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)


Introduction.
In order to discuss the numerical solution of the second-order VIDEs we consider the following linear integrodifferential equation: y (2) (t) = q(t) + 1 i=0 p i (t)y (i) ( with y(0) = y 0 , y (1) (0) = y 1 , ( where q : I → R, p i : I → R, and k i : 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 = t 0 < t 1 < ••• < t N = T be the mesh for the interval I, and set σ n := t n ,t n+1 , h n := t n+1 − t n , n= 0, 1,...,N − 1, h = max h n : 0 ≤ n ≤ N − 1 (mesh diameter), Z N := t n : n = 1, 2,...,N − 1 , Z N = Z N ∪{T }. (1.3) Let π m+d be the set of (real) polynomials of degree not exceeding m+d, where m ≥ 1 and d ≥ −1 are given integers.The solution y to the initial-value problem (1.1), (1.2) will be approximated by an element u in the polynomial spline space, a n,r t − t n d+r , ( where From (1.5), we see that the element u ∈ S (d) m+d (Z N ) is well defined provided the coefficients {a n,r } r =1,...,m are known.In order to determine these coefficients, we consider a set of collocation parameters {c j } j=1,...,m , where 0 < c 1 < ••• < c m ≤ 1, and define the set of collocation points as X n , with X n := t n,j := t n + c j h n , j = 1, 2,...,m . ( The approximate solution u ∈ S (d) m+d (Z N ) will be determined by imposing the condition that u satisfies the following initial-value problem on X(N): u (2) with u(0) = y 0 , u (1) (0) = y 1 , (1.9) with a uniform the mesh sequence { N }, h n = h, for all n = 0, 1,...,N −1, but for small h, (1.8) has a unique solution {a n,j } j=1,...,m , for all n = 0, 1,...,N − 1.
Therefore, the modified collocation equation (1.8) can be rewritten as u n t n,j = q t n,j + p 0 t n,j u n t n,j + p 1 t n,j u n t n,j where (1.11) In most applications the integrals appearing in (1.11) cannot be evaluated analytically, so we seek suitable quadrature formulas as follows: (1.12) where µ 0 and µ 1 are two given positive integers, {d l } and {d j,l } are two sets of parameters satisfying the following conditions: and w l , w j,l denote the quadrature weights.The corresponding quadrature error terms are defined as (1.15) By the above quadrature process (1.10) is a fully discretized collocation method which defines an element û ∈ S (d) m+d (Z N ) (which, in general, will be different from the approximate solution u determined by the exact collocation equation (1.10)).The full discretized version of the collocation equation (1.10) is û n t n,j = q t n,j + p 0 t n,j ûn t n,j + p 1 t n,j û n t n,j (1.16) Similarly, the approximate discretized solution û ∈ S (d) m+d (Z N ) has the following form: The collocation equations (1.10) and (1.16) represent recursive systems, for each n = 0, 1,...,N − 1, which yield the coefficients {a n,r } r =1,...,m and { ân,r } r =1,...,m , respectively.Once the coefficients are known, the values of u, û together with their derivatives are determined on σ n by (1.5) and (1.17), respectively.
On each of the N subintervals of I we have to solve an m × m system of linear equations.On the first subinterval σ 0 , d + 1 additional equations are furnished by the d + 1 initial conditions (1.6) and (1.18).

