A novel method for solving second order fractional eigenvalue problems

The paper presents a new numerical method for solving eigenvalue problems for fractional differential equations. It combines two techniques: the method of external excitation (MEE) and the backward substitution method (BSM). The first one is a mathematical model of physical measurements when a mechanical, electrical or acoustic system is excited by some source and resonant frequencies can be determined by using the growth of the amplitude of oscillations near these frequencies. The BSM consists of replacing the original equation by an approximate equation which has an exact analytic solution with a set of free parameters. These free parameters are determined by the use of the collocation procedure. Some examples are given to demonstrate the validity and applicability of the new method and a comparison is made with the existing results. The numerical results show that the proposed method is of a high accuracy and is efficient for solving of a wide class of eigenvalue problems.

[1]  Mehdi Dehghan,et al.  Application of the collocation method for solving nonlinear fractional integro-differential equations , 2014, J. Comput. Appl. Math..

[2]  Pedro R. S. Antunes,et al.  An Augmented-RBF Method for Solving Fractional Sturm-Liouville Eigenvalue Problems , 2015, SIAM J. Sci. Comput..

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

[4]  Jun-Sheng Duan,et al.  Eigenvalue problems for fractional ordinary differential equations , 2013 .

[5]  M. L. Morgado,et al.  Nonpolynomial collocation approximation of solutions to fractional differential equations , 2013 .

[6]  Arvet Pedas,et al.  Numerical solution of nonlinear fractional differential equations by spline collocation methods , 2014, J. Comput. Appl. Math..

[7]  Fathi M. Allan,et al.  An efficient algorithm for solving higher-order fractional Sturm-Liouville eigenvalue problems , 2014, J. Comput. Phys..

[8]  Xiuying Li,et al.  Approximate analytical solutions of nonlocal fractional boundary value problems , 2015 .

[9]  E. H. Doha,et al.  A NEW JACOBI OPERATIONAL MATRIX: AN APPLICATION FOR SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS , 2012 .

[10]  Sergiy Yu. Reutskiy.  The method of external excitation for solving generalized Sturm-Liouville problems , 2010, J. Comput. Appl. Math..

[11]  R. Herrmann.  Fractional Calculus: An Introduction for Physicists , 2011 .

[12]  M. Rehman,et al.  A numerical method for solving boundary value problems for fractional differential equations , 2012 .

[13]  E. H. Doha,et al.  EFFICIENT CHEBYSHEV SPECTRAL METHODS FOR SOLVING MULTI-TERM FRACTIONAL ORDERS DIFFERENTIAL EQUATIONS , 2011 .

[14]  Sergiy Yu. Reutskiy,et al.  The backward substitution method for multipoint problems with linear Volterra-Fredholm integro-differential equations of the neutral type , 2016, J. Comput. Appl. Math..

[15]  Neville J. Ford,et al.  A nonpolynomial collocation method for fractional terminal value problems , 2015, J. Comput. Appl. Math..

[16]  Mustafa Gülsu,et al.  Numerical approach for solving fractional Fredholm integro-differential equation , 2013, Int. J. Comput. Math..

[17]  F. A. Seiler,et al.  Numerical Recipes in C: The Art of Scientific Computing , 1989 .

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

[19]  Rahmat Ali Khan,et al.  The Legendre wavelet method for solving fractional differential equations , 2011 .

[20]  Arvet Pedas,et al.  Spline collocation for nonlinear fractional boundary value problems , 2014, Appl. Math. Comput..

[21]  N. Ford,et al.  Analysis of Fractional Differential Equations , 2002 .

[22]  Arvet Pedas,et al.  Modified spline collocation for linear fractional differential equations , 2015, J. Comput. Appl. Math..

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

[24]  Yury F. Luchko,et al.  Algorithms for the fractional calculus: A selection of numerical methods , 2005 .

[25]  Wei Zhang,et al.  Legendre wavelets method for solving fractional integro-differential equations , 2015, Int. J. Comput. Math..

[26]  E. A. Rawashdeh,et al.  Numerical solution of fractional integro-differential equations by collocation method , 2006, Appl. Math. Comput..

[27]  Saeid Abbasbandy,et al.  Homotopy analysis method for multiple solutions of the fractional Sturm-Liouville problems , 2010, Numerical Algorithms.

[28]  Ali H. Bhrawy,et al.  A quadrature tau method for fractional differential equations with variable coefficients , 2011, Appl. Math. Lett..

[29]  S. Reutskiy.  A Meshless Method for Nonlinear, Singular and Generalized Sturm-Liouville Problems , 2008 .

[30]  M. Anwar,et al.  A collocation-shooting method for solving fractional boundary value problems , 2010 .

[31]  K. Parand,et al.  Application of Bessel functions for solving differential and integro-differential equations of the fractional order ☆ , 2014 .

[32]  Sergiy Yu. Reutskiy,et al.  A method of particular solutions for multi-point boundary value problems , 2014, Appl. Math. Comput..

[33]  Ali H Bhrawy,et al.  A shifted Legendre spectral method for fractional-order multi-point boundary value problems , 2012 .

[34]  Arvet Pedas,et al.  Piecewise polynomial collocation for linear boundary value problems of fractional differential equations , 2012, J. Comput. Appl. Math..

[35]  Dumitru Baleanu,et al.  On shifted Jacobi spectral approximations for solving fractional differential equations , 2013, Appl. Math. Comput..

[36]  William H. Press,et al.  Numerical recipes in C , 2002 .

[37]  Saeid Abbasbandy,et al.  An Adaptive Pseudospectral Method for Fractional Order Boundary Value Problems , 2012 .