Collocation Methods for A Class of Volterra Integral Functional Equations with Multiple Proportional Delays

In this paper, we apply the collocation methods to a class of Volterra inte- gral functional equations with multiple proportional delays (VIFEMPDs). We shall present the existence, uniqueness and regularity properties of analytic solutions for this type of equations, and then analyze the convergence orders of the collocation solutions and give corresponding error estimates. The numerical results verify our theoretical analysis. AMS subject classifications: 65R20, 34K06, 34K28

[1]  Hehu Xie,et al.  Analysis of collocation solutions for a class of functional equations with vanishing delays , 2011 .

[2]  Qiya Hu.  Multilevel correction for discrete collocation solutions of Volterra integral equations with delay arguments , 1999 .

[3]  Ishtiaq Ali,et al.  Spectral methods for pantograph-type differential and integral equations with multiple delays , 2009 .

[4]  COLLOCATION METHODS FOR PANTOGRAPH-TYPE VOLTERRA FUNCTIONAL EQUATIONS WITH MULTIPLE DELAYS , 2008 .

[5]  Tao Tang,et al.  Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel , 2010, Math. Comput..

[6]  T. Koshy.  Catalan Numbers with Applications , 2008 .

[7]  Hermann Brunner,et al.  Discontinuous Galerkin approximations for Volterra integral equations of the first kind , 2009 .

[8]  Qun Lin,et al.  Geometric meshes in collocation methods for Volterra integral equations with proportional delays , 2001 .

[9]  Arieh Iserles,et al.  On the generalized pantograph functional-differential equation , 1993, European Journal of Applied Mathematics.

[10]  A. Bellen,et al.  Numerical methods for delay differential equations , 2003 .

[11]  Wayne H. Enright,et al.  Interpolating Runge-Kutta methods for vanishing delay differential equations , 1995, Computing.

[12]  K. Atkinson.  The Numerical Solution of Integral Equations of the Second Kind , 1997 .

[13]  Hermann Brunner.  On the discretization of differential and Volterra integral equations with variable delay , 1997 .

[14]  Emiko Ishiwata,et al.  On the Attainable Order of Collocation Methods for Delay Differential Equations with Proportional Delay , 2000 .

[15]  George J. Fix,et al.  Analysis of finite element approximation and quadrature of Volterra integral equations , 1997 .