Spline collocation method for integro-differential equations with weakly singular kernels

In the first part of this paper we study the regularity properties of solutions to initial or boundary-value problems of Fredholm integro-differential equations with weakly singular or other nonsmooth kernels. We then use these results in the analysis of a piecewise polynomial collocation method for solving such problems numerically. Presented numerical examples display that theoretical results are in good accordance with actual convergence rates of proposed algorithms.

[1]  Ian H. Sloan,et al.  Optimal order spline methods for nonlinear differential and integro-differential equations , 1999 .

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

[3]  Hosseini Seyed Mohammad,et al.  Tau numerical solution of Fredholm integro-differential equations with arbitrary polynomial bases , 2003 .

[4]  Collocation approximations for weakly singular Volterra integro‐differential equations 1 , 2003 .

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

[6]  Tao Tang,et al.  Superconvergence of numerical solutions to weakly singular Volterra integro-differential equations , 1992 .

[7]  G. Leaf,et al.  COLLOCATION METHODS FOR INTEGRO-DIFFERENTIAL EQUATIONS* , 1977 .

[8]  Raul Kangro,et al.  Superconvergence in the Maximum Norm of a Class of Piecewise Polynomial Collocation Methods for Solving Linear Weakly Singular Volterra Integro-Differential Equations , 2003 .

[9]  Qiya Hu,et al.  Interpolation correction for collocation solutions of Fredholm integro-differential equations , 1998, Math. Comput..

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

[11]  W. Volk.  The iterated Galerkin method for linear integro-differential equations , 1988 .

[12]  M. R. Osborne,et al.  The numerical solution of differential equations , 1961 .

[13]  Hermann Brunner.  Collocation Methods for Volterra Integral and Related Functional Differential Equations: Epilogue , 2004 .

[14]  A. Spence,et al.  Orthogonal collocation for a nonlinear integro-differential equation , 1998 .

[15]  Khosrow Maleknejad,et al.  Taylor polynomial solution of high-order nonlinear Volterra-Fredholm integro-differential equations , 2003, Appl. Math. Comput..

[16]  Abdul-Majid Wazwaz,et al.  A reliable algorithm for solving boundary value problems for higher-order integro-differential equations , 2001, Appl. Math. Comput..

[17]  Spline collocation methods for weakly singular Volterra integro-differential equations , 2003 .

[18]  Hermann Brunner,et al.  A Spline Collocation Method for Linear Volterra Integro-Differential Equations with Weakly Singular Kernels , 2001 .