메뉴 건너뛰기




Volumn 2004, Issue , 2010, Pages 1-30

The Analysis of Fractional Differential Equations

Author keywords

[No Author keywords available]

Indexed keywords


EID: 84863322756     PISSN: 00758434     EISSN: 16179692     Source Type: Book Series    
DOI: None     Document Type: Chapter
Times cited : (483)

References (194)
  • 1
    • 84941377590 scopus 로고
    • Auflösung einer mechanischen Aufgabe
    • Abel, N.H.: Auflösung einer mechanischen Aufgabe. J. Reine Angew. Math. 1, 153–157 (1826)
    • (1826) J. Reine Angew. Math. , Issue.1 , pp. 153-157
    • Abel, N.H.1
  • 2
    • 85008014734 scopus 로고
    • Handbook of Mathematical Functions, 2nd printing with corrections. National Bureau of Standards, Washington (1964); republished by Dover
    • Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions, 2nd printing with corrections. National Bureau of Standards, Washington (1964); republished by Dover, New York (1965)
    • (1965) New York
    • Abramowitz, M.1    Stegun, I.A.2
  • 4
    • 77949264980 scopus 로고    scopus 로고
    • A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions
    • Agarwal, R.P., Benchohra, M., Hamani, S.: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math. 109, 973–1033 (2010)
    • (2010) Acta Appl. Math. , vol.109 , pp. 973-1033
    • Agarwal, R.P.1    Benchohra, M.2    Hamani, S.3
  • 5
    • 33947164737 scopus 로고    scopus 로고
    • Fractional-order dynamical models of love
    • Ahmad, W.M., El-Khazali, R.: Fractional-order dynamical models of love. Chaos Solitons Fractals 33, 1367–1375 (2007)
    • (2007) Chaos Solitons Fractals , vol.33 , pp. 1367-1375
    • Ahmad, W.M.1    El-Khazali, R.2
  • 6
    • 0010999805 scopus 로고
    • Einführung in die Theorie der Gammafunktion. Teubner
    • English translation: The Gamma Function. Holt, Rinehart and Winston, New York
    • Artin, E.: Einführung in die Theorie der Gammafunktion. Teubner, Leipzig (1931); English translation: The Gamma Function. Holt, Rinehart and Winston, New York (1964)
    • (1964) Leipzig , vol.1931
    • Artin, E.1
  • 8
    • 0009933441 scopus 로고    scopus 로고
    • A threshold phenomenon in the propagation of a point source initiated flame
    • Audounet, J., Giovangigli, V., Roquejoffre, J.-M.: A threshold phenomenon in the propagation of a point source initiated flame. Physica D 121, 295–316 (1998)
    • (1998) Physica D , vol.121 , pp. 295-316
    • Audounet, J.1    Giovangigli, V.2    Roquejoffre, J.-M.3
  • 9
    • 0026124343 scopus 로고
    • Fractional order state equations for the control of viscoelastically damped structures
    • Bagley, R.L., Calico, R.A.: Fractional order state equations for the control of viscoelastically damped structures. J. Guid. Contr. Dynam. 14, 304–311 (1991)
    • (1991) J. Guid. Contr. Dynam. , vol.14 , pp. 304-311
    • Bagley, R.L.1    Calico, R.A.2
  • 10
    • 0001342751 scopus 로고    scopus 로고
    • On the existence of the order domain and the solution of distributed order equations – Part I
    • Bagley, R.L., Torvik, P.J.: On the existence of the order domain and the solution of distributed order equations – Part I. Int. J. Appl. Math. 2, 865–882 (2000)
    • (2000) Int. J. Appl. Math. , vol.2 , pp. 865-882
    • Bagley, R.L.1    Torvik, P.J.2
  • 11
    • 0001342751 scopus 로고    scopus 로고
    • On the existence of the order domain and the solution of distributed order equations – Part II
    • Bagley, R.L., Torvik, P.J.: On the existence of the order domain and the solution of distributed order equations – Part II. Int. J. Appl. Math. 2, 965–987 (2000)
    • (2000) Int. J. Appl. Math. , vol.2 , pp. 965-987
    • Bagley, R.L.1    Torvik, P.J.2
  • 12
    • 34648831966 scopus 로고    scopus 로고
    • Fractional-order anisotropic diffusion for image denoising
    • Bai, J., Feng, X.-C.: Fractional-order anisotropic diffusion for image denoising. IEEE Trans. Image Process. 16, 2492–2502 (2007)
    • (2007) IEEE Trans. Image Process. , vol.16 , pp. 2492-2502
    • Bai, J.1    Feng, X.-C.2
  • 14
    • 0002502625 scopus 로고
    • On the motion of a sphere in a viscous liquid
    • Basset, A.B.: On the motion of a sphere in a viscous liquid. Philos. Trans. R. Soc. A 179, 43–63 (1888)
    • (1888) Philos. Trans. R. Soc. A , vol.179 , pp. 43-63
    • Basset, A.B.1
  • 15
    • 0345432546 scopus 로고
    • On the descent of a sphere in a viscous liquid
    • Basset, A.B.: On the descent of a sphere in a viscous liquid. Q. J. Pure Appl. Math. 41, 369–381 (1910)
    • (1910) Q. J. Pure Appl. Math. , vol.41 , pp. 369-381
    • Basset, A.B.1
  • 16
    • 58249103422 scopus 로고    scopus 로고
    • Boundary value problems for differential equations with fractional order
    • Benchohra, M., Hamani, S., Ntouyas, S.K.: Boundary value problems for differential equations with fractional order. Surv. Math. Appl. 3, 1–12 (2008)
    • (2008) Surv. Math. Appl. , vol.3 , pp. 1-12
    • Benchohra, M.1    Hamani, S.2    Ntouyas, S.K.3
  • 18
    • 34247394106 scopus 로고    scopus 로고
    • On systems of linear fractional differential equations with constant coefficients
    • Bonilla, B., Rivero, M., Trujillo, J.J.: On systems of linear fractional differential equations with constant coefficients. Appl. Math. Comput. 187, 68–78 (2007)
    • (2007) Appl. Math. Comput. , vol.187 , pp. 68-78
    • Bonilla, B.1    Rivero, M.2    Trujillo, J.J.3
  • 19
    • 0003855437 scopus 로고
    • Vandenhoeck & Ruprecht, Göttingen
    • Braß, H.: Quadraturverfahren. Vandenhoeck & Ruprecht, Göttingen (1977)
    • (1977) Quadraturverfahren
    • Braß, H.1
  • 21
    • 0040575075 scopus 로고    scopus 로고
    • The piecewise polynomial collocation method for nonlinear weakly singular Volterra equations
    • Brunner, H., Pedas, A., Vainikko, G.: The piecewise polynomial collocation method for nonlinear weakly singular Volterra equations. Math. Comput. 68, 1079–1095 (1999)
    • (1999) Math. Comput. , vol.68 , pp. 1079-1095
    • Brunner, H.1    Pedas, A.2    Vainikko, G.3
  • 23
    • 84941377590 scopus 로고
    • Auflösung einer mechanischen Aufgabe
    • Abel, N.H.: Auflösung einer mechanischen Aufgabe. J. Reine Angew. Math. 1, 153–157 (1826)
    • (1826) J. Reine Angew. Math. , Issue.1 , pp. 153-157
    • Abel, N.H.1
  • 24
    • 84941377590 scopus 로고
    • Auflösung einer mechanischen Aufgabe
    • Abel, N.H.: Auflösung einer mechanischen Aufgabe. J. Reine Angew. Math. 1, 153–157 (1826)
    • (1826) J. Reine Angew. Math. , Issue.1 , pp. 153-157
    • Abel, N.H.1
  • 25
    • 51649182765 scopus 로고
    • Linear models of dissipation in anelastic solids
    • Caputo, M., Mainardi, F.: Linear models of dissipation in anelastic solids. Rivista del Nuovo Cimento 1, 161–198 (1971)
    • (1971) Rivista Del Nuovo Cimento , vol.1 , pp. 161-198
