Consider the first-order Volterra integro-differential equation (VIDE):

with initial condition $y(0)=y_{0}$. Here, the given functions $f:I\times R\rightarrow R$ and $K:S\times R\rightarrow R$ (with $S:=\{(t,s):0\leq s\leq t\leq T\}$ ), are supposed to be sufficiently smooth for the initial-value problem for VIDE (1.1) to have a unique solution $y\in C^{\alpha}(I)$, with $\alpha\in N$ (see[3], [6]).

VIDE-s of the above form will be solved numerically in certain polynomial spline spaces. In order to describe these approximating spaces let $\Pi_{N}:0=t_{0}<t_{1}<<\ldots<t_{N}=T$ (with $t_{n}=t_{n}^{(N)}$ ) be a mesh for the given interval $I$. and set

Moreover, let $\mathscr{P}_{k}$ denote the space of (real) polynomials of degree not exceeding $k$. We then define, for given integers $m$ and $d$ with $m\geq 1$ and $d\geq-1$,

to be the space of polynomial splines of degree $m+d$ whose elements possess the knots $Z_{N}$ and are $d$ times continually differentiable on $I$. If $d=-1$, then the elements of $S_{m-1}^{(-1)}\left(Z_{N}\right)$ may have jump discontinues at the knots $Z_{N^{\prime}}$.

In many papers, the problem of approximating the exact solution of initialvalue problem for $\operatorname{VIDE}(1.1)$, has been solved by collocation method in polynomial splines spaces $S_{m}^{(0)}\left(Z_{N}\right)$ and $S_{m}^{(1)}\left(Z_{N}\right)$ (see [1], [2], [3]) or in polynomial splines space $S_{d+1}^{(d)}\left(Z_{N}\right)$ (see[6]). In this paper we shall construct an approximate solution in the space of polynomial spline functions $S_{m+d}^{(d)}\left(Z_{N}\right)$, with $m\geq 1$ and $d\geq 0$. This approximation $u\in S_{m+d}^{(d)}\left(Z_{N}\right)$ will be determined by collocation methods. The attainable order of global and local convergence of these methods is analyzed in detail.

We shall assume in the following that mesh sequence $\left(\Pi_{N}\right)_{n\geq 1}$ is quasiuniform, that is, there exists a finite constant $\gamma$ independent of $N$ such that:

In [7] M. Micula and G. Micula proved that an element $u\in S_{m+d}^{(d)}\left(Z_{N}\right)$ has for all $n=0,\ldots,N-1$ and for all $t\in\sigma_{n}$ the following form:

where:

From (2.1) we have that on element $u\in S_{m+d}^{(d)}\left(Z_{N}\right)$ is well defined when we know the coefficients $\left\{a_{n,r}\right\}_{r=\overline{1,m}}$ for all $n=0,\ldots,N-1$. In order to determine these coefficients we consider the set of collocation parameters $\left\{c_{j}\right\}_{j=1,m}$, where $0\leq c_{1}<\ldots<c_{m}\leq 1$, and we define the set of collocation points by:

The approximate solution $u\in S_{m+d}^{(d)}\left(Z_{N}\right)$ will be determined imposing the condition that $u$ satisfy the $\operatorname{VIDE}(1.1)$ on $X(N)$ and the initial condition, i.e.:
(2.3) $u^{\prime}(t)=f(t,u(t))+\int_{0}^{t}K(t,s,u(s))\mathrm{d}s$, for all $t\in X(N)$, with $u(0)=y_{0}$. The exact collocation equation (2.3) may be written in the form:

where:

denotes the lag term.
For $h$ small enought it is easy to show that system (2.4) has a unique solution $\left\{a_{n,j}\right\}_{j=1,\bar{m}}$ for all $n=0,\ldots,N-1$.

For linear the version of (1.1)

the collocation equation assumes the form:

where:

We phrase our convergence results for the linear equation (2.6); a remark on the extension of these results to the general case (1.1) will follow each of the proofs.

In most applications the integrals (2.7) occurring in the exact collocation (2.6) cannot be evaluated analytically, and one is forced to resort to employing suitable quadrature formulas for their approximation. In the following we suppose that these integrals are approximated by quadrature formulas of the form:
(2.8) $\hat{\phi}_{n,i}^{(j)}\left[u_{i}\right]:=\begin{cases}\sum_{l=1}^{\mu_{1}}w_{l}K\left(t_{n,j},t_{i}+d_{l}h_{i}\right)u_{i}\left(t_{i}+d_{l}h_{i}\right)&\text{ if }i=0,\ldots,n-1,\\ \sum_{l=1}^{\mu_{0}}w_{j,l}K\left(t_{n,j},t_{n}+d_{j,l}h_{n}\right)u_{n}\left(t_{n}+d_{j,l}h_{n}\right).&\text{ if }i=n;\end{cases}$ where $\mu_{0}$ and $\mu_{1}$ are two given positive integers; $\left\{d_{l}\right\},\left\{d_{j,l}\right\}$ are two sets of parameters satisfying, respectively:

and $w_{l},w_{j,l}$ denote the quadrature weights.
The corresponding error term are defined by:

Hence, the fully discretization version of the collocation equation (2.6) is given by:

