scholarly journals Tension spline collocation methods for singularly perturbed Volterra integro-differential and Volterra integral equations

2002 ◽  
Vol 140 (1-2) ◽  
pp. 381-402 ◽  
Author(s):  
Vilmoš Horvat ◽  
Mladen Rogina
Keyword(s):  
Integral Equations ◽  
Volterra Integral Equations ◽  
Singularly Perturbed ◽  
Collocation Methods ◽  
Spline Collocation ◽  
Tension Spline

Collocation methods for second-kind Volterra integral equations with weakly singular kernels

1994 ◽  
Vol 124 (2) ◽  
pp. 199-210 ◽  
Author(s):  
Teresa Diogo ◽  
Sean McKee ◽  
Tao Tang
Keyword(s):  
Integral Equations ◽  
Convergence Rates ◽  
Singular Term ◽  
Analytic Solutions ◽  
Volterra Integral Equations ◽  
Collocation Methods ◽  
Spline Collocation ◽  
Weakly Singular ◽  
Singular Kernels ◽  
Optimal Convergence Rates

In this paper it is shown that the use of uniform meshes leads to optimal convergence rates provided that the analytic solutions of a particular class of Volterra integral equations (VIEs) are smooth. If the exact solutions are not smooth, however, suitable transformations can be made so that the new VIEs possess smooth solutions. Spline collocation methods with uniform meshes applied to these new VIEs are then shown to be able to yield optimal (global) convergence rates. The general theory is applied to a typical case, i.e. the integral kernels consisting of the singular term (t − s) −½.

Sign in / Sign up

Export Citation Format

Share Document