238940 scopus 로고    scopus 로고
    • Application of the collocation method for solving nonlinear fractional integro-differential equations
    • Eslahchi M.R., Dehghan M., Parvizi M. Application of the collocation method for solving nonlinear fractional integro-differential equations. J. Comput. Appl. Math. 2014, 257:105-128.
    • (2014) J. Comput. Appl. Math. , vol.257 , pp. 105-128
    • Eslahchi, M.R.1    Dehghan, M.2    Parvizi, M.3
  • 41
    • 84896885047 scopus 로고    scopus 로고
    • Numerical solution for a class of fractional convection-diffusion equation using the flatlet oblique multiwavelets
    • Dehghan M., Abdi-mazraeh S., Lakestani M. Numerical solution for a class of fractional convection-diffusion equation using the flatlet oblique multiwavelets. J. Vib. Control 2014, 20:913-924.
    • (2014) J. Vib. Control , vol.20 , pp. 913-924
    • Dehghan, M.1    Abdi-mazraeh, S.2    Lakestani, M.3
  • 42
    • 84896826819 scopus 로고    scopus 로고
    • A method for obtaining the operational matrix of the fractional Jacobi functions and applications
    • Kayedi-Bardeh A., Eslahchi M.R., Dehghan M. A method for obtaining the operational matrix of the fractional Jacobi functions and applications. J. Vib. Control 2014, 20:736-748.
    • (2014) J. Vib. Control , vol.20 , pp. 736-748
    • Kayedi-Bardeh, A.1    Eslahchi, M.R.2    Dehghan, M.3
  • 43
    • 84875833942 scopus 로고    scopus 로고
    • Numerical solution of a class of fractional optimal control problems via the Legendre orthonormal basis combined with the operational matrix and the Gauss quadrature rule
    • Lotfi A., Yousefi S.A., Dehghan M. Numerical solution of a class of fractional optimal control problems via the Legendre orthonormal basis combined with the operational matrix and the Gauss quadrature rule. J. Comput. Appl. Math. 2013, 250:143-160.
    • (2013) J. Comput. Appl. Math. , vol.250 , pp. 143-160
    • Lotfi, A.1    Yousefi, S.A.2    Dehghan, M.3
  • 45
    • 84861893481 scopus 로고    scopus 로고
    • A new Jacobi operational matrix: an application for solving fractional differential equations
    • Doha E.H., Bhrawy A.H., Ezz-Eldien S.S. A new Jacobi operational matrix: an application for solving fractional differential equations. Appl. Math. Model. 2012, 36:4931-4943.
    • (2012) Appl. Math. Model. , vol.36 , pp. 4931-4943
    • Doha, E.H.1    Bhrawy, A.H.2    Ezz-Eldien, S.S.3
  • 47
    • 77952938132 scopus 로고    scopus 로고
    • Operational matrices of Bernstein polynomials and their applications
    • Yousefi S.A., Behroozifar M. Operational matrices of Bernstein polynomials and their applications. Int. J. Syst. Sci. 2010, 32:709-716.
    • (2010) Int. J. Syst. Sci. , vol.32 , pp. 709-716
    • Yousefi, S.A.1    Behroozifar, M.2
  • 48
    • 84894381039 scopus 로고    scopus 로고
    • Dirac's formalism combined with complex Fourier operational matrices to solve initial and boundary value problems
    • Labecca W., Guimaraes O., Piqueira J.R.C. Dirac's formalism combined with complex Fourier operational matrices to solve initial and boundary value problems. Commun. Nonlinear Sci. Numer. Simul. 2014, 19(8):2614-2623.
    • (2014) Commun. Nonlinear Sci. Numer. Simul. , vol.19 , Issue.8 , pp. 2614-2623
    • Labecca, W.1    Guimaraes, O.2    Piqueira, J.R.C.3
  • 49
    • 23744472619 scopus 로고    scopus 로고
    • Legendre wavelets method for the nonlinear Volterra-Fredholm integral equations
    • Razzaghi M., Yousefi S. Legendre wavelets method for the nonlinear Volterra-Fredholm integral equations. Math. Comput. Simul. 2005, 70:1-8.
    • (2005) Math. Comput. Simul. , vol.70 , pp. 1-8
    • Razzaghi, M.1    Yousefi, S.2
  • 50
    • 36048943025 scopus 로고    scopus 로고
    • Numerical solution of integro-differential equations by using CAS wavelet operational matrix of integration
    • Danfu H., Xufeng S. Numerical solution of integro-differential equations by using CAS wavelet operational matrix of integration. Appl. Math. Comput. 2007, 194:460-466.
