On fully discrete collocation methods for solving weakly singular integro‐differential equations

Abstract

In order to find approximate solutions of Volterra and Fredholm integro‐differential equations by collocation methods it is necessary to compute certain integrals that determine the required algebraic systems. Those integrals usually can not be computed exactly and if the kernels of the integral operators are not smooth, simple quadrature formula approximations of the integrals do not preserve the convergence rate of the collocation method. In the present paper fully discrete analogs of collocation methods where non‐smooth integrals are replaced by appropriate quadrature formulas approximations, are considered and corresponding error estimates are derived. Presented numerical examples display that theoretical results are in a good accordance with the actual convergence rates of the proposed algorithms.

Keywords:

• weakly singular integro‐differential equation
• collocation method
• fully discrete collocation method
• graded grid

Notes

The work of E. Tamme was supported by Estonian Science Foundation Grant No. 7353. The work of R. Kangro was supported by Estonian Science Foundation Grant N0. 7044.