Modified spline collocation for linear fractional differential equations

We propose and analyze a class of high order methods for the numerical solution of initial value problems for linear multi-term fractional differential equations involving Caputo-type fractional derivatives. Using an integral equation reformulation of the initial value problem we first regularize the solution by a suitable smoothing transformation. After that we solve the transformed equation by a piecewise polynomial collocation method on a mildly graded or uniform grid. Optimal global convergence estimates are derived and a superconvergence result for a special choice of collocation parameters is established. Theoretical results are verified by some numerical examples.

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

[2]  M. Kolk,et al.  Numerical solution of Volterra integral equations with weakly singular kernels which may have a boundary singularity , 2009 .

[3]  Hermann Brunner,et al.  Nonpolynomial Spline Collocation for Volterra Equations with Weakly Singular Kernels , 1983 .

[4]  Hermann Brunner,et al.  Piecewise Polynomial Collocation Methods for Linear Volterra Integro-Differential Equations with Weakly Singular Kernels , 2001, SIAM J. Numer. Anal..

[5]  Arvet Pedas,et al.  Spline collocation methods for linear multi-term fractional differential equations , 2011, J. Comput. Appl. Math..

[6]  N. Ford,et al.  Higher order numerical methods for solving fractional differential equations , 2014 .

[7]  Min Huang,et al.  Superconvergence of the Iterated Hybrid Collocation Method for Weakly Singular Volterra Integral Equations , 2006 .

[8]  Ivan P. Gavrilyuk,et al.  Collocation methods for Volterra integral and related functional equations , 2006, Math. Comput..

[9]  G. Vainikko,et al.  High-Order Methods for Volterra Integral Equations with General Weak Singularities , 2009 .

[10]  Kai Diethelm,et al.  An efficient parallel algorithm for the numerical solution of fractional differential equations , 2011 .

[11]  Xiaohua Ma,et al.  Numerical solution of fractional integro-differential equations by a hybrid collocation method , 2013, Appl. Math. Comput..

[12]  Arvet Pedas,et al.  Numerical solution of Volterra integral equations with singularities , 2013 .

[13]  Sean McKee,et al.  Collocation methods for second-kind Volterra integral equations with weakly singular kernels , 1994 .

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

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

[16]  R. Agarwal,et al.  A Survey on Existence Results for Boundary Value Problems of Nonlinear Fractional Differential Equations and Inclusions , 2010 .

[17]  H. Brunner,et al.  The numerical solution of Volterra equations , 1988 .

[18]  Chengming Huang,et al.  Spectral collocation method for linear fractional integro-differential equations , 2014 .

[19]  Hermann Brunner,et al.  The piecewise polynomial collocation method for nonlinear weakly singular Volterra equations , 1999, Math. Comput..

[20]  Gennadi Vainikko,et al.  Multidimensional Weakly Singular Integral Equations , 1993 .

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

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

[23]  Annamaria Palamara Orsi,et al.  A new approach to the numerical solution of weakly singular Volterra integral equations , 2004 .

[24]  Arvet Pedas,et al.  Smoothing Transformation and Piecewise Polynomial Collocation for Weakly Singular Volterra Integral Equations , 2004, Computing.

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

[26]  Jingjun Zhao,et al.  Collocation methods for fractional integro-differential equations with weakly singular kernels , 2013, Numerical Algorithms.

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

[28]  Yuesheng Xu,et al.  A Hybrid Collocation Method for Volterra Integral Equations with Weakly Singular Kernels , 2003, SIAM J. Numer. Anal..

[29]  Arvet Pedas,et al.  On the convergence of spline collocation methods for solving fractional differential equations , 2011, J. Comput. Appl. Math..

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

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

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