On the fractional Cornu spirals

Abstract This paper introduces two fractional integral sine and cosine curves. Based on such formulation a series of fractional plane and space Cornu spirals are represented. These curves can be used for modeling diffraction phenomena under the light of the fractional calculus tools. The adoption of symbolic packages simplifies the model implementation showing that present-day computational resources lead to a straightforward implementation of fractional differential equations.

[1]  T. Hartley,et al.  The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science , 2016 .

[2]  S. Das.  Functional Fractional Calculus , 2011 .

[3]  Karabi Biswas,et al.  Fractional-Order Devices , 2017 .

[4]  S. Holm,et al.  A unifying fractional wave equation for compressional and shear waves. , 2010, The Journal of the Acoustical Society of America.

[5]  Virginia Kiryakova,et al.  The Chronicles of Fractional Calculus , 2017 .

[6]  P. Ufimtsev.  Fundamentals of the Physical Theory of Diffraction , 2007 .

[7]  Atınç Pirti,et al.  Transrapid and the transition curve as sinusoid , 2016 .

[8]  J. Machado,et al.  Introduction to Fractional Differential Equations , 2018, Nonlinear Systems and Complexity.

[9]  A. Hitachi.  Cornu's spiral in the Fresnel regime studied using ultrasound: a phase study. , 2010, The Journal of the Acoustical Society of America.

[10]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[11]  Y. Povstenko.  Linear Fractional Diffusion-Wave Equation for Scientists and Engineers , 2015 .

[12]  Stevan Pilipović,et al.  Fractional Calculus with Applications in Mechanics: Wave Propagation, Impact and Variational Principles , 2014 .

[13]  J. F. Gómez‐Aguilar,et al.  A physical interpretation of fractional calculus in observables terms: analysis of the fractional time constant and the transitory response , 2014 .

[14]  Sverre Holm,et al.  Deriving fractional acoustic wave equations from mechanical and thermal constitutive equations , 2013, Comput. Math. Appl..

[15]  D. Paganin,et al.  Generalised Cornu spirals: an experimental study using hard x-rays. , 2016, Optics express.

[16]  N. Engheia.  On the role of fractional calculus in electromagnetic theory , 1997 .

[17]  Debasmita Mondal,et al.  Experimental studies on realization of fractional inductors and fractional‐order bandpass filters , 2015, Int. J. Circuit Theory Appl..

[18]  Alfred Gray,et al.  Modern differential geometry of curves and surfaces with Mathematica (2. ed.) , 1998 .

[19]  Q. Naqvi,et al.  Electromagnetic Fields and Waves in Fractional Dimensional Space , 2012 .

[20]  Hosein Nasrolahpour,et al.  A note on fractional electrodynamics , 2012, Commun. Nonlinear Sci. Numer. Simul..

[21]  D. Walton,et al.  The use of Cornu spirals in drawing planar curves of controlled curvature , 1989 .

[22]  I. Podlubny.  Fractional differential equations : an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications , 1999 .

[23]  J. A. Tenreiro Machado,et al.  Fractional order junctions , 2015, Commun. Nonlinear Sci. Numer. Simul..

[24]  David von Seggern,et al.  CRC Standard Curves and Surfaces with Mathematica , 2016 .

[25]  V. Castaño,et al.  Phase change in a diffracted wave: a Cornu spiral perspective. , 2010, Optics Letters.

[26]  F. Mainardi.  Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models , 2010 .