Global convergence criteria.
If the given functions q, p i , and k i (i = 0, 1) are of class m + d on their domain of definition, then the VIDE (1.1) has a unique solution y, which is of class m + d + 2. Let ϕ n be the restriction of the function ϕ on I in the subinterval σ n , for n = 0, 1,...,N − 1, and the norm is defined by (2.1) The following theorem describes the order of global convergence of u ∈ S Theorem 2.1.Let q, p i , and k i be m + d times continuously differentiable in the respective domains I and S. For d ∈ {0, 1, 2}, every choice of collocation parameters and all quasiuniform mesh sequences { N } with sufficiently small h > 0, the following hold: where C k 's are finite constants independent of h; (ii) if the quadrature formulas (1.12) and (1.13) satisfy that for i = 0, 1,...,n− 1, and for j = 1,...,m, whenever the integrand is sufficiently smooth, then for all the discretized approximation solution û ∈ S (d) m+d (Z N ) defined by (1.17), the following relationships hold: where s 0 = min{r 0 + 1,r 1 } + 1, s 1 = min{s 0 ,m + d + 1}, and Q k , Ĉk are finite constants independent of h.
Proof.We will prove the theorem by induction using the same technique as in [2,3,9].
(i) For n = 0, 1,...,N − 1 and for all t = t n + τh n ∈ σ n (τ ∈ (0, 1]), the exact solution y can be obtained by Taylor series expansion: where (2.10) m+d (Z N ) to the subinterval σ n is given by (1.5), then by using (2.9), the error function on this subinterval has the form where D n is an m × m matrix, F n is an m × (d + 1) matrix, E n is a (d + 1) vector, and β n , r n , q n,i , p n,i are m vectors whose elements are given by n−1 t n , β n r = β n,r as defined in (2.12), (2.15) Note that the matrix D n defined by the coefficient of {β n,j } on the left-hand side of (2.13) and (2.14) is invertible whenever h n is sufficiently small.This follows from the assumptions of the theorem and the fact that for h n = 0 the determinant of this matrix is essentially a Vandermonde determinant.Therefore, for sufficiently small h n > 0, the matrix D n possesses a uniformly bounded inverse.
By the hypothesis on p i and k i , the 1 -norm of the vector r 0 is uniformly bounded.Hence, for p = m + d + 1, (2.13) leads to (2.16) From (2.11) we have the following result: Suppose, for j = 0, 1,...,n− 1, This completes the first assertion of the theorem.

3.
Local superconvergence on Z N .In many applications one is especially interested in obtaining higher-order approximations at the mesh points Z N .There arises the question as to whether there exist particular sets of collocation parameters leading to a discrete convergence order (on Z N ) that is higher than the global order.
In the subsequent analysis, the integrals play a critical role.
Theorem 3.1.Suppose that the given functions q, p i , and k i (i = 0, 1) in the VIDE (1.1) are m + p times continuously differentiable on their respective domains I and S (d+1 < p ≤ m and d ∈ {0, 1, 2}), and assume that the collocation parameters {c j } j=1,...,m are chosen so that J k = 0 for k = 0, 1,...,p − 1, and J p ≠ 0.Then, for all quasiuniform mesh sequences { N } with sufficiently small h > 0, the following hold: is the approximate solution defined by the exact collocation equation (1.8) and y is the exact solution of (1.1)It is well known that the orthogonality conditions J k = 0 (k = 0,...,p − 1) imply that the degree of precision of the m-point interpolatory quadrature formula on [0, 1] based on the abscissas {c j } j=1,...,m is m + p − 1.Since this degree of precision cannot exceed the value 2m − 1, we always have p ≤ m.Moreover, since the local order is required to be greater than or equal to the global order (m + d + 1), it is necessary that d + 2 ≤ p.
The following corollary deals with some important special cases and its proof relies on the above theorem.
As we can see from the above tables, the collocation spline method yields very good approximations when d = 1, 2. However, for d = 0, the method performs poorly.But for the case d = 3, the method converges if the collocation parameters are uniformly distributed and diverges if the parameters are Radau II points, which can be seen from Tables 4. 1, 4.2, 4.3, and 4.4.
:= u := u(t)| t∈σn := u n (t) ∈ π m+d , n = 0, 1,...,N − 1, for j = 0, 1,...,d, t n ∈ Z N , (1.4) that is, by a polynomial spline function of degree m + d which possesses the knots Z N and is d times continuously differentiable on I.If d = −1, then the elements of S (−1) m−1 (Z N ) may have jump discontinuities at the knots Z N .Initial value problems, such as (1.1) and (1.2), have often been solved by collocation method in polynomial spines spaces S According to M. Miculá and G. Micula [10], an element u ∈ S (d) m+d (Z N ) has the following form: for all n = 0, 1,...,N − 1 and t ∈ σ n , (i) the exact collocation equation (1.10) defines a unique approximation u∈ S (d) m+d (Z N ) and the resulting error function e := y − u satisfies

Example 4 . 1 .
Consider the following integrodifferential equation of second order: