Abstract
Discussions under this title were held during a special session in frames of the International Conference “Fractional Differentiation and Applications” (ICFDA ’14) held in Catania (Italy), 23-25 June 2014, see details at http://www.icfda14.dieei.unict.it/. Along with the presentations made during this session, we include here some contributions by the participants sent afterwards and also by few colleagues planning but failed to attend.
The intention of this special session was to continue the useful traditions from the first conferences on the Fractional Calculus (FC) topics, to pose open problems, challenging hypotheses and questions “where to go”, to discuss them and try to find ways to resolve.
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).10.1007/978-1-4614-4036-9Search 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).Search 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).10.1007/978-3-642-30898-7Search 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).10.1002/9781118577530Search in Google Scholar
[5] S. Al-Azawi, Some Results in Fractional Calculus. LAP Lambert Acad. Publ. (2011).Search 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.Search 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).10.1142/8180Search 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), 092101.10.1063/1.4819502Search 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.2014.11.012Search 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.Search 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.76.041105Search 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.2009.2036643Search 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_8Search 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).Search 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).Search 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.Search 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.2007.1900Search in Google Scholar
[18] O. Furdui, Limits, Series, and Fractional Part Integrals. Problem Books in Mathematics, Springer (2013).10.1007/978-1-4614-6762-5Search 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.Search 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.Search 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.Search in Google Scholar
[22] R. Gorenflo, Afterthoughts on interpretation of fractional derivatives and integrals. In: [97] (1998), 589-591.Search 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). 10.1007/978-3-662-43930-2Search in Google Scholar
[24] A. Hanyga, Multi-dimensional solutions of space-fractional diffusion equations. Proc. Royal Society A 457, No 2016 (2001), 2993-3005.Search 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.2012.05.001Search in Google Scholar PubMed
[26] R. Herrmann, Fractional Calculus: An Introduction for Physicists. World Scientific Publ. Co., Singapore (2011).10.1142/8072Search in Google Scholar
[27] R. Herrmann, Fractional Calculus - Introduction for Physicists. World Scientific, 2nd edition (2014).10.1142/8934Search 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.Search in Google Scholar
[29] O.C. Ibe, Elements of Random Walk and Diffusion Processes. Wiley (2013).10.1002/9781118618059Search 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.Search 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.Search 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.Search 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.Search 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.Search 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.Search 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). Search 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), 062102.10.1063/1.3430552Search 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).Search in Google Scholar
[39] G. Jumarie, Fractional Differential Calculus for Non-differentiable Functions: Mechanics, Geometry, Stochastics, Information Theory. LAP Lambert Academic Publishing (2014).Search 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).Search in Google Scholar
[41] S. Kempfle, Modelling viscous damped oscillations by fractional differential operators. In: [97] (1998), 592-593.Search in Google Scholar
[42] V. Kiryakova, A long standing conjecture failed?. In: [97] (1998), 579-588.Search 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.Search 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/9780199234868.001.0001Search 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.Search 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.Search in Google Scholar
[47] J.S. Leszczyanski, An Introduction to Fractional Mechanics. Czestochowa University of Technology, Czestochowa (2011).Search 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.Search 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.Search in Google Scholar
[50] Yu. Luchko, Fractional Schrödinger equation for a particle moving in a potential well. J. Mathematical Physics 54, No 1 (2013), 012111.10.1063/1.4777472Search in Google Scholar
[51] Y. Luo, and Y.-Q. Chen, Fractional Order Motion Controls. JohnWiley & Sons, New York (2012). 10.1002/9781118387726Search 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.2014.02.023Search 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.Search in Google Scholar
[54] R.L. Magin, X. Feng, and D. Baleanu, Fractional calculus in NMR. Magnetic Resonance Engineering 34 (2009), 16-23.Search in Google Scholar
[55] F. Mainardi, Considerations on fractional calculus, interpretations and applications. In: [97] (1998), 594-597.Search 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/p871Search in Google Scholar
[57] T.S. Margulies, Flows, Energetics, and Waves: Mathematical Applications: Physical Sciences and Engineering Analysis. CreateSpace Independent Publishing Platform (2014).Search in Google Scholar
[58] A.M. Mathai, Jacobians of Matrix Transformations and Functions of Matrix Argument. World Scientific Publishing, New York (1997).10.1142/3438Search 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).Search 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.Search in Google Scholar
[61] A.M. Mathai, Fractional integral operators involving many matrix variables. Linear Algebra and Its Applications 446 (2014), 196-215.Search 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.Search 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).Search 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.2005.11.015Search in Google Scholar
[65] M.M. Meerschaert, and A. Sikorskii, Stochastic Models for Fractional Calculus. Walter de Gruyter, Berlin (2011).10.1515/9783110258165Search 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). 10.1007/978-3-642-39010-4Search 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/BFb0067119Search 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.91.018302Search 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.94.065003Search in Google Scholar PubMed
