Fractional Sturm-Liouville problem

In this paper, we define some Fractional Sturm-Liouville Operators (FSLOs) and introduce two classes of Fractional Sturm-Liouville Problems (FSLPs) namely regular and singular FSLP. The operators defined here are different from those defined in the literature in the sense that the operators defined here contain left and right Riemann-Liouville and left and right Caputo fractional derivatives. For both classes we investigate the eigenvalue and eigenfunction properties of the FSLOs. In the class of regular FSLPs, we discuss two types of FSLPs. As an operator for the class of singular FSLPs, we introduce a Fractional Legendre Equation (FLE) and discuss its solution. It is shown that the Legendre Polynomials resulting from an FLE are the same as those obtained from the integer order Legendre equation; however, the eigenvalues of the two equations differ. Using the Legendre integral transform we demonstrate some applications of our results by solving two fractional differential equations, one ordinary and the other partial. It is our hope that this paper will initiate new research in the area of FSLPs and many of its variations.

[1]  R. Courant,et al.  Methods of Mathematical Physics , 1962 .

[2]  Mark M Meerschaert,et al.  FRACTIONAL PEARSON DIFFUSIONS. , 2013, Journal of mathematical analysis and applications.

[3]  M. D’Ovidio.  From Sturm–Liouville problems to fractional and anomalous diffusions , 2010, 1011.1424.

[4]  Rahmat Darzi,et al.  Introducing an Iterative Method for Solving a Special FDE , 2009 .

[5]  Qasem M. Al-Mdallal,et al.  An efficient method for solving fractional Sturm–Liouville problems , 2009 .

[6]  M. Sørensen,et al.  The Pearson Diffusions: A Class of Statistically Tractable Diffusion Processes , 2007 .

[7]  O. Agrawal,et al.  Regular Fractional Sturm-Liouville Problem with Generalized Derivatives of Order in (0, 1) , 2013 .

[8]  Igor M. Sokolov,et al.  Physics of Fractal Operators , 2003 .

[9]  I. Podlubny.  Fractional differential equations , 1998 .

[10]  Vedat Suat Erturk,et al.  Computing Eigenelements of Sturm–Liouville Problems of Fractional Order via Fractional Differential Transform Method , 2011 .

[11]  A. Luo,et al.  Fractional Dynamics and Control , 2011 .

[12]  K. Diethelm,et al.  Fractional Calculus: Models and Numerical Methods , 2012 .

[13]  L. Debnath,et al.  Integral Transforms and Their Applications, Second Edition , 2006 .

[14]  Qasem M. Al-Mdallal,et al.  On the numerical solution of fractional Sturm–Liouville problems , 2010, Int. J. Comput. Math..

[16]  Om Prakash Agrawal,et al.  Generalized Variational Problems and Euler-Lagrange equations , 2010, Comput. Math. Appl..

[17]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[18]  O. Agrawal.  Some generalized fractional calculus operators and their applications in integral equations , 2012 .

[19]  T. Atanacković,et al.  Generalized wave equation in nonlocal elasticity , 2009 .

[20]  F. Mainardi,et al.  Fractals and fractional calculus in continuum mechanics , 1997 .

[21]  R. Magin.  Fractional Calculus in Bioengineering , 2006 .

[22]  O. P. Agrawal,et al.  On a Regular Fractional Sturm-Liouville Problem with derivatives of order in (0,1) , 2012, Proceedings of the 13th International Carpathian Control Conference (ICCC).

[23]  Jiangang Qi,et al.  Eigenvalue problems of the model from nonlocal continuum mechanics , 2011 .

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