Fractional Calculus: Quo Vadimus? (Where are we Going?)

References

[1] S. Abbas, M. Benchohra, and G.M. N’Guérékata, Topics in Fractional Differential Equations. Ser. Developments in Mathematics, Vol. 27, Springer, New York (2012).10.1007/978-1-4614-4036-9Suche in Google Scholar

[2] S. Abbas, M. Benchohra, and G.M. N’Guérékata, Advanced Fractional Differential and Integral Equations. Ser. Mathematics Research Developments, Nova Science Publishers (2014).Suche in Google Scholar

[3] M.H. Annaby, and Z.S. Mansour, q-Fractional Calculus and Equations. Lecture Notes in Mathematics, Vol. 2056, Springer, Heidelberg (2012).10.1007/978-3-642-30898-7Suche in Google Scholar

[4] T.M. Atanackovic, S. Pilipovic, B. Stankovic, and D. Zorica, Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes. Wiley-ISTE (2014).10.1002/9781118577530Suche in Google Scholar

[5] S. Al-Azawi, Some Results in Fractional Calculus. LAP Lambert Acad. Publ. (2011).Suche in Google Scholar

[6] B. Baeumer, and R.L. Magin, Stochastic solutions for fractional Cauchy problems. Fractional Calculus and Applied Analysis 4, No 4 (2001), 481-500.Suche in Google Scholar

[7] D. Baleanu, K. Diethelm, E. Scalas, and J.J. Trujillo, Fractional Calculus: Models and Numerical Methods. Ser. on Complexity, Nonlinearity and Chaos, World Scientific Publishing Company, Singapore (2012).10.1142/8180Suche in Google Scholar

[8] S.S. Bayin, On the consistency of the solutions of the space fractional Schrödinger equation. J. Mathematical Physics 54, No 9 (2013), 092101.10.1063/1.4819502Suche in Google Scholar

[9] M. Caputo and M. Fabrizio, Damage and fatigue by a fractional derivative model. Journal of Computational Physics (In press); doi:10.1016/j.jcp.2014.11.012.10.1016/j.jcp.2014.11.012Suche in Google Scholar

[10] Á. Cartea, and D. del-Castillo-Negrete, Fractional diffusion models of option prices in markets with jumps. Physica A: Stat. Mech. and its Appl. 374, No 2 (2007), 749-763.Suche in Google Scholar

[11] Á. Cartea, and D. del-Castillo-Negrete, Fluid limit of the continuoustime random walk with general Lévy jump distribution functions. Physical Review E 76, No 4 (2007), 041105.10.1103/PhysRevE.76.041105Suche in Google Scholar PubMed

[12] M. Cugnet, J. Sabatier, S. Laruelle, S. Grugeon, B. Sahut, A. Oustaloup, J.-M. Tarascon, On Lead-Acid-Battery Resistance and Cranking-Capability Estimation. IEEE Transactions on Industrial Electronics 57, No 3 (2010), 909-91710.1109/TIE.2009.2036643Suche in Google Scholar

[13] A. Chevalier, D. Copot, C.M. Ionescu, J.A.T. Machado, R. De Keyser, Emerging tools for quantifying unconscious analgesia: Fractional order impedance models. In: Discontinuity and Complexity in Nonlinear Physical Systems, Eds. J.A. Tenreiro Machado, D. Baleanu, A.C.J. Luo, Ser. Nonlinear Systems and Complexity, Vol. 6, Springer (2014), 135-149.10.1007/978-3-319-01411-1_8Suche in Google Scholar

[14] S. Cohen, A. Kuznetsov, A.E. Kyprianou, and V. Rivero, Lévy Matters II. Recent Progress in Theory and Applications: Fractional Lévy Fields, and Scale Functions. Lecture Notes in Mathematics, Springer, Berlin (2013).Suche in Google Scholar

[15] S. Das, and I. Pan, Fractional Order Signal Processing: Introductory Concepts and Applications. SpringerBriefs in Applied Sciences and Technology, Springer, Heidelberg (2012).Suche in Google Scholar

[16] K. Diethelm, A fractional calculus based model for the simulation of an outbreak of dengue fever. Nonlinear Dynamics 71, No 4 (2013), 613-619.Suche in Google Scholar

[17] A.I.J. Forrester, A. Sóbester, and A.J. Keane, Multi-fidelity optimization via surrogate modelling. Proc. Royal Society A 463, (2007), 3251-3269.10.1098/rspa.2007.1900Suche in Google Scholar

[18] O. Furdui, Limits, Series, and Fractional Part Integrals. Problem Books in Mathematics, Springer (2013).10.1007/978-1-4614-6762-5Suche in Google Scholar

[19] J.J. GadElkarim, R.L. Magin, M.M.Meerschaert, S. Capuani, M. Palombo, A. Kumar, and A.D. Leow, Fractional order generalization of anomalous diffusion as a multidimensional extension of the transmission line equation. IEEE J. on Emerging and Selected Topics in Circuits and Systems 3, No 3 (2013), 432-441.Suche in Google Scholar

[20] R. Garra, E. Orsingher, and F. Polito, Fractional Klein-Gordon equations and related stochastic processes. J. of Statistical Physics 155, No 4 (2014), 777-809.Suche in Google Scholar

[21] R.K. Gazizov, and A.A. Kasatkin, Construction of exact solutions for fractional order differential equations by the invariant subspace method. Computers and Math. with Appl. 66, No 5 (2013), 576-584.Suche in Google Scholar

[22] R. Gorenflo, Afterthoughts on interpretation of fractional derivatives and integrals. In: [97] (1998), 589-591.Suche in Google Scholar

[23] R. Gorenflo, A.A. Kilbas, F. Mainardi, S.V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications. Ser. Springer Monographs in Mathematics, Springer (2014). 10.1007/978-3-662-43930-2Suche in Google Scholar

[24] A. Hanyga, Multi-dimensional solutions of space-fractional diffusion equations. Proc. Royal Society A 457, No 2016 (2001), 2993-3005.Suche in Google Scholar

[25] A. Hanyga, and M. Seredynska, Anisotropy in high-resolution diffusion-weighted MRI and anomalous diffusion. J. of Magnetic Resonance 220, (2012), 85-93.10.1016/j.jmr.2012.05.001Suche in Google Scholar PubMed

[26] R. Herrmann, Fractional Calculus: An Introduction for Physicists. World Scientific Publ. Co., Singapore (2011).10.1142/8072Suche in Google Scholar

[27] R. Herrmann, Fractional Calculus - Introduction for Physicists. World Scientific, 2nd edition (2014).10.1142/8934Suche in Google Scholar

[28] R. Hilfer, Threefold introduction to fractional derivatives. In: Anomalous Transport: Foundations and Applications, Ch. 2, Sect. 3, R. Klages et al. (Eds.) (2008), 47-59.Suche in Google Scholar

[29] O.C. Ibe, Elements of Random Walk and Diffusion Processes. Wiley (2013).10.1002/9781118618059Suche in Google Scholar

[30] C. Ingo, R. L. Magin, L. Colon-Perez, W. Triplett, and T. H. Mareci, On random walks and entropy in diffusion-weighted magnetic resonance imaging studies of neural tissue. Magnetic Resonance in Medicine 71, No 2 (2014), 617-627.Suche in Google Scholar

[31] C.M. Ionescu, and R. De Keyser, Relations between fractional order model parameters and lung pathology in chronic obstructive pulmonary disease. IEEE Trans. on Biomedical Engineering 56, No 4 (2009), 978-987.Suche in Google Scholar

[32] C.M. Ionescu, I. Muntean, J.T. Machado, R. De Keyser, and M. Abrudean, A theoretical study on modelling the respiratory tract with ladder networks by means of intrinsic fractal geometry. IEEE Trans. on Biomedical Engineering 57, No 2 (2010), 246-253.Suche in Google Scholar

[33] C.M. Ionescu, J.T. Machado, and R. De Keyser, Modeling of the lung impedance using a fractional order ladder network with constant phase elements. IEEE Trans. on Biomedical Circuits and Systems 5, No 1 (2011), 83-89.Suche in Google Scholar

[34] C.M. Ionescu, R. De Keyser, J. Sabatier, A. Oustaloup, and F. Levron, Low frequency constant-phase behaviour in the respiratory impedance. Biomedical Signal Processing and Control 6, No 1 (2011), 197-208.Suche in Google Scholar

[35] C.M. Ionescu, The phase constancy in neural dynamics. IEEE Trans. on Systems, Man and Cybernetics, Part A: Systems and Humans 42, No 6 (2012), 1543-1551.Suche in Google Scholar

[36] C.M. Ionescu, The Human Respiratory System: An Analysis of the Interplay between Anatomy, Structure, Breathing and Fractal Dynamics. Series in BioEngineering, Springer (2013). Suche in Google Scholar

[37] M. Jeng, S.-L.-Y. Xu, E. Hawkins, and J.M. Schwarz, On the nonlocality of the fractional Schrödinger equation. J. Mathematical Physics 51, No 6 (2010), 062102.10.1063/1.3430552Suche in Google Scholar

[38] Z. Jiao, Y.-Q. Chen, and I. Podlubny, Distributed-Order Dynamic Systems: Stability, Simulation, Applications and Perspectives. Ser. SpringerBriefs in Electr. and Computer Eng., Springer, London (2012).Suche in Google Scholar

[39] G. Jumarie, Fractional Differential Calculus for Non-differentiable Functions: Mechanics, Geometry, Stochastics, Information Theory. LAP Lambert Academic Publishing (2014).Suche in Google Scholar

[40] T. Kaczorek, and L. Sajewski, The Realization Problem for Positive and Fractional Systems. Ser. Studies in Systems, Decision and Control, Vol. 1, Springer (2014).Suche in Google Scholar

[41] S. Kempfle, Modelling viscous damped oscillations by fractional differential operators. In: [97] (1998), 592-593.Suche in Google Scholar

[42] V. Kiryakova, A long standing conjecture failed?. In: [97] (1998), 579-588.Suche in Google Scholar

[43] V. Kiryakova, A brief story about the operators of the generalized fractional calculus. Fract. Calc. Appl. Anal. 11, No 2 (2008), 203-220, at http://www.math.bas.bg/∼fcaa.Suche in Google Scholar

[44] J. Klafter, and I. M. Sokolov, First Steps in Random Walks: From Tools to Applications. Oxford University Press, Oxford (2011).10.1093/acprof:oso/9780199234868.001.0001Suche in Google Scholar

[45] M. Kwasnicki, Eigenvalues of the fractional Laplace operator in the interval. J. of Functional Analysis 262, No 5 (2012), 2379-2402.Suche in Google Scholar

[46] L. Lao, and E. Orsingher, Hyperbolic and fractional hyperbolic Brownian motion with some applications. Stochastics 79, No 6 (2007), 505-522.Suche in Google Scholar

[47] J.S. Leszczyanski, An Introduction to Fractional Mechanics. Czestochowa University of Technology, Czestochowa (2011).Suche in Google Scholar

[48] R.A. Leo, G. Sicuro, and P. Tempesta, A theorem on the existence of symmetries of fractional PDEs. Comptes Rendus Mathematique 352, No 3 (2014), 219-222.Suche in Google Scholar

[49] F. Lindgren, H. Rue, and J. Lindström, An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach. J. Royal Statistical Society: Ser. B (Stat. Methodology) 73, No 4 (2011), 423-498.Suche in Google Scholar

[50] Yu. Luchko, Fractional Schrödinger equation for a particle moving in a potential well. J. Mathematical Physics 54, No 1 (2013), 012111.10.1063/1.4777472Suche in Google Scholar

[51] Y. Luo, and Y.-Q. Chen, Fractional Order Motion Controls. JohnWiley & Sons, New York (2012). 10.1002/9781118387726Suche in Google Scholar

[52] A. Maachou, R. Malti, P. Melchior, J.-L. Battaglia, A. Oustaloup and B. Hay, Nonlinear thermal system identification using fractional Volterra series. Control Engineering Practice 29, (2014), 50-60.10.1016/j.conengprac.2014.02.023Suche in Google Scholar

[53] R.L. Magin, O. Abdullah, D. Baleanu, and X.J. Zhou, Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation. J. of Magnetic Resonance 190, No 2 (2008), 255-270.Suche in Google Scholar

[54] R.L. Magin, X. Feng, and D. Baleanu, Fractional calculus in NMR. Magnetic Resonance Engineering 34 (2009), 16-23.Suche in Google Scholar

[55] F. Mainardi, Considerations on fractional calculus, interpretations and applications. In: [97] (1998), 594-597.Suche in Google Scholar

[56] A.B. Malinowska, and D.F.M. Torres, Introduction to the Fractional Calculus of Variations. Imperial College Press, Singapore (2012).10.1142/p871Suche in Google Scholar

[57] T.S. Margulies, Flows, Energetics, and Waves: Mathematical Applications: Physical Sciences and Engineering Analysis. CreateSpace Independent Publishing Platform (2014).Suche in Google Scholar

[58] A.M. Mathai, Jacobians of Matrix Transformations and Functions of Matrix Argument. World Scientific Publishing, New York (1997).10.1142/3438Suche in Google Scholar

[59] A.M. Mathai and H.J. Haubold, Erdélyi-Kober fractional integrals from a statistical point of view I-IV. E-prints, available at Cornell University, arXiv (2012).Suche in Google Scholar

[60] A.M. Mathai, Fractional integral operators in the complex matrixvariate case. Linear Algebra and Its Applications 439, No 10 (2013), 2901-2913.Suche in Google Scholar

[61] A.M. Mathai, Fractional integral operators involving many matrix variables. Linear Algebra and Its Applications 446 (2014), 196-215.Suche in Google Scholar

[62] A.C. McBride, Fractional powers of a class of ordinary differential operators. Proc. London Math. Society 3, No 45 (1982), 519-546.Suche in Google Scholar

[63] A.C. McBride, G.F. Roach (Eds.), Fractional Calculus (Proc. Workshop held at Ross Priory, Univ. of Strathclyde (Glasgow), Aug. 1984). Ser. Pitman Res. Notes in Math. 138, Pitman, Boston-London- Melbourne (1985).Suche in Google Scholar

[64] M.M. Meerschaert, J. Mortensen, and S.W. Wheatcraft, Fractional vector calculus for fractional advection-dispersion. Physica A: Stat. Mechanics and its Appl. 367, No C (2006), 181-190.10.1016/j.physa.2005.11.015Suche in Google Scholar

[65] M.M. Meerschaert, and A. Sikorskii, Stochastic Models for Fractional Calculus. Walter de Gruyter, Berlin (2011).10.1515/9783110258165Suche in Google Scholar

[66] V. Méndez, D. Campos, and F. Bartumeus, Anomalous Diffusion, Front Propagation and Random Searches. Ser. Stochastic Foundations in Movement Ecology, Springer (2014). 10.1007/978-3-642-39010-4Suche in Google Scholar

[67] M. Mikol`as, On the recent trends in the development, theory and applications of fractional calculus. In: [96] (1975), 357-375.10.1007/BFb0067119Suche in Google Scholar

[68] D. del-Castillo-Negrete, B.A. Carreras, and V.E. Lynch, Front dynamics in reaction-diffusion systems with Levy flights: A fractional diffusion approach. Physical Review Letters 91, No 1 (2003), 018302.10.1103/PhysRevLett.91.018302Suche in Google Scholar PubMed

[69] D. del-Castillo-Negrete, B.A. Carreras, and V.E. Lynch, Nondiffusive transport in plasma turbulence: A fractional diffusion approach. Physical Review Letters 94, No 6 (2005), 065003.10.1103/PhysRevLett.94.065003Suche in Google Scholar PubMed

[70] D. del-Castillo-Negrete, Fractional diffusion models of nonlocal transport. Physics of Plasmas 13, No 8 (2006), 082308.10.1063/1.2336114Suche in Google Scholar

[71] D. del-Castillo-Negrete, P. Mantica, V. Naulin, and J.J. Rasmussen, Fractional diffusion models of non-local perturbative transport: numerical results and application to JET experiments. Nuclear Fusion 48, No 7 (2008), 75009.10.1088/0029-5515/48/7/075009Suche in Google Scholar

[72] D. del-Castillo-Negrete, V.Yu. Gonchar, and A.V. Chechkin, Fluctuation-driven directed transport in the presence of Lévy flights. Physica A: Stat. Mechanics and its Appl. 387, No 27 (2008), 6693-6704.Suche in Google Scholar

[73] D. del-Castillo-Negrete, Truncation effects in superdiffusive front propagation with Lévy flights. Physical Review E 79, No 3 (2009), 018302.10.1103/PhysRevE.79.031120Suche in Google Scholar PubMed

[74] D. del-Castillo-Negrete, Non-diffusive, non-local transport in fluids and plasmas. Nonlinear Progress in Geophysics 17 (2010), 795-807.Suche in Google Scholar

[75] R.R. Nigmatullin, and Y.E. Ryabov, Cole-Davidson dielectric relaxation as a self-similar relaxation process. Physics of the Solid State 39, No 1 (1997), 87-90.Suche in Google Scholar

[76] K. Nishimoto (Ed.), Fractional Calculus and Its Applications (Proc. Internat. Conf. held at Nihon Univ., Tokyo, 1989). College of Eng.- Nihon University, Tokyo (1990).Suche in Google Scholar

[77] I. Nourdin, Selected Aspects of Fractional Brownian Motion. Bocconi & Springer Series, Springer, Milano (2012).10.1007/978-88-470-2823-4Suche in Google Scholar

[78] V.V. Novikov, K.W. Wojciechowski, O.A. Komkova, and T. Thiel, Anomalous relaxation in dielectrics. Equations with fractional derivatives. Materials Science-Poland 23, No 4 (2005), 977-984.Suche in Google Scholar

[79] K.B. Oldham, An introduction to the fractional calculus and some applications. In: [97] (1998), 598-609.Suche in Google Scholar

[80] M.D. Ortigueira, Fractional Calculus for Scientists and Engineers. Lecture Notes in Electrical Engineering, Springer, Dordrecht, Heidelberg (2011).10.1007/978-94-007-0747-4Suche in Google Scholar

[81] T.J. Osler, Open questions for research. In: [96] (1975), 376-381.10.1007/BFb0067120Suche in Google Scholar

[82] A. Oustaloup, F. Levron, F. Nanot, and B. Mathieu, Frequency band complex non integer differentiator: characterization and synthesis. IEEE Transactions on Circuits and Systems I 47, No 1 (2000), 25-39.Suche in Google Scholar

[83] A. Oustaloup, and X. Moreau, Mechanical version of the Crone suspension, In: Advances in the Theory of Control, Signals and Systems with Physical Modeling (Eds. J. Lévine, Ph. Müllhaupt), Springer (2010), 99-112.10.1007/978-3-642-16135-3_9Suche in Google Scholar

[84] A. Oustaloup, From diversity to unexpected dynamic performances: A simplified presentation almost in layman’s terms. In: Proc. of the 2nd IEEE CCCA’12, Plenary Lecture, 6-8 December (2012).Suche in Google Scholar

[85] A. Oustaloup, P. Lanusse, J. Sabatier and P. Melchior, CRONE Control: Principles, extensions and Applications. Journal of Applied Nonlinear Dynamics 25, No 3 (2013), 207-223.Suche in Google Scholar

[86] A. Oustaloup, Diversity and Non-integer Differentiation for System Dynamics. Wiley (2014).10.1002/9781118760864Suche in Google Scholar

[87] M. D’Ovidio, and R. Garra, Multidimensional fractional advectiondispersion equations and related stochastic processes. Electronic Journal of Probability 19, No 61 (2014), 1-31.Suche in Google Scholar

[88] M. D’Ovidio, E. Orsingher, B.Toaldo, Fractional telegraph-type equations and hyperbolic Brownian motion. Statistics and Probability Letters 89 (2014), 131-137.Suche in Google Scholar