    • Caputo, M.1    Mainardi, F.2
  • 26
    • 18444384521 scopus 로고    scopus 로고
    • Statistical origins of fractional derivatives in viscoelasticity
    • Chatterjee, A.: Statistical origins of fractional derivatives in viscoelasticity. J. Sound Vib. 284, 1239–1245 (2005)
    • (2005) J. Sound Vib. , vol.284 , pp. 1239-1245
    • Chatterjee, A.1
  • 27
    • 34249883566 scopus 로고    scopus 로고
    • The final value theorem revisited – infinite limits and irrational functions
    • Chen, J., Lundberg, K.H., Davison, D.E., Bernstein, D.S.: The final value theorem revisited – infinite limits and irrational functions. IEEE Contr. Syst. Mag. 27 (3), 97–99 (2007)
    • (2007) IEEE Contr. Syst. Mag. , vol.27 , Issue.3 , pp. 97-99
    • Chen, J.1    Lundberg, K.H.2    Davison, D.E.3    Bernstein, D.S.4
  • 31
    • 79959753668 scopus 로고    scopus 로고
    • Image processing by means of a linear integro-differential equation
    • Hamza, M.H. (ed.), ACTA Press, Calgary
    • Cuesta, E., Finat Codes, J.: Image processing by means of a linear integro-differential equation. In: Hamza, M.H. (ed.) Visualization, Imaging, and Image Processing 2003, Paper 91. ACTA Press, Calgary (2003)
    • (2003) Visualization, Imaging, and Image Processing 2003, Paper , pp. 91
    • Cuesta, E.1    Finat Codes, J.2
  • 32
    • 84966251516 scopus 로고
    • Asymptotic expansions for product integration
    • de Hoog, F., Weiss, R.: Asymptotic expansions for product integration. Math. Comput. 27, 295–306 (1973)
    • (1973) Math. Comput. , vol.27 , pp. 295-306
    • de Hoog, F.1    Weiss, R.2
  • 33
    • 34249335010 scopus 로고    scopus 로고
    • Short memory principle and a predictor-corrector approach for fractional differential equations
    • Deng, W.: Short memory principle and a predictor-corrector approach for fractional differential equations. J. Comput. Appl. Math. 206, 174–188 (2007)
    • (2007) J. Comput. Appl. Math. , vol.206 , pp. 174-188
    • Deng, W.1
  • 34
    • 0001618393 scopus 로고    scopus 로고
    • An algorithm for the numerical solution of differential equations of fractional order
    • Diethelm, K.: An algorithm for the numerical solution of differential equations of fractional order. Electron. Trans. Numer. Anal. 5, 1–6 (1997)
    • (1997) Electron. Trans. Numer. Anal. , vol.5 , pp. 1-6
    • Diethelm, K.1
  • 35
    • 30444454946 scopus 로고    scopus 로고
    • Predictor-corrector strategies for single-and multi-term fractional differential equations
    • Lipitakis, E.A. (ed.), pp., LEA Press, Athens
    • Diethelm, K.: Predictor-corrector strategies for single-and multi-term fractional differential equations. In: Lipitakis, E.A. (ed.) Proceedings of the 5th Hellenic-European Conference on Computer Mathematics and Its Applications, pp. 117–122. LEA Press, Athens (2002)
    • (2002) Proceedings of the 5Th Hellenic-European Conference on Computer Mathematics and Its Applications , pp. 117-122
    • Diethelm, K.1
  • 36
    • 0348158242 scopus 로고    scopus 로고
    • mE methods
    • mE methods. Computing 71, 305–319 (2003)
    • (2003) Computing , vol.71 , pp. 305-319
    • Diethelm, K.1
  • 37
    • 77956152726 scopus 로고    scopus 로고
    • Smoothness properties of solutions of Caputo-type fractional differential equations
    • Diethelm, K.: Smoothness properties of solutions of Caputo-type fractional differential equations. Fract. Calc. Appl. Anal. 10, 151–160 (2007)
    • (2007) Fract. Calc. Appl. Anal. , vol.10 , pp. 151-160
    • Diethelm, K.1
  • 38
    • 42149141688 scopus 로고    scopus 로고
    • An investigation of some nonclassical methods for the numerical approximation of Caputo-type fractional derivatives
    • Diethelm, K.: An investigation of some nonclassical methods for the numerical approximation of Caputo-type fractional derivatives. Numer. Algorithms 47, 361–390 (2008)
    • (2008) Numer. Algorithms , vol.47 , pp. 361-390
    • Diethelm, K.1
  • 39
    • 70450163815 scopus 로고    scopus 로고
    • Multi-term fractional differential equations, multi-order fractional differential systems and their numerical solution
    • Diethelm, K.: Multi-term fractional differential equations, multi-order fractional differential systems and their numerical solution. J. Eur. Syst. Autom. 42, 665–676 (2008)
    • (2008) J. Eur. Syst. Autom. , vol.42 , pp. 665-676
    • Diethelm, K.1
  • 40
    • 77956153293 scopus 로고    scopus 로고
    • On the separation of solutions of fractional differential equations
    • Diethelm, K.: On the separation of solutions of fractional differential equations. Fract. Calc. Appl. Anal. 11, 259–268 (2008)
    • (2008) Fract. Calc. Appl. Anal. , vol.11 , pp. 259-268
    • Diethelm, K.1
  • 41
    • 76049087706 scopus 로고    scopus 로고
    • An improvement of a nonclassical numerical method for the computation of fractional derivatives
    • Diethelm, K.: An improvement of a nonclassical numerical method for the computation of fractional derivatives. J. Vib. Acoust. 131, 014502 (2009)
    • (2009) J. Vib. Acoust. , vol.131
    • Diethelm, K.1
  • 42
    • 0012570827 scopus 로고    scopus 로고
    • Numerical solution methods for distributed order differential equations
    • Diethelm, K., Ford, N.J.: Numerical solution methods for distributed order differential equations. Fract. Calc. Appl. Anal. 4, 531–542 (2001)
    • (2001) Fract. Calc. Appl. Anal. , vol.4 , pp. 531-542
    • Diethelm, K.1    Ford, N.J.2
  • 43
    • 0037081673 scopus 로고    scopus 로고
    • Analysis of fractional differential equations
    • Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. J. Math. Anal. Appl. 265, 229–248 (2002)
    • (2002) J. Math. Anal. Appl. , vol.265 , pp. 229-248
    • Diethelm, K.1    Ford, N.J.2
  • 44
    • 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 42, 490–507 (2002)
    • (2002) BIT , vol.42 , pp. 490-507
    • Diethelm, K.1    Ford, N.J.2
  • 45
    • 3042709180 scopus 로고    scopus 로고
    • Multi-order fractional differential equations and their numerical solution
    • Diethelm, K., Ford, N.J.: Multi-order fractional differential equations and their numerical solution. Appl. Math. Comput. 154, 621–640 (2004)
    • (2004) Appl. Math. Comput. , vol.154 , pp. 621-640
    • Diethelm, K.1    Ford, N.J.2
  • 46
    • 58549120748 scopus 로고    scopus 로고
    • Numerical analysis for distributed order differential equations
    • Diethelm, K., Ford, N.J.: Numerical analysis for distributed order differential equations. J. Comput. Appl. Math. 225, 96–104 (2009)
    • (2009) J. Comput. Appl. Math. , vol.225 , pp. 96-104
    • Diethelm, K.1    Ford, N.J.2
  • 47
    • 85009408983 scopus 로고    scopus 로고
    • Volterra integral equations and fractional calculus: Do neighbouring solutions intersect?
    • in press
    • Diethelm, K., Ford, N.J.: Volterra integral equations and fractional calculus: do neighbouring solutions intersect? J. Integr. Equ. Appl. (in press)
    • J. Integr. Equ. Appl.
    • Diethelm, K.1    Ford, N.J.2
  • 48
    • 0036650479 scopus 로고    scopus 로고
    • A predictor-corrector approach for the numerical solution of fractional differential equations
    • Diethelm, K., Ford, N.J., Freed, A.D.: A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dynam. 29, 3–22 (2002)
    • (2002) Nonlinear Dynam , vol.29 , pp. 3-22
    • Diethelm, K.1    Ford, N.J.2    Freed, A.D.3
  • 49
    • 4043121080 scopus 로고    scopus 로고
    • Detailed error analysis for a fractional Adams method
    • Diethelm, K., Ford, N.J., Freed A.D.: Detailed error analysis for a fractional Adams method. Numer. Algorithms 36, 31–52 (2004)
    • (2004) Numer. Algorithms , vol.36 , pp. 31-52
    • Diethelm, K.1    Ford, N.J.2    Freed, A.D.3
  • 51
    • 0010186169 scopus 로고    scopus 로고
    • The FracPECE subroutine for the numerical solution of differential equations of fractional order
    • Heinzel, S., Plesser, T. (eds.), pp., Gesellschaft für wissenschaftliche Datenverarbeitung, Göttingen
    • Diethelm, K., Freed, A.D.: The FracPECE subroutine for the numerical solution of differential equations of fractional order. In: Heinzel, S., Plesser, T. (eds.) Forschung und wissenschaftliches Rechnen: Beiträge zum Heinz-Billing-Preis 1998, pp. 57–71. Gesellschaft für wissenschaftliche Datenverarbeitung, Göttingen (1999)
    • (1999) Forschung Und Wissenschaftliches Rechnen: Beiträge Zum Heinz-Billing-Preis 1998 , pp. 57-71
    • Diethelm, K.1    Freed, A.D.2
  • 52
    • 33748663764 scopus 로고    scopus 로고
    • Numerical solution of linear multi-term initial value problems of fractional order
    • Diethelm, K., Luchko, Y.: Numerical solution of linear multi-term initial value problems of fractional order. J. Comput. Anal. Appl. 6, 243–263 (2004)
    • (2004) J. Comput. Anal. Appl. , vol.6 , pp. 243-263
    • Diethelm, K.1    Luchko, Y.2
  • 53
    • 0043044718 scopus 로고    scopus 로고
    • Numerical solution of fractional order differential equations by extrapolation
    • Diethelm, K., Walz, G.: Numerical solution of fractional order differential equations by extrapolation. Numer. Algorithms 16, 231–253 (1997)
    • (1997) Numer. Algorithms , vol.16 , pp. 231-253
    • Diethelm, K.1    Walz, G.2
  • 54
    • 64549102866 scopus 로고    scopus 로고
    • A numerical approach for Joulin’s model of a point source initiated flame
    • Diethelm, K., Weilbeer, M.: A numerical approach for Joulin’s model of a point source initiated flame. Fract. Calc. Appl. Anal. 7, 191–212 (2004)
    • (2004) Fract. Calc. Appl. Anal. , vol.7 , pp. 191-212
    • Diethelm, K.1    Weilbeer, M.2
  • 55
    • 24944498872 scopus 로고
    • Weakly singular discrete Gronwall inequalities
    • Dixon, J., McKee, S.: Weakly singular discrete Gronwall inequalities. Z. Angew. Math. Mech. 66, 535–544 (1986)
    • (1986) Z. Angew. Math. Mech. , vol.66 , pp. 535-544
    • Dixon, J.1    McKee, S.2
  • 56
    • 85072819567 scopus 로고
    • Anleitung zum praktischen Gebrauch der Laplace-Transformation und der Z-Transformation, 6th edn
    • Doetsch, G.: Anleitung zum praktischen Gebrauch der Laplace-Transformation und der Z-Transformation, 6th edn. Oldenbourg, München (1989)
    • (1989) Oldenbourg, München
    • Doetsch, G.1
  • 58
    • 0003031851 scopus 로고
    • Fractional derivatives and the Cauchy problem for differential equations of fractional order
    • in Russian
    • Dzherbashyan, M.M., Nersesian, A.B.: Fractional derivatives and the Cauchy problem for differential equations of fractional order. Izv. Akad. Nauk Arm. SSR, Mat. 3, 3–29 (1968) (in Russian)
    • (1968) Izv. Akad. Nauk Arm. SSR, Mat. , vol.3 , pp. 3-29
    • Dzherbashyan, M.M.1    Nersesian, A.B.2
  • 59
    • 0037113861 scopus 로고    scopus 로고
    • The numerical solution of linear multi-term fractional differential equations: Systems of equations
    • Edwards, J.T., Ford, N.J., Simpson, A.C.: The numerical solution of linear multi-term fractional differential equations: systems of equations. J. Comput. Appl. Math. 148, 401–418 (2002)
    • (2002) J. Comput. Appl. Math. , vol.148 , pp. 401-418
    • Edwards, J.T.1    Ford, N.J.2    Simpson, A.C.3
  • 60
    • 0011092359 scopus 로고    scopus 로고
    • A comparison of Adomian’s decomposition method and Runge–Kutta methods for approximate solution of some predator prey model equations. Numerical Analysis Report 309
    • Edwards, J.T., Roberts, J.A., Ford, N.J.: A comparison of Adomian’s decomposition method and Runge–Kutta methods for approximate solution of some predator prey model equations. Numerical Analysis Report 309, Manchester Centre for Computational Mathematics (1997)
    • (1997) Manchester Centre for Computational Mathematics
    • Edwards, J.T.1    Roberts, J.A.2    Ford, N.J.3
  • 61
    • 21344497166 scopus 로고
    • An asymptotic analysis of two algorithms for certain Hadamard finite-part integrals
    • Elliott, D.: An asymptotic analysis of two algorithms for certain Hadamard finite-part integrals. IMA J. Numer. Anal. 13, 445–462 (1993)
    • (1993) IMA J. Numer. Anal. , vol.13 , pp. 445-462
    • Elliott, D.1
  • 62
    • 0003864328 scopus 로고
    • Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G. (eds.), vol., McGraw-Hill, New York
    • Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G. (eds.): Higher Transcendental Functions, vol. I. McGraw-Hill, New York (1953)
    • (1953) Higher Transcendental Functions , vol.1
  • 63
    • 85013305936 scopus 로고
    • (eds.): Higher Transcendental Functions, vol. III. McGraw-Hill, New York
    • Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G. (eds.): Higher Transcendental Functions, vol. III. McGraw-Hill, New York (1955)
    • (1955) Tricomi, F.G
    • Erdelyi, A.1    Magnus, W.2    Oberhettinger, F.3
  • 64
    • 85072827605 scopus 로고    scopus 로고
    • A critical review of the Adomian method. Tech. Report CUED/F-INFENG/TR. 277
    • Fitzgerald, W.J., Leung, C.M.: A critical review of the Adomian method. Tech. Report CUED/F-INFENG/TR. 277. University of Cambridge (1996)
    • (1996) University of Cambridge
    • Fitzgerald, W.J.1    Leung, C.M.2
  • 65
    • 33646341532 scopus 로고    scopus 로고
    • Comparison of numerical methods for fractional differential equations
    • Ford, N.J., Connolly, J.A.: Comparison of numerical methods for fractional differential equations. Commun. Pure Appl. Anal. 5, 289–307 (2006)
    • (2006) Commun. Pure Appl. Anal. , vol.5 , pp. 289-307
    • Ford, N.J.1    Connolly, J.A.2
  • 66
    • 67349259080 scopus 로고    scopus 로고
    • Systems-based decomposition schemes for the approximate solution of multi-term fractional differential equations
    • Ford, N.J., Connolly, J.A.: Systems-based decomposition schemes for the approximate solution of multi-term fractional differential equations. J. Comput. Appl. Math. 229, 382–391 (2009)
    • (2009) J. Comput. Appl. Math. , vol.229 , pp. 382-391
    • Ford, N.J.1    Connolly, J.A.2
  • 68
    • 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. 5, 203–215 (2006)
    • (2006) Biomech. Model. Mechanobiol. , vol.5 , pp. 203-215
    • Freed, A.D.1    Diethelm, K.2
  • 69
    • 10644269006 scopus 로고    scopus 로고
    • Fractional-order viscoelasticity (FOV): Constitutive development using the fractional calculus (first annual report)
    • NASA Glenn Research Center, Cleveland
    • Freed, A.D., Diethelm, K., Luchko, Y.: Fractional-order viscoelasticity (FOV): constitutive development using the fractional calculus (first annual report). Technical Memorandum 2002-211914, NASA Glenn Research Center, Cleveland (2002)
    • (2002) Technical Memorandum , vol.2002
    • Freed, A.D.1    Diethelm, K.2    Luchko, Y.3
  • 70
    • 56949091229 scopus 로고    scopus 로고
    • Fractional Adams–Moulton methods
    • Galeone, L., Garrappa, R.: Fractional Adams–Moulton methods. Math. Comput. Simul. 79, 1358–1367 (2008)
    • (2008) Math. Comput. Simul. , vol.79 , pp. 1358-1367
    • Galeone, L.1    Garrappa, R.2
  • 71
    • 44949282247 scopus 로고
    • Damping description involving fractional operators
    • Gaul, L., Klein, P., Kempfle, S.: Damping description involving fractional operators. Mech. Syst. Signal Process. 5, 81–88 (1991)
    • (1991) Mech. Syst. Signal Process. , vol.5 , pp. 81-88
    • Gaul, L.1    Klein, P.2    Kempfle, S.3
  • 73
    • 0000983674 scopus 로고
    • On a generalization of the Fourier series
    • in Russian
    • Gel’fond, A.O., Leont’ev, A.F.: On a generalization of the Fourier series. Mat. Sb. (N. S.) 29, 477–500 (1951) (in Russian)
    • (1951) Mat. Sb. (N. S.) , vol.29 , pp. 477-500
    • Gel’Fond, A.O.1    Leont’Ev, A.F.2
  • 74
    • 17844388644 scopus 로고
    • A generalization of linear laws of deformation and its application to the problems of internal friction
    • in Russian
    • Gerasimov, A.N.: A generalization of linear laws of deformation and its application to the problems of internal friction. Prikl. Mat. Mekh. 12, 251–260 (1948) (in Russian)
    • (1948) Prikl. Mat. Mekh. , vol.12 , pp. 251-260
    • Gerasimov, A.N.1
  • 75
    • 0028878140 scopus 로고
    • A fractional calculus approach to self-similar protein dynamics
    • Glöckle, W.G., Nonnenmacher, T.F.: A fractional calculus approach to self-similar protein dynamics. Biophys. J. 68, 46–53 (1995)
    • (1995) Biophys. J. , vol.68 , pp. 46-53
    • Glöckle, W.G.1    Nonnenmacher, T.F.2
  • 76
    • 0345375537 scopus 로고    scopus 로고
    • Let us teach this generalization of the final-value theorem
    • Gluskin, E.: Let us teach this generalization of the final-value theorem. Eur. J. Phys. 24, 591–597 (2003)
    • (2003) Eur. J. Phys. , vol.24 , pp. 591-597
    • Gluskin, E.1
  • 77
    • 0001983732 scopus 로고    scopus 로고
    • Fractional calculus: Some numerical methods
    • Carpinteri, A., Mainardi, F. (eds.), pp., Springer, Wien
    • Gorenflo, R.: Fractional calculus: some numerical methods. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics, pp. 277–290. Springer, Wien (1997)
    • (1997) Fractals and Fractional Calculus in Continuum Mechanics , pp. 277-290
    • Gorenflo, R.1
  • 78
    • 0032647143 scopus 로고    scopus 로고
    • Discrete random walk models for symmetric Lévy-Feller diffusion processes
    • Gorenflo, R., De Fabritiis, G., Mainardi, F.: Discrete random walk models for symmetric Lévy-Feller diffusion processes. Physica A 269, 79–89 (1999)
    • (1999) Physica A , vol.269 , pp. 79-89
    • Gorenflo, R.1    de Fabritiis, G.2    Mainardi, F.3
  • 79
    • 84941377590 scopus 로고
    • Auflösung einer mechanischen Aufgabe
    • Abel, N.H.: Auflösung einer mechanischen Aufgabe. J. Reine Angew. Math. 1, 153–157 (1826)
    • (1826) J. Reine Angew. Math. , Issue.1 , pp. 153-157
    • Abel, N.H.1
  • 81
    • 0001983732 scopus 로고    scopus 로고
    • Fractional calculus: Integral and differential equations of fractional order
    • Carpinteri, A., Mainardi, F. (eds.), pp., Springer, Wien
    • Gorenflo, R., Mainardi, F.: Fractional calculus: integral and differential equations of fractional order. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics, pp. 223–276. Springer, Wien (1997)
    • (1997) Fractals and Fractional Calculus in Continuum Mechanics , pp. 223-276
    • Gorenflo, R.1    Mainardi, F.2
  • 82
    • 0003144812 scopus 로고
    • On ultraslow and intermediate processes
    • Rusev, P., Dimovski, I., Kiryakova, V. (eds.), pp., Science Culture Technology, Singapore
    • Gorenflo, R., Rutman, R.: On ultraslow and intermediate processes. In: Rusev, P., Dimovski, I., Kiryakova, V. (eds.) Transform Methods and Special Functions, pp. 61–81. Science Culture Technology, Singapore (1995)
    • (1995) Transform Methods and Special Functions , pp. 61-81
    • Gorenflo, R.1    Rutman, R.2
  • 83
    • 0141996364 scopus 로고    scopus 로고
    • Fully discrete random walks for space-time fractional diffusion equations
    • Gorenflo, R., Vivoli, A.: Fully discrete random walks for space-time fractional diffusion equations. Signal Process. 83, 2411–2420 (2003)
    • (2003) Signal Process , vol.83 , pp. 2411-2420
    • Gorenflo, R.1    Vivoli, A.2
  • 84
    • 36849117395 scopus 로고
    • On creep and relaxation
    • Gross, B.: On creep and relaxation. J. Appl. Phys. 18, 212–221 (1947)
    • (1947) J. Appl. Phys. , vol.18 , pp. 212-221
    • Gross, B.1
  • 85
    • 0000238491 scopus 로고
    • Über “begrenzte” Derivationen und deren Anwendung
    • Grünwald, A.K.: Über “begrenzte” Derivationen und deren Anwendung. Z. Angew. Math. Phys. 12, 441–480 (1867)
    • (1867) Z. Angew. Math. Phys. , vol.12 , pp. 441-480
    • Grünwald, A.K.1
  • 86
    • 85072824494 scopus 로고
    • Lectures on Cauchy’s Problem in Linear Partial Differential Equations. Yale Univ. Press, New Haven (1923); reprinted by Dover
    • Hadamard, J.: Lectures on Cauchy’s Problem in Linear Partial Differential Equations. Yale Univ. Press, New Haven (1923); reprinted by Dover, New York (1952)
    • (1952) New York
    • Hadamard, J.1
  • 87
    • 8844268216 scopus 로고    scopus 로고
    • Some existence theorems on differential equations of generalized order through a fixed-point theorem
    • Hadid, S.B., Ta’ani, A.A., Momani, S.M.: Some existence theorems on differential equations of generalized order through a fixed-point theorem. J. Fract. Calc. 9, 45–49 (1996)
    • (1996) J. Fract. Calc. , vol.9 , pp. 45-49
    • Hadid, S.B.1    Ta’Ani, A.A.2    Momani, S.M.3
  • 91
    • 0032308350 scopus 로고    scopus 로고
    • Approximate solution of non linear differential equations with convolution product nonlinearities
    • He, J.H.: Approximate solution of non linear differential equations with convolution product nonlinearities. Comput. Methods Appl. Mech. Eng. 167, 69–73 (1998)
    • (1998) Comput. Methods Appl. Mech. Eng. , vol.167 , pp. 69-73
    • He, J.H.1
  • 92
    • 34250668369 scopus 로고    scopus 로고
    • Variational iteration method – some recent results and new interpretations
    • He, J.H.: Variational iteration method – some recent results and new interpretations. J. Comput. Appl. Math. 207, 3–17 (2007)
    • (2007) J. Comput. Appl. Math. , vol.207 , pp. 3-17
    • He, J.H.1
  • 98
    • 0035995563 scopus 로고    scopus 로고
    • The curious history of Faà di Bruno’s formula
    • Johnson, W.P.: The curious history of Faà di Bruno’s formula. Am. Math. Mon. 109, 217–234 (2002)
    • (2002) Am. Math. Mon. , vol.109 , pp. 217-234
    • Johnson, W.P.1
  • 99
    • 84950202291 scopus 로고
    • Point-source initiation of lean spherical flames of light reactants: An asymptotic theory
    • Joulin, G.: Point-source initiation of lean spherical flames of light reactants: an asymptotic theory. Combust. Sci. Technol. 43, 99–113 (1985)
    • (1985) Combust. Sci. Technol. , vol.43 , pp. 99-113
    • Joulin, G.1
  • 101
    • 1242327688 scopus 로고    scopus 로고
    • Differential equations of fractional order. Methods, results and problems
    • Kilbas, A.A., Trujillo, J.J.: Differential equations of fractional order. Methods, results and problems. I. Appl. Anal. 78, 153–192 (2001)
    • (2001) I. Appl. Anal. , vol.78 , pp. 153-192
    • Kilbas, A.A.1    Trujillo, J.J.2
  • 102
    • 25144514177 scopus 로고    scopus 로고
    • Differential equations of fractional order. Methods, results and problems. II
    • Kilbas, A.A., Trujillo, J.J.: Differential equations of fractional order. Methods, results and problems. II. Appl. Anal. 81, 435–493 (2002)
    • (2002) Appl. Anal. , vol.81 , pp. 435-493
    • Kilbas, A.A.1    Trujillo, J.J.2
  • 103
    • 76449114769 scopus 로고    scopus 로고
    • The multi-index Mittag-Leffler functions as an important class of special functions of fractional calculus
    • Kiryakova, V.: The multi-index Mittag-Leffler functions as an important class of special functions of fractional calculus. Comput. Math. Appl. 59, 1885–1895 (2010)
    • (2010) Comput. Math. Appl. , vol.59 , pp. 1885-1895
    • Kiryakova, V.1
  • 104
    • 74149091657 scopus 로고    scopus 로고
    • The special functions of fractional calculus as generalized fractional calculus operators of some basic functions
    • Kiryakova, V.: The special functions of fractional calculus as generalized fractional calculus operators of some basic functions. Comput. Math. Appl. 59, 1128–1141 (2010)
    • (2010) Comput. Math. Appl. , vol.59 , pp. 1128-1141
    • Kiryakova, V.1
  • 105
    • 84889299911 scopus 로고    scopus 로고
    • Klages, R., Radons, G., Sokolov, I.M. (eds.), Wiley-VCH, Weinheim
    • Klages, R., Radons, G., Sokolov, I.M. (eds.): Anomalous Transport: Foundations and Applications. Wiley-VCH, Weinheim (2008)
    • (2008) Anomalous Transport: Foundations and Applications
  • 106
    • 77956168580 scopus 로고    scopus 로고
    • Fractional differential equations: α-entire solutions, regular and irregular singularities
    • Kochubei, A.N.: Fractional differential equations: α-entire solutions, regular and irregular singularities. Fract. Calc. Appl. Anal. 12, 135–158 (2009)
    • (2009) Fract. Calc. Appl. Anal. , vol.12 , pp. 135-158
    • Kochubei, A.N.1
  • 107
    • 0022700808 scopus 로고
    • Polynomial operators, Stieltjes convolution, and fractional calculus in hereditary mechanics
    • Koeller, R.C.: Polynomial operators, Stieltjes convolution, and fractional calculus in hereditary mechanics. Acta Mech. 58, 251–264 (1986)
    • (1986) Acta Mech , vol.58 , pp. 251-264
    • Koeller, R.C.1
  • 110
    • 0038053920 scopus 로고    scopus 로고
    • Mathematical justification of a nonlinear integro-differential equation for the propagation of spherical flames
    • Lederman, C., Roquejoffre, J.-M., Wolanski, N.: Mathematical justification of a nonlinear integro-differential equation for the propagation of spherical flames. C. R. Math. Acad. Sci. Paris 334, 569–574 (2002)
    • (2002) C. R. Math. Acad. Sci. Paris , vol.334 , pp. 569-574
    • Lederman, C.1    Roquejoffre, J.-M.2    Wolanski, N.3
  • 111
    • 34548052741 scopus 로고    scopus 로고
    • Mehauté, A., Tenreiro Machado, J.A., Trigeassou, J.C., Sabatier, J. (eds.), Ubooks, Neusäß
    • Le Mehauté, A., Tenreiro Machado, J.A., Trigeassou, J.C., Sabatier, J. (eds.): Fractional Differentiation and Its Applications. Ubooks, Neusäß (2005)
    • (2005) Fractional Differentiation and Its Applications
  • 112
    • 0001936243 scopus 로고
    • Theory of differentiation with an arbitrary index
    • in Russian
    • Letnikov, A.V.: Theory of differentiation with an arbitrary index. Mat. Sb. 3, 1–66 (1868) (in Russian)
    • (1868) Mat. Sb. , vol.3 , pp. 1-66
    • Letnikov, A.V.1
  • 113
    • 67349162538 scopus 로고    scopus 로고
    • Comparison of homotopy analysis method and homotopy perturbation method through an evolution equation
    • Liang, S., Jeffrey, D.J.: Comparison of homotopy analysis method and homotopy perturbation method through an evolution equation. Commun. Nonlinear Sci. Numer. Simulat. 14, 4057–4064 (2009)
    • (2009) Commun. Nonlinear Sci. Numer. Simulat. , vol.14 , pp. 4057-4064
    • Liang, S.1    Jeffrey, D.J.2
  • 116
    • 0002588201 scopus 로고
    • Memoire sur quelques questions de géometrie et de mécanique, et sur un nouveau gentre pour resoudre ces questions
    • Liouville J.: Memoire sur quelques questions de géometrie et de mécanique, et sur un nouveau gentre pour resoudre ces questions. J. Ecole Polytech. 13, 1–69 (1832)
    • (1832) J. Ecole Polytech. , vol.13 , pp. 1-69
    • Liouville, J.1
  • 117
  • 118
    • 19744367302 scopus 로고    scopus 로고
    • Wave field simulation for heterogeneous porous media with singular memory drag force
    • Lu, J.-F., Hanyga, A.: Wave field simulation for heterogeneous porous media with singular memory drag force. J. Comput. Phys. 208, 651–674 (2005)
    • (2005) J. Comput. Phys. , vol.208 , pp. 651-674
    • Lu, J.-F.1    Hanyga, A.2
  • 119
    • 33646366905 scopus 로고    scopus 로고
    • Chaotic dynamics of the fractional-order Lü system and its synchronization
    • Lu, J.G.: Chaotic dynamics of the fractional-order Lü system and its synchronization. Phys. Lett. A 354, 305–311 (2006)
    • (2006) Phys. Lett. A , vol.354 , pp. 305-311
    • Lu, J.G.1
  • 120
    • 84962571892 scopus 로고
    • Runge-Kutta theory for Volterra and Abel integral equations of the second kind
    • Lubich, C.: Runge-Kutta theory for Volterra and Abel integral equations of the second kind. Math. Comput. 41, 87–102 (1983)
    • (1983) Math. Comput. , vol.41 , pp. 87-102
    • Lubich, C.1
  • 121
    • 0000717432 scopus 로고
    • Discretized fractional calculus
    • Lubich, C.: Discretized fractional calculus. SIAM J. Math. Anal. 17, 704–719 (1986)
    • (1986) SIAM J. Math. Anal. , vol.17 , pp. 704-719
    • Lubich, C.1
  • 122
    • 77952646760 scopus 로고    scopus 로고
    • A survey of weighted polynomial approximation with exponential weights
    • Lubinsky, D.S.: A survey of weighted polynomial approximation with exponential weights. Surv. Approx. Theory 3, 1–105 (2007)
    • (2007) Surv. Approx. Theory , vol.3 , pp. 1-105
