The stability of collocation methods for VIDEs of second order

Abstract

Simplest results presented here are the stability criteria of collocation methods for the second-order Volterra integro differential equation (VIDE) by polynomial spline functions. The polynomial spline collocation method is stable if all eigenvalues of a matrix are in the unit disk and all eigenvalues with |λ|=1 belong to a 1×1 Jordan block. Also many other conditions are derived depending upon the choice of collocation parameters used in the solution procedure

Convergence analysis of spectral methods for high-order nonlinear Volterra integro-differential equations

“…The numerical results obtained here are compared with the numerical results obtained by spline collocation methods with uniformly distributed collocation parameters in Rawashdeh et al (2005). It is seen from Table 5 that the results of the present method are very superior to those of methods in Rawashdeh et al (2005). y(0) = 1, y (1)…”

“…4. In addition, we give a comparison between our methods and other related methods in the literature (Aguilar and Brunner 1988;Rawashdeh et al 2005;Yüzbaşi et al 2011).…”

“…This demonstrates that our methods provide an accurate approximation and yields the exponential rate of convergence. Example 5.3 The third case is the second-order linear Volterra integro-differential equations proposed in Rawashdeh et al (2005): with q(t) chosen so that the exact solution is y(t) = sin 4t.…”