[89] I. Pan, and S. Das, Intelligent Fractional Order Systems and Control. An Introduction. Ser. Studies in Computational Intelligence 438, Springer (2013).10.1007/978-3-642-31549-7Suche in Google Scholar

[90] M. Pellet, P. Melchior, Y. Abdelmoumen, and A. Oustaloup, Fractional thermal model of the lungs using Havriliak-Negami function. In: Proc. of the 7th ASME/IEEE MESA’11, Washington, DC, USA, August 29-31 (2011).Suche in Google Scholar

[91] P. Perdikaris and G.E. Karniadakis, Fractional-order viscoelasticity in one-dimensional blood flow models. Annals of Biomedical Engineering 42, No 5 (2012), 1012-1023.Suche in Google Scholar

[92] I. PetrÁš, Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation. Ser. Nonlinear Physical Science, Springer, Heidelberg (2011).Suche in Google Scholar

[93] J.K. Popović, M.T. Atanacković, A.S. Pilipović, M.R. Rapaić, S. Pilipović, and T.M. Atanacković, A new approach to the compartmental analysis in pharmacokinetics: Fractional time evolution of diclofenac. J. of Pharmacokinetics Pharmacodynamics 37, No 2 (2010), 119-134.Suche in Google Scholar

[94] J.K. Popović, M.T. Atanacković, A.S. Pilipović, M.R. Rapaić, S. Pilipović, and T.M. Atanacković, Erratum to: A new approach to the compartmental analysis in pharmacokinetics: fractional time evolution of diclofenac. J. of Pharmacokinetics Pharmacodynamics 38, No 1 (2010), 163-164.Suche in Google Scholar

[95] S.D. Roscani, D.A. Tarzia, A generalized Neumann solution for the two-phase fractional Lamé-Clapeyron-Stefan problem. arXiv:1405.5928v1 (2014), 042401.Suche in Google Scholar

[96] B. Ross (Editor), Fractional Calculus and Its Applications (Proc. of the Intern. Conf. held at the University of New Haven, June 1974). Lecture Notes in Mathematics No 457, Springer, Berlin (1975).10.1007/BFb0067095Suche in Google Scholar

[97] P. Rusev, I. Dimovski, V. Kiryakova (Eds.), Transform Methods & Special Functions, Varna’96 (Proc. 2nd Intern. Workshop, 23-30 Aug. 1996, Varna). Inst. Math. Inform. - Bulg. Acad. Sci, Sofia (1998).Suche in Google Scholar

[98] J. Sabatier, M. Merveillaut, R. Malti, and A. Oustaloup, How to impose physically coherent initial conditions to a fractional system?. Com. in Nonlin. Science and Num. Sim. 15, No 5 (2010), 1318-1326.Suche in Google Scholar

[99] J. Sabatier, C. Farges, and J.C. Trigeassou, Fractional systems state space description: some wrong ideas and proposed solutions. Journal of Vibration and Control 20, No 7 (2014), 1076-1084.Suche in Google Scholar

[100] J. Sabatier, and C. Farges, Long memory models: a first solution to the infinite energy storage ability of linear time-invariant fractional models. In: 19th IFAC World Congress, Cape Town, South Africa (2014).Suche in Google Scholar

[101] S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach Sci. Publ., Yverdon etc. (1993).Suche in Google Scholar

[102] H. Sheng, Y.-Q. Chen, and T.-S. Qiu, Fractional Processes and Fractional-Order Signal Processing: Techniques and Applications. Ser. Signals and Communication Technology, Springer, London (2012).10.1007/978-1-4471-2233-3Suche in Google Scholar

[103] L. Sommacal, P. Melchior, J.-M. Cabelguen, A. Oustaloup, and A.J. Ijspeert, Fractional multi-models of the gastrocnemius frog muscle. Journal of Vibration and Control 14, No 9-10 (2008), 1415-1430.10.1177/1077546307087440Suche in Google Scholar

[104] H.M. Srivastava, Open questions for further researches on fractional calculus and its applications. In: [76] (1990), 281-284.Suche in Google Scholar