    • Lubinsky, D.S.1
  • 123
    • 0012659515 scopus 로고    scopus 로고
    • An operational method for solving fractional differential equations with the Caputo derivatives
    • Luchko, Y., Gorenflo, R.: An operational method for solving fractional differential equations with the Caputo derivatives, Acta Math. Vietnamica 24, 207–233 (1999)
    • (1999) Acta Math. Vietnamica , vol.24 , pp. 207-233
    • Luchko, Y.1    Gorenflo, R.2
  • 125
    • 0001983732 scopus 로고    scopus 로고
    • Fractional calculus: Some basic problems in continuum and statistical mechanics
    • Carpinteri, A., Mainardi, F. (eds.), pp., Springer, Wien
    • Mainardi, F.: Fractional calculus: some basic problems in continuum and statistical mechanics. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics, pp. 291–348. Springer, Wien (1997)
    • (1997) Fractals and Fractional Calculus in Continuum Mechanics , pp. 291-348
    • Mainardi, F.1
  • 127
    • 0000145521 scopus 로고    scopus 로고
    • On Mittag-Leffler-type functions in fractional evolution processes
    • Mainardi, F., Gorenflo, R.: On Mittag-Leffler-type functions in fractional evolution processes. J. Comput. Appl. Math. 118, 283–299 (2000)
    • (2000) J. Comput. Appl. Math. , vol.118 , pp. 283-299
    • Mainardi, F.1    Gorenflo, R.2
  • 128
    • 0034502929 scopus 로고    scopus 로고
    • Fractional calculus and continuous-time finance II: The waiting-time distribution
    • Mainardi, F., Raberto, M., Gorenflo, R., Scalas, E.: Fractional calculus and continuous-time finance II: the waiting-time distribution. Physica A 287, 468–481 (2000)
    • (2000) Physica A , vol.287 , pp. 468-481
    • Mainardi, F.1    Raberto, M.2    Gorenflo, R.3    Scalas, E.4
  • 129
    • 0000631760 scopus 로고
    • Sur les dérivées et sur les différences des fonctions de variables réelles
    • Marchaud, A.: Sur les dérivées et sur les différences des fonctions de variables réelles. J. Math. 6, 337–425 (1927)
    • (1927) J. Math. , vol.6 , pp. 337-425
    • Marchaud, A.1
  • 130
    • 0019606817 scopus 로고
    • Differintegral interpolation from a bandlimited signal’s samples
    • Marks, R.J. II, Hall, M.W.: Differintegral interpolation from a bandlimited signal’s samples. IEEE Trans. Acoust. Speech Signal Process. 29, 872–877 (1981)
    • (1981) IEEE Trans. Acoust. Speech Signal Process. , vol.29 , pp. 872-877
    • Marks, R.J.1    Hall, M.W.2
  • 133
    • 85072833819 scopus 로고    scopus 로고
    • (eds.): Fractional Differential Systems: Models, Methods, and Applications. SMAI, Paris
    • Matignon, D., Montseny, G. (eds.): Fractional Differential Systems: Models, Methods, and Applications. SMAI, Paris (1998)
    • (1998) Montseny, G
    • Matignon, D.1
  • 134
    • 0001044887 scopus 로고
    • Relaxation in filled polymers: A fractional calculus approach
    • Metzler, R., Schick, W., Kilian, H.-G., Nonnenmacher, T.F.: Relaxation in filled polymers: a fractional calculus approach. J. Chem. Phys. 103, 7180–7186 (1995)
    • (1995) J. Chem. Phys. , vol.103 , pp. 7180-7186
    • Metzler, R.1    Schick, W.2    Kilian, H.-G.3    Nonnenmacher, T.F.4
  • 136
    • 33746839018 scopus 로고
    • Une généralisation de l’intégrale de Laplace-Abel
    • Mittag-Leffler, G.M.: Une généralisation de l’intégrale de Laplace-Abel. C. R. Acad. Sci. Paris (Ser. II) 136, 537–539 (1903)
    • (1903) C. R. Acad. Sci. Paris (Ser. II) , vol.136 , pp. 537-539
    • Mittag-Leffler, G.M.1
  • 138
    • 8844274113 scopus 로고    scopus 로고
    • Local and global uniqueness theorems on differential equations of noninteger order via Bihari’s and Gronwall’s inequalities
    • Momani, S.M.: Local and global uniqueness theorems on differential equations of noninteger order via Bihari’s and Gronwall’s inequalities. Rev. Téc. Fac. Ing. Univ. Zulia 23, 66–69 (2000)
    • (2000) Rev. Téc. Fac. Ing. Univ. Zulia , vol.23 , pp. 66-69
    • Momani, S.M.1
  • 139
    • 34247395044 scopus 로고    scopus 로고
    • Homotopy perturbation method for nonlinear partial differential equations of fractional order
    • Momani, S., Odibat, Z.: Homotopy perturbation method for nonlinear partial differential equations of fractional order. Phys. Lett. A 365, 345–350 (2007)
    • (2007) Phys. Lett. A , vol.365 , pp. 345-350
    • Momani, S.1    Odibat, Z.2
  • 141
    • 21344465606 scopus 로고
    • On the Riemann–Liouville fractional calculus and some recent applications
    • Nonnenmacher, T.F., Metzler, R.: On the Riemann–Liouville fractional calculus and some recent applications. Fractals 3, 557–566 (1995)
    • (1995) Fractals , vol.3 , pp. 557-566
    • Nonnenmacher, T.F.1    Metzler, R.2
  • 142
    • 0000266650 scopus 로고
    • A new general law of deformation
    • Nutting, P.G.: A new general law of deformation. J. Franklin Inst. 191, 679–685 (1921)
    • (1921) J. Franklin Inst. , vol.191 , pp. 679-685
    • Nutting, P.G.1
  • 143
    • 0343020693 scopus 로고
    • A general stress–strain–time formula
    • Nutting, P.G.: A general stress–strain–time formula. J. Franklin Inst. 235, 513–524 (1943)
    • (1943) J. Franklin Inst. , vol.235 , pp. 513-524
    • Nutting, P.G.1
  • 144
    • 74149090385 scopus 로고    scopus 로고
    • Analytic study on linear systems of fractional differential equations
    • Odibat, Z.M.: Analytic study on linear systems of fractional differential equations. Comput. Math. Appl. 59, 1171–1183 (2010)
    • (2010) Comput. Math. Appl. , vol.59 , pp. 1171-1183
    • Odibat, Z.M.1
  • 145
    • 39449122795 scopus 로고    scopus 로고
    • Generalized differential transform method: Application to differential equations of fractional order
    • Odibat, Z., Momani, S., Erturk, V.S.: Generalized differential transform method: application to differential equations of fractional order. Appl. Math. Comput. 197, 467–477 (2008)
    • (2008) Appl. Math. Comput. , vol.197 , pp. 467-477
    • Odibat, Z.1    Momani, S.2    Erturk, V.S.3
  • 147
    • 0016941699 scopus 로고
    • Diffusion in a semi-infinite region with nonlinear surface dissipation
    • Olmstead, W.E., Handelsman, R.A.: Diffusion in a semi-infinite region with nonlinear surface dissipation. SIAM Rev. 18, 275–291 (1976)
    • (1976) SIAM Rev , vol.18 , pp. 275-291
    • Olmstead, W.E.1    Handelsman, R.A.2
  • 148
    • 77956143817 scopus 로고
    • Über Integration von gewöhnlichen Differentialgleichungen durch Reihen. Sitz.-Ber. Heidelberger Akademie der Wissenschaften A Math.-Nat
    • Perron, O.: Über Integration von gewöhnlichen Differentialgleichungen durch Reihen. Sitz.-Ber. Heidelberger Akademie der Wissenschaften A Math.-Nat. Klasse Abhandlung 2 (1919)
    • (1919) Klasse Abhandlung , pp. 2
    • Perron, O.1
  • 152
    • 0003767029 scopus 로고
    • Fractional-order systems and fractional-order controllers. Technical report UEF-03-94, Institute for Experimental Physics, Slovak
    • Podlubny, I.: Fractional-order systems and fractional-order controllers. Technical report UEF-03-94, Institute for Experimental Physics, Slovak Acad. Sci. (1994)
    • (1994) Acad. Sci.
    • Podlubny, I.1
  • 154
    • 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. 5, 367–386 (2002)
    • (2002) Fract. Calc. Appl. Anal. , vol.5 , pp. 367-386
    • Podlubny, I.1
  • 155
    • 0000169312 scopus 로고
    • Application of fractional-order derivatives to calculation of heat load intensity change in blast furnace walls
    • Podlubny, I., Dorcak, L., Misanek, J.: Application of fractional-order derivatives to calculation of heat load intensity change in blast furnace walls. Trans. Tech. Univ. Košice 5, 137–144 (1995)
    • (1995) Trans. Tech. Univ. Košice , vol.5 , pp. 137-144
    • Podlubny, I.1    Dorcak, L.2    Misanek, J.3
  • 157
    • 84941377590 scopus 로고
    • Auflösung einer mechanischen Aufgabe
    • Abel, N.H.: Auflösung einer mechanischen Aufgabe. J. Reine Angew. Math. 1, 153–157 (1826)
    • (1826) J. Reine Angew. Math. , Issue.1 , pp. 153-157
    • Abel, N.H.1
  • 158
    • 50549176610 scopus 로고
    • The initial-and final-value theorems in Laplace transform theory
    • Rasof, B.: The initial-and final-value theorems in Laplace transform theory. J. Franklin Inst. 274, 165–177 (1962)
    • (1962) J. Franklin Inst. , vol.274 , pp. 165-177
    • Rasof, B.1
  • 159
    • 0002588102 scopus 로고
    • Nonlinear dynamical systems: On the accuracy of Adomian’s decomposition method
    • Répaci, A.: Nonlinear dynamical systems: on the accuracy of Adomian’s decomposition method. Appl. Math. Lett. 3(4), 35–39 (1990)
    • (1990) Appl. Math. Lett. , vol.3 , Issue.4 , pp. 35-39
    • Répaci, A.1
  • 160
    • 84941377590 scopus 로고
    • Auflösung einer mechanischen Aufgabe
    • Abel, N.H.: Auflösung einer mechanischen Aufgabe. J. Reine Angew. Math. 1, 153–157 (1826)
    • (1826) J. Reine Angew. Math. , Issue.1 , pp. 153-157
    • Abel, N.H.1
  • 161
    • 0042762748 scopus 로고    scopus 로고
    • Ignition and propagation in an integro-differential model for spherical flames
    • Roquejoffre, J.-M., Vázquez, J.-L.: Ignition and propagation in an integro-differential model for spherical flames. Discrete Contin. Dyn. Syst. Ser. B 2, 379–387 (2002)
    • (2002) Discrete Contin. Dyn. Syst. Ser. B , vol.2 , pp. 379-387
    • Roquejoffre, J.-M.1    Vázquez, J.-L.2
  • 162
    • 23644455602 scopus 로고
    • The development of fractional calculus 1695–1900
    • Ross, B.: The development of fractional calculus 1695–1900. Hist. Math. 4, 75–89 (1977)
    • (1977) Hist. Math. , vol.4 , pp. 75-89
    • Ross, B.1
  • 163
    • 77955238264 scopus 로고    scopus 로고
    • Reflections on two parallel ways in the progress of fractional calculus in mechanics of solids
    • Rossikhin, Yu.A.: Reflections on two parallel ways in the progress of fractional calculus in mechanics of solids. Appl. Mech. Rev. 63, 010701 (2010)
    • (2010) Appl. Mech. Rev. , vol.63
    • Rossikhin, Y.A.1
  • 164
    • 77954346342 scopus 로고    scopus 로고
    • Comparative analysis of viscoelastic models involving fractional derivatives of different orders
    • Rossikhin, Yu.A., Shitikova, M.V.: Comparative analysis of viscoelastic models involving fractional derivatives of different orders. Fract. Calc. Appl. Anal. 10, 111–121 (2007)
    • (2007) Fract. Calc. Appl. Anal. , vol.10 , pp. 111-121
    • Rossikhin, Y.A.1    Shitikova, M.V.2
  • 166
    • 84885710343 scopus 로고    scopus 로고
    • Sabatier, J., Agrawal, O.P., Tenreiro Machado, J.A. (eds.), Springer, Dordrecht
    • Sabatier, J., Agrawal, O.P., Tenreiro Machado, J.A. (eds.): Advances in Fractional Calculus. Springer, Dordrecht (2007)
    • (2007) Advances in Fractional Calculus
  • 168
    • 0041014183 scopus 로고
    • Integral representation of remainders
    • Sard, A.: Integral representation of remainders. Duke Math. J. 15, 333–345 (1948)
    • (1948) Duke Math. J. , vol.15 , pp. 333-345
    • Sard, A.1
  • 169
    • 0034275979 scopus 로고    scopus 로고
    • Fractional calculus and continuous-time finance
    • Scalas, E., Gorenflo, R., Mainardi, F.: Fractional calculus and continuous-time finance. Physica A 284, 376–384 (2000)
    • (2000) Physica A , vol.284 , pp. 376-384
    • Scalas, E.1    Gorenflo, R.2    Mainardi, F.3
  • 170
    • 1842531858 scopus 로고    scopus 로고
    • Uncoupled continuous-time random walks: Analytic solution and limiting behaviour of the master equation
    • Scalas, E., Gorenflo, R., Mainardi, F.: Uncoupled continuous-time random walks: analytic solution and limiting behaviour of the master equation. Phys. Rev. E 69, 011107 (2004)
    • (2004) Phys. Rev. E , vol.69
    • Scalas, E.1    Gorenflo, R.2    Mainardi, F.3
  • 171
    • 26444463136 scopus 로고    scopus 로고
    • On a critique of a numerical scheme for the calculation of fractionally damped dynamical systems
    • Schmidt, A., Gaul, L.: On a critique of a numerical scheme for the calculation of fractionally damped dynamical systems. Mech. Res. Commun. 33, 99–107 (2006)
    • (2006) Mech. Res. Commun. , vol.33 , pp. 99-107
    • Schmidt, A.1    Gaul, L.2
  • 172
    • 0006981852 scopus 로고
    • The rheological law underlying the Nutting equation
    • Scott Blair, G.W., Reiner, M.: The rheological law underlying the Nutting equation. Appl. Sci. Res. 2, 225–234 (1951)
    • (1951) Appl. Sci. Res. , vol.2 , pp. 225-234
    • Scott Blair, G.W.1    Reiner, M.2
  • 173
    • 34249763957 scopus 로고
    • Asymptotic formulae for zeros of a function of Mittag-Leffler’s type
    • in Russian
    • Sedletskij, A.M.: Asymptotic formulae for zeros of a function of Mittag-Leffler’s type. Anal. Math. 20, 117–132 (1994) (in Russian)