    • (2007) Appl. Math. Comput. , vol.194 , pp. 460-466
    • Danfu, H.1    Xufeng, S.2
  • 51
    • 84887117347 scopus 로고    scopus 로고
    • Solution of nonlinear Fredholm integro-differential equations using a hybrid of block pulse functions and normalized Bernstein polynomials
    • Behiry S.H. Solution of nonlinear Fredholm integro-differential equations using a hybrid of block pulse functions and normalized Bernstein polynomials. J. Comput. Appl. Math. 2014, 260:258-265.
    • (2014) J. Comput. Appl. Math. , vol.260 , pp. 258-265
    • Behiry, S.H.1
  • 52
    • 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
  • 53
    • 84888857306 scopus 로고    scopus 로고
    • Bernstein operational matrix of fractional derivatives and its applications
    • Saadatmandi A. Bernstein operational matrix of fractional derivatives and its applications. Appl. Math. Model. 2014, 38:1365-1372.
    • (2014) Appl. Math. Model. , vol.38 , pp. 1365-1372
    • Saadatmandi, A.1
  • 54
    • 84884141435 scopus 로고    scopus 로고
    • Chebyshev operational matrix method for solving multi-order fractional ordinary differential equations
    • Atabakzadeh M.H., Akrami M.H., Erjaee G.H. Chebyshev operational matrix method for solving multi-order fractional ordinary differential equations. Appl. Math. Model. 2013, 37:8903-8911.
    • (2013) Appl. Math. Model. , vol.37 , pp. 8903-8911
    • Atabakzadeh, M.H.1    Akrami, M.H.2    Erjaee, G.H.3
  • 55
    • 84866046515 scopus 로고    scopus 로고
    • The operational matrix of fractional integration for shifted Chebyshev polynomials
    • Bhrawy A.H., Alofi A.S. The operational matrix of fractional integration for shifted Chebyshev polynomials. Appl. Math. Lett. 2013, 26:25-31.
    • (2013) Appl. Math. Lett. , vol.26 , pp. 25-31
    • Bhrawy, A.H.1    Alofi, A.S.2
  • 56
    • 79960999102 scopus 로고    scopus 로고
    • A numerical technique for solving fractional optimal control problems
    • Lotfi A., Dehghan M., Yousefi S.A. A numerical technique for solving fractional optimal control problems. Comput. Math. Appl. 2011, 62:1055-1067.
    • (2011) Comput. Math. Appl. , vol.62 , pp. 1055-1067
    • Lotfi, A.1    Dehghan, M.2    Yousefi, S.A.3
  • 57
    • 84907996080 scopus 로고    scopus 로고
    • A new operational matrix of fractional integration for shifted Jacobi polynomials
    • Bhrawy A.H., Tharwat M.M., Alghamdi M.A. A new operational matrix of fractional integration for shifted Jacobi polynomials. Bull. Malays. Math. Sci. Soc. (2) 2014, 37(4).
    • (2014) Bull. Malays. Math. Sci. Soc. (2) , vol.37 , Issue.4
    • Bhrawy, A.H.1    Tharwat, M.M.2    Alghamdi, M.A.3
  • 58
    • 0742271179 scopus 로고    scopus 로고
    • On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials
    • Doha E.H. On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials. J. Phys. A, Math. Gen. 2004, 37:657-675.
    • (2004) J. Phys. A, Math. Gen. , vol.37 , pp. 657-675
    • Doha, E.H.1
  • 60
    • 85162564447 scopus 로고    scopus 로고
    • The fundamental solutions for multi-term modified power law wave equations in a finite domain
    • Jiang H., Liu F., Meerschaert M.M., et al. The fundamental solutions for multi-term modified power law wave equations in a finite domain. Electron. J. Math. Anal. Appl. 2013, 1:1.
    • (2013) Electron. J. Math. Anal. Appl. , vol.1 , pp. 1
    • Jiang, H.1    Liu, F.2    Meerschaert, M.M.3
  • 61
    • 56749132453 scopus 로고    scopus 로고
    • Analytical time-domain Greens functions for power-law media
    • Kelly J.K., McGough R.J., Meerschaert M.M. Analytical time-domain Greens functions for power-law media. J. Acoust. Soc. Am. 2008, 124:2861-2872.
    • (2008) J. Acoust. Soc. Am. , vol.124 , pp. 2861-2872
    • Kelly, J.K.1    McGough, R.J.2    Meerschaert, M.M.3
  • 62
    • 69249214155 scopus 로고    scopus 로고
    • Numerical methods for fractional partial differential equations with Riesz space fractional derivatives
    • Yang Q., Liu F., Turner I. Numerical methods for fractional partial differential equations with Riesz space fractional derivatives. Appl. Math. Model. 2010, 34:200-218.
    • (2010) Appl. Math. Model. , vol.34 , pp. 200-218
    • Yang, Q.1    Liu, F.2    Turner, I.3
  • 63
    • 84875877772 scopus 로고    scopus 로고
    • Fractional wave equation and damped waves
    • Luchko Y. Fractional wave equation and damped waves. J. Math. Phys. 2013, 54:031505.
