Polynomial spline collocation methods for second-order Volterra integrodifferential equations

where q : I → R, pi : I → R, and ki : D → R (i = 0,1) (with D := {(t,s) : 0 ≤ s ≤ t ≤ T}) are given functions and are assumed to be (at least) continuous in the respective domains. For more details of these equations, many other interesting methods for the approximated solution and stability procedures are available in earlier literatures [1, 3, 4, 5, 6, 7, 8, 11]. The above equation is usually known as basis test equation and is suggested by Brunner and Lambert [5]. Since then it has been widely used for analyzing the solution and stability properties of various methods. Second-order VIDEs of the above form (1.1) will be solved numerically using polynomial spline spaces. In order to describe these approximating polynomial spline spaces, let ∏ N : 0= t0 < t1 < ···< tN = T be the mesh for the interval I, and set σn := [ tn,tn+1 ] , hn := tn+1−tn, n= 0,1, . . . ,N−1, h =max{hn : 0≤n≤N−1} (mesh diameter), ZN := { tn :n= 1,2, . . . ,N−1 } , ZN = ZN ∪{T}. (1.3)