[105] H.M. Srivastava, R.K. Raina, Xiao-Jun Yang, Special Functions in Fractional Calculus and Related Fractional Differintegral Equations. World Scientific Publ. Co. (2014).10.1142/8936Suche in Google Scholar

[106] V.E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Ser. Nonlinear Physical Science, Springer, Beijing-Heidelberg (2011). Suche in Google Scholar

[107] J. Tenreiro Machado, V. Kiryakova, and F. Mainardi, Recent history of fractional calculus. Commun. in Nonlinear Sci. and Numerical Simulations 16, No 3 (2011), 1140-1153; doi:10.1016/j.cnsns.2010.05.027.10.1016/j.cnsns.2010.05.027Suche in Google Scholar

[108] R. Toledo-Hernandez, V. Rico-Ramirez, G.A. Iglesias-Silva, and U. M. Diwekar, A fractional calculus approach to the dynamic optimization of biological reactive systems. Part I: Fractional models for biological reactions. Chemical Engineering Science 117, No 27 (2014), 217-228.Suche in Google Scholar

[109] J.C. Trigeassou, and N. Maamri, State-space modelling of fractional differential equations and the initial condition problem. In: IEEE SSD’09, Djerba, Tunisia (2009).Suche in Google Scholar

[110] V.V. Uchaikin, Fractional Derivatives for Physicists and Engineers. Volume I: Background and Theory, Volume II: Applications. Ser. Nonlinear Physical Science, Springer, and Higher Education Press (2012).Suche in Google Scholar

[111] D. Valério, and J.S. da Costa, Variable-order fractional derivatives and their numerical approximations. Signal Processing 91 (2011), 470-483.Suche in Google Scholar

[112] D. Valério, and J.S. da Costa, An Introduction to Fractional Control. IET, Stevenage (2012).10.1049/PBCE091ESuche in Google Scholar

[113] D. Verotta, Fractional compartmental models and multi-term Mittag- Leffler response functions. J. Pharmacokinetics Pharmacodynamics 37, No 2 (2010), 209-215.Suche in Google Scholar

[114] S. Victor, P. Melchior, and A. Oustaloup, Robust path tracking using flatness for fractional linear MIMO systems: A thermal application. Computers And Mathematics With Applications 59, No 5 (2010), 1667-1678.Suche in Google Scholar

[115] S. Victor, R. Malti, H. Garnier, and A. Oustaloup, Parameter and differentiation order estimation in fractional models. Automatica 49, No 4 (2013), 926-935.Suche in Google Scholar

[116] V.R. Voller, F. Falcini, and R. Garra, Fractional Stefan problems exhibiting lumped and distributed latent-heat memory effects. Physical Review E 87, No 4 (2013), 042401.10.1103/PhysRevE.87.042401Suche in Google Scholar PubMed

[117] X.J. Yang, Local Fractional Functional Analysis and Its Applications. Asian Academic Publisher Limited, Hong Kong (2011).Suche in Google Scholar

[118] X.J. Yang, Advanced Local Fractional Calculus and Its Applications. World Science Publisher, New York (2012).Suche in Google Scholar

[119] N. Yousfi, P. Melchior, P. Lanusse, N. Derbel, and A. Oustaloup, Decentralized CRONE control of nonsquare multivariable systems in path-tracking design. Nonlinear Dynamics 76, No 1 (2013), 447-457. Suche in Google Scholar

[120] M. Zayernouri and G.E. Karniadakis, Fractional Sturm-Liouville eigen-problems: Theory and numerical approximations. J. Comput. Phys. 252 (2013), 495-517.Suche in Google Scholar

[121] Y. Zhou, Basic Theory of Fractional Differential Equations. World Scientific Publishing Company (2014).10.1142/9069Suche in Google Scholar

[122] M. Zubair, M.J. Mughal, and Q.A. Naqvi, Electromagnetic Fields and Waves in Fractional Dimensional Space. Ser. SpringerBriefs in Appl. Sci. and Technology, Springer, Heidelberg (2012). 10.1007/978-3-642-25358-4Suche in Google Scholar