    • (2013) J. Math. Phys. , vol.54 , pp. 031505
    • Luchko, Y.1
  • 64
    • 84867573206 scopus 로고    scopus 로고
    • The analytical solution and numerical solution of the fractional diffusion-wave equation with damping
    • Chen J., Liu F., Anh V., Shen S., Liu Q., Liao C. The analytical solution and numerical solution of the fractional diffusion-wave equation with damping. Appl. Math. Comput. 2012, 219:1737-1748.
    • (2012) Appl. Math. Comput. , vol.219 , pp. 1737-1748
    • Chen, J.1    Liu, F.2    Anh, V.3    Shen, S.4    Liu, Q.5    Liao, C.6
  • 65
    • 33644585230 scopus 로고
    • Collocation and residual correction
    • Oliveira Collocation and residual correction. Numer. Math. 1980, 36:27-31.
    • (1980) Numer. Math. , vol.36 , pp. 27-31
    • Oliveira1
  • 66
    • 25144500559 scopus 로고    scopus 로고
    • Numerical solution of the general form linear Fredholm-Volterra integro-differential equations by the Tau method with an error estimation
    • Shahmorad S. Numerical solution of the general form linear Fredholm-Volterra integro-differential equations by the Tau method with an error estimation. Appl. Math. Comput. 2005, 167:1418-1429.
    • (2005) Appl. Math. Comput. , vol.167 , pp. 1418-1429
    • Shahmorad, S.1
  • 67
    • 0034289836 scopus 로고    scopus 로고
    • On the history of multivariate polynomial interpolation
    • Gasca M., Sauer T. On the history of multivariate polynomial interpolation. J. Comput. Appl. Math. 2000, 122:23-35.
    • (2000) J. Comput. Appl. Math. , vol.122 , pp. 23-35
    • Gasca, M.1    Sauer, T.2
  • 69
    • 67349156357 scopus 로고    scopus 로고
    • Numerical solution of the system of nonlinear Volterra integro-differential equations with nonlinear differential part by the operational Tau method and error estimation
    • Abbasbandy S., Taati A. Numerical solution of the system of nonlinear Volterra integro-differential equations with nonlinear differential part by the operational Tau method and error estimation. J. Comput. Appl. Math. 2009, 231(1):106-113.
    • (2009) J. Comput. Appl. Math. , vol.231 , Issue.1 , pp. 106-113
    • Abbasbandy, S.1    Taati, A.2
  • 70
    • 84864290662 scopus 로고    scopus 로고
    • An efficient algorithm for solving multi-pantograph equation systems
    • Yüzbaşi S. An efficient algorithm for solving multi-pantograph equation systems. Comput. Math. Appl. 2012, 64(4):589-603.
    • (2012) Comput. Math. Appl. , vol.64 , Issue.4 , pp. 589-603
    • Yüzbaşi, S.1
  • 71
    • 83055181136 scopus 로고    scopus 로고
    • Richardson extrapolation combined with the sequential splitting procedure and θ-method
    • Zlatev Z., Faragó I., Havasi Á. Richardson extrapolation combined with the sequential splitting procedure and θ-method. Cent. Eur. J. Math. 2012, 10(1):159-172.
    • (2012) Cent. Eur. J. Math. , vol.10 , Issue.1 , pp. 159-172
    • Zlatev, Z.1    Faragó, I.2    Havasi, Á.3
  • 72
    • 34250387053 scopus 로고
    • The convergence rates of expansions in Jacobi polynomials
    • Main M., Delves L.M. The convergence rates of expansions in Jacobi polynomials. Numer. Math. 1977, 27:219-225.
    • (1977) Numer. Math. , vol.27 , pp. 219-225
    • Main, M.1    Delves, L.M.2
  • 73
    • 84887094660 scopus 로고    scopus 로고
    • Improved matrix transform method for the Riesz space fractional reaction dispersion equation
    • Zhang Y.X., Ding H.F. Improved matrix transform method for the Riesz space fractional reaction dispersion equation. Comput. Math. Appl. 2014, 260:266-280.
    • (2014) Comput. Math. Appl. , vol.260 , pp. 266-280
    • Zhang, Y.X.1    Ding, H.F.2
  • 74
    • 84879751962 scopus 로고    scopus 로고
    • Stability and convergence of an implicit difference approximation for the space Riesz fractional reaction-dispersion equation
    • Chen J., Liu F. Stability and convergence of an implicit difference approximation for the space Riesz fractional reaction-dispersion equation. Numer. Math. J. Chinese Univ. English Ser. 2007, 16:253-264.
    • (2007) Numer. Math. J. Chinese Univ. English Ser. , vol.16 , pp. 253-264
    • Chen, J.1    Liu, F.2


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