[70] D. del-Castillo-Negrete, Fractional diffusion models of nonlocal transport. Physics of Plasmas 13, No 8 (2006), 082308.10.1063/1.2336114Search 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), 75009.10.1088/0029-5515/48/7/075009Search 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.Search 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.79.031120Search 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.Search 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.Search 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).Search in Google Scholar
[77] I. Nourdin, Selected Aspects of Fractional Brownian Motion. Bocconi & Springer Series, Springer, Milano (2012).10.1007/978-88-470-2823-4Search 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.Search in Google Scholar
[79] K.B. Oldham, An introduction to the fractional calculus and some applications. In: [97] (1998), 598-609.Search in Google Scholar
[80] M.D. Ortigueira, Fractional Calculus for Scientists and Engineers. Lecture Notes in Electrical Engineering, Springer, Dordrecht, Heidelberg (2011).10.1007/978-94-007-0747-4Search in Google Scholar
[81] T.J. Osler, Open questions for research. In: [96] (1975), 376-381.10.1007/BFb0067120Search 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.Search 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_9Search 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).Search 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.Search in Google Scholar
[86] A. Oustaloup, Diversity and Non-integer Differentiation for System Dynamics. Wiley (2014).10.1002/9781118760864Search 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.Search 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.Search 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).10.1007/978-3-642-31549-7Search 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).Search 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.Search in Google Scholar
[92] I. PetrÁš, Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation. Ser. Nonlinear Physical Science, Springer, Heidelberg (2011).Search 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.Search 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.Search 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.Search 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/BFb0067095Search 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).Search 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.Search 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.Search 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).Search 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).Search 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).10.1007/978-1-4471-2233-3Search 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), 1415-1430.10.1177/1077546307087440Search in Google Scholar
[104] H.M. Srivastava, Open questions for further researches on fractional calculus and its applications. In: [76] (1990), 281-284.Search 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).10.1142/8936Search 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). Search 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.2010.05.027Search 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.Search 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).Search 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).Search 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.Search in Google Scholar
[112] D. Valério, and J.S. da Costa, An Introduction to Fractional Control. IET, Stevenage (2012).10.1049/PBCE091ESearch 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.Search 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.Search 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.Search 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.87.042401Search in Google Scholar PubMed
[117] X.J. Yang, Local Fractional Functional Analysis and Its Applications. Asian Academic Publisher Limited, Hong Kong (2011).Search in Google Scholar
[118] X.J. Yang, Advanced Local Fractional Calculus and Its Applications. World Science Publisher, New York (2012).Search 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. Search 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.Search in Google Scholar
[121] Y. Zhou, Basic Theory of Fractional Differential Equations. World Scientific Publishing Company (2014).10.1142/9069Search 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). 10.1007/978-3-642-25358-4Search in Google Scholar
© 2015 Diogenes Co., Sofia
Articles in the same Issue
- Contents
- Fcaa Related News, Events and Books (Fcaa–Volume 18–2–2015)
- New Results from Old Investigation: A Note on Fractional M-Dimensional Differential Operators. The Fractional Laplacian
- Pollutant Reduction of a Turbocharged Diesel Engine Using a Decentralized Mimo Crone Controller
- Experimental Implications of Bochner-Levy-Riesz Diffusion
- Fractional Diffusion on Bounded Domains
- On a System of Fractional Differential Equations with Coupled Integral Boundary Conditions
- A Numerical Approach for Fractional Order Riccati Differential Equation Using B-Spline Operational Matrix
- Solving Fractional Delay Differential Equations: A New Approach
- Formal Consistency Versus Actual Convergence Rates of Difference Schemes for Fractional-Derivative Boundary Value Problems
- Asymptotic Stability Of Dynamic Equations With Two Fractional Terms: Continuous Versus Discrete Case
- Analysis of Natural and Artificial Phenomena Using Signal Processing and Fractional Calculus
- Fractional Approach for Estimating Sap Velocity in Trees
- Fractional Calculus: Quo Vadimus? (Where are we Going?)
Articles in the same Issue
- Contents
- Fcaa Related News, Events and Books (Fcaa–Volume 18–2–2015)
- New Results from Old Investigation: A Note on Fractional M-Dimensional Differential Operators. The Fractional Laplacian
- Pollutant Reduction of a Turbocharged Diesel Engine Using a Decentralized Mimo Crone Controller
- Experimental Implications of Bochner-Levy-Riesz Diffusion
- Fractional Diffusion on Bounded Domains
- On a System of Fractional Differential Equations with Coupled Integral Boundary Conditions
- A Numerical Approach for Fractional Order Riccati Differential Equation Using B-Spline Operational Matrix
- Solving Fractional Delay Differential Equations: A New Approach
- Formal Consistency Versus Actual Convergence Rates of Difference Schemes for Fractional-Derivative Boundary Value Problems
- Asymptotic Stability Of Dynamic Equations With Two Fractional Terms: Continuous Versus Discrete Case
- Analysis of Natural and Artificial Phenomena Using Signal Processing and Fractional Calculus
- Fractional Approach for Estimating Sap Velocity in Trees
- Fractional Calculus: Quo Vadimus? (Where are we Going?)