Fundamental Results of Conformable Sturm-Liouville Eigenvalue Problems

We suggest a regular fractional generalization of the well-known Sturm-Liouville eigenvalue problems. The suggested model consists of a fractional generalization of the Sturm-Liouville operator using conformable derivative and with natural boundary conditions on bounded domains. We establish fundamental results of the suggested model. We prove that the eigenvalues are real and simple and the eigenfunctions corresponding to distinct eigenvalues are orthogonal and we establish a fractional Rayleigh Quotient result that can be used to estimate the first eigenvalue. Despite the fact that the properties of the fractional Sturm-Liouville problem with conformable derivative are very similar to the ones with the classical derivative, we find that the fractional problem does not display an infinite number of eigenfunctions for arbitrary boundary conditions. This interesting result will lead to studying the problem of completeness of eigenfunctions for fractional systems.

[1]  W. Marsden.  I and J , 2012 .

[2]  C. Basaran.  Preface , 1934, The Yale Journal of Biology and Medicine.

[3]  Malgorzata Klimek.  Fractional Sturm-Liouville Problem in Terms of Riesz Derivatives , 2015, RRNR.

[4]  George E. Karniadakis,et al.  Fractional Sturm-Liouville eigen-problems: Theory and numerical approximation , 2013, J. Comput. Phys..

[5]  H. Kober.  ON FRACTIONAL INTEGRALS AND DERIVATIVES , 1940 .

[6]  Delfim F. M. Torres,et al.  A Conformable Fractional Calculus on Arbitrary Time Scales , 2015, 1505.03134.

[7]  Thabet Abdeljawad,et al.  On the Definitions of Nabla Fractional Operators , 2012 .

[8]  M. Al-Refai.  BASIC RESULTS ON NONLINEAR EIGENVALUE PROBLEMS OF FRACTIONAL ORDER , 2012 .

[9]  Dumitru Baleanu,et al.  New exact solutions of Burgers’ type equations with conformable derivative , 2017 .

[10]  Mahmoud H. Annaby,et al.  q-Fractional Calculus and Equations , 2012 .

[11]  A. Atangana,et al.  New Fractional Derivatives with Nonlocal and Non-Singular Kernel: Theory and Application to Heat Transfer Model , 2016, 1602.03408.

[12]  Delfim F. M. Torres,et al.  Variational calculus with conformable fractional derivatives , 2016, IEEE/CAA Journal of Automatica Sinica.

[13]  T. Abdeljawad.  Dual identities in fractional difference calculus within Riemann , 2011, 1112.5795.

[14]  T. Kaczorek,et al.  Fractional Differential Equations , 2015 .

[15]  Dumitru Baleanu,et al.  Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels , 2016 .

[16]  Thabet Abdeljawad,et al.  On conformable fractional calculus , 2015, J. Comput. Appl. Math..

[17]  Om Prakash Agrawal,et al.  Fractional Sturm-Liouville problem , 2013, Comput. Math. Appl..

[18]  D. Baleanu,et al.  On fractional derivatives with exponential kernel and their discrete versions , 2016, 1606.07958.

[19]  D. Anderson,et al.  Undetermined coefficients for local fractional differential equations , 2016 .

[20]  Udita N. Katugampola.  New approach to a generalized fractional integral , 2010, Appl. Math. Comput..

[21]  M. Sababheh,et al.  A new definition of fractional derivative , 2014, J. Comput. Appl. Math..

[22]  P. Eloe,et al.  Initial value problems in discrete fractional calculus , 2008 .

[23]  Zhaosheng Feng,et al.  Positive solutions of fractional differential equations with derivative terms , 2012 .

[24]  Ercan Çelik,et al.  Solutions of Sequential Conformable Fractional Differential Equations around an Ordinary Point and Conformable Fractional Hermite Differential Equation , 2015 .

[25]  Udita N. Katugampola.  A NEW APPROACH TO GENERALIZED FRACTIONAL DERIVATIVES , 2011, 1106.0965.

[26]  Jiangang Qi,et al.  Spectral problems for fractional differential equations from nonlocal continuum mechanics , 2014, Advances in Difference Equations.

[27]  M. Caputo,et al.  A new Definition of Fractional Derivative without Singular Kernel , 2015 .