    • (1994) Anal. Math. , vol.20 , pp. 117-132
    • Sedletskij, A.M.1
  • 174
    • 55349118252 scopus 로고    scopus 로고
    • Numerical results for the generalized Mittag-Leffler functions
    • Seybold, H.J., Hilfer, R.: Numerical results for the generalized Mittag-Leffler functions. Fract. Calc. Appl. Anal. 8, 127–139 (2005)
    • (2005) Fract. Calc. Appl. Anal. , vol.8 , pp. 127-139
    • Seybold, H.J.1    Hilfer, R.2
  • 175
    • 55349106156 scopus 로고    scopus 로고
    • Numerical algorithm for calculating the generalized Mittag-Leffler function
    • Seybold, H.J., Hilfer, R.: Numerical algorithm for calculating the generalized Mittag-Leffler function. SIAM J. Numer. Anal. 47, 69–88 (2008)
    • (2008) SIAM J. Numer. Anal. , vol.47 , pp. 69-88
    • Seybold, H.J.1    Hilfer, R.2
  • 177
    • 33745906525 scopus 로고    scopus 로고
    • Galerkin projections and finite elements for fractional order derivatives
    • Singh, S.J., Chatterjee, A.: Galerkin projections and finite elements for fractional order derivatives. Nonlinear Dynam. 45, 183–206 (2006)
    • (2006) Nonlinear Dynam , vol.45 , pp. 183-206
    • Singh, S.J.1    Chatterjee, A.2
  • 180
    • 58149510910 scopus 로고    scopus 로고
    • Taş, K., Tenreiro Machado, J.A., Baleanu, D. (eds.), Springer, Dordrecht
    • Taş, K., Tenreiro Machado, J.A., Baleanu, D. (eds.): Mathematical Methods in Engineering. Springer, Dordrecht (2007)
    • (2007) Mathematical Methods in Engineering
  • 181
    • 34250696383 scopus 로고    scopus 로고
    • On the convergence of He’s variational iteration method
    • Tatari, M., Dehghan, M.: On the convergence of He’s variational iteration method. J. Comput. Appl. Math. 207, 121–128 (2007)
    • (2007) J. Comput. Appl. Math. , vol.207 , pp. 121-128
    • Tatari, M.1    Dehghan, M.2
  • 182
    • 67650122083 scopus 로고    scopus 로고
    • Comments on stability analysis of a class of nonlinear fractional-order systems
    • Tavazoei, M.S.: Comments on stability analysis of a class of nonlinear fractional-order systems. IEEE Trans. Circ. Syst. II 56, 519–520 (2009)
    • (2009) IEEE Trans. Circ. Syst. II , vol.56 , pp. 519-520
    • Tavazoei, M.S.1
  • 183
    • 67649624652 scopus 로고    scopus 로고
    • Stability preservation analysis for frequency-based methods in numerical simulation of fractional-order systems
    • Tavazoei, M.S., Haeri, M., Bolouki, S., Siami, M.: Stability preservation analysis for frequency-based methods in numerical simulation of fractional-order systems. SIAM J. Numer. Anal. 47, 321–328 (2008)
    • (2008) SIAM J. Numer. Anal. , vol.47 , pp. 321-328
    • Tavazoei, M.S.1    Haeri, M.2    Bolouki, S.3    Siami, M.4
  • 184
    • 0021439912 scopus 로고
    • On the appearance of the fractional derivative in the behavior of real materials
    • Torvik, P.J., Bagley, R.L.: On the appearance of the fractional derivative in the behavior of real materials. J. Appl. Mech. 51, 294–298 (1984)
    • (1984) J. Appl. Mech. , vol.51 , pp. 294-298
    • Torvik, P.J.1    Bagley, R.L.2
  • 185
    • 0036869926 scopus 로고    scopus 로고
    • Treatment of dynamic systems with fractional derivatives without evaluating memory-integrals
    • Trinks, C., Ruge, P.: Treatment of dynamic systems with fractional derivatives without evaluating memory-integrals. Comput. Mech. 29, 471–476 (2002)
    • (2002) Comput. Mech. , vol.29 , pp. 471-476
    • Trinks, C.1    Ruge, P.2
  • 186
    • 70749123687 scopus 로고    scopus 로고
    • Method of Fractional Derivatives
    • Artishok Publ. House, in Russian
    • Uchaikin, V.V.: Method of Fractional Derivatives. Artishok Publ. House, Ul’yanovsk (2008) (in Russian)
    • (2008) Ul’yanovsk
    • Uchaikin, V.V.1
  • 187
    • 77953541437 scopus 로고    scopus 로고
    • Fractional compartmental models and multi-term Mittag-Leffler response functions
    • Verotta, D.: Fractional compartmental models and multi-term Mittag-Leffler response functions. J. Pharmacokinet. Pharmacodyn. 37, 209–215 (2010)
    • (2010) J. Pharmacokinet. Pharmacodyn. , vol.37 , pp. 209-215
    • Verotta, D.1
  • 189
    • 84979309618 scopus 로고
    • Zur Theorie und Anwendung des Iterationsverfahrens
    • Weissinger, J.: Zur Theorie und Anwendung des Iterationsverfahrens. Math. Nachr. 8, 193–212 (1952)
    • (1952) Math. Nachr. , vol.8 , pp. 193-212
    • Weissinger, J.1
  • 191
    • 0001799540 scopus 로고
    • α(x)
    • α(x). Acta Math. 29, 191–201 (1905)
    • (1905) Acta Math , vol.29 , pp. 191-201
    • Wiman, A.1
  • 192
    • 0039293970 scopus 로고
    • α(x)
    • α(x). Acta Math. 29, 217–234 (1905)
    • (1905) Acta Math , vol.29 , pp. 217-234
    • Wiman, A.1
  • 193
    • 0033433331 scopus 로고    scopus 로고
    • Analytic continuation of operators. Applications: From number theory and group theory to quantum field and string theories
    • Woon, S.C.: Analytic continuation of operators. Applications: from number theory and group theory to quantum field and string theories. Rev. Math. Phys. 11, 463–501 (1999)
    • (1999) Rev. Math. Phys. , vol.11 , pp. 463-501
    • Woon, S.C.1
  • 194
    • 0036554885 scopus 로고    scopus 로고
    • A numerical scheme for dynamic systems containing fractional derivatives
    • Yuan, L., Agrawal, O.P.: A numerical scheme for dynamic systems containing fractional derivatives. J. Vib. Acoust. 124, 321–324 (2002)
    • (2002) J. Vib. Acoust. , vol.124 , pp. 321-324
    • Yuan, L.1    Agrawal, O.P.2


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.