One can observe that the approximation $\hat{u}\in S_{m+d}^{(d)}\left(Z_{N}\right)$, given by the fully discretized collocation equations (2.10) will, in general, be different from the approximation $u\in S_{m+d}^{(d)}\left(Z_{N}\right)$ given by the exact collocation equations (2.6). For all $n=0,\ldots,N-1$ and for all $t\in\sigma_{n}$ the approximation $\hat{u}\in S_{m+d}^{(d)}\left(Z_{N}\right)$ has the form:

with:

Equations (2.6) and (2.10) represent, for each $n=0,1,\ldots,N-1$ a recursive system which will give the unknowns $\left\{a_{n,r}\right\}_{r=\overline{1,m}}$, respectively $\left\{\hat{a}_{n,r}\right\}_{r=1,m}$. Since
this solutions have been found, the values of $u$ and $\hat{u}$ together with their derivatives on $\sigma_{n}$ are determined by the formula (2.1), respectively, by the formula (2.11).

If the given functions $p,q$ and $K$ are of class $m+d$ on their domain of definition, then the VIDE (2.5) has a unique solution $y$, which is of class $m+d+1$. For a function $\varphi$ defined on $I$ we shall denote by $\varphi_{n}$ the restriction of $\varphi$ to the subinterval $\sigma_{n}$, for all $n=0,1,\ldots,N-1$, and we shall use the following norm:

Concerning the convergence of the method described above we give the following theorems:

THEOREM 3.1 Let $p,q$ and $K$ in (2.5) be $m+d$ times continuously differentiable on their respective domains $I$ and $S$. Then, for every choice of the collocation parameters $\left\{c_{j}\right\}_{j=\overline{1,m}}$ with $0<c_{1}<c_{2}<\ldots<c_{m}\leq 1$ and for all quasi-uniform mesh sequences $\left\{\Pi_{N}\right\}$ with sufficiently small $h>0$, we have:
(i) the exact collocation equation (2.6) defines a unique approximation $u\in S_{m+d}^{(d)}\left(Z_{N}\right)$, and the resulting error function $e:=y-u$ satisfies:

where $C_{k}$ are finite constants independent of $h$;
(ii) if the quadrature formulas (2.8) satisfy:

and, for $j=1,\ldots,m$,

whenever the integrand is a sufficiently smooth function, then for the approximation $\hat{u}\in S_{m+d}^{(d)}\left(Z_{N}\right)$ defined by the discretized collocation equation (2.10), the following relations hold:

and

where $s^{\prime}=\min\left\{r_{0}+1,r_{1}\right\}+1,s=\min\left\{s^{\prime},m+d+1\right\}$ and $Q_{k},\hat{C}_{k}$ are finite constants independent of $h$.

Proof. We shall prove it by induction using the same technique as in [4] or in [5].
(i) For $n=0,1,\ldots,N-1$ and for all $t=t_{n}+\tau h_{n}\in\sigma_{n}\quad(\tau\in(0,1])$ the exact solution $y$ can be developed in Taylor series:

where:

So, by (2.1) and (3.7) we have:

where:

Taking into account that $y$ is a solution of $\operatorname{VIDE}$ (2.5) and $u\in S_{m+d}^{(d)}\left(Z_{N}\right)$ satisfies the exact collocation equation (2.6) and employing the expression (3.8) for $e_{n}$, we are led to:

where, we have introduced the abbreviations $p_{n,j}:=p\left(t_{n}+c_{j}h_{n}\right)$ and $k_{n,j}(\cdot)=K\left(t_{n,j},\cdot\right)$.
Relation (3.9) can be written:

where $D_{n}\in\mathscr{M}_{m\times m},F_{n}\in\mathscr{M}_{m\times(d+1)},E_{n}\in\mathscr{M}_{(d+1)\times(d+1)}\left(\mathscr{M}_{\alpha\times\gamma}\right.$ denote the set of matrices with $\alpha$ lines and $\gamma$ columns) and $\beta_{n},r_{n},q_{n,i}$ are the column vectors. The explicit form of the matrices and the vectors results from (3.9).

For $n=0$, by (2.1) and (3.10) we obtain:

From the assumptions of the theorem it results that the vector $r_{0}$ is bounded and for sufficiently small $h_{0}>0$ the matrix $D_{0}$ possesses a uniformly bounded inverse. Hence, for $p=m+d+1$ we have:

and from (3.8) it results:

Deriving relation (3.8) $k$ times $(k=1,2,\ldots,m+d\}$ and using (3.12) we obtain:

Suppose now that, for all $j=0,1,\ldots,n-1$

hold and prove that (3.15) holds for $j=n$.
By (3.9), (3.10), (3.15) and the assumptions of the theorem it follows that for sufficiently small $h_{n}>0$ the matrix $D_{n}$ possess a uniformly bounded inverse, $\left\|E_{n}\right\|_{1}=O\left(h^{m+d+1}\right),\left\|q_{n,i}\right\|_{1}=O\left(h^{m+d+1}\right)(i=0,1,\ldots,n-1)$ and $\left\|r_{n}\right\|_{1}$ is bound. Thus, from (3.10) for $p=m+d+1$, we obtain:
and from (3.8) it results:

for all $\tau\in(0,1]$ and $k=0,1,\ldots,m+d$
Evaluations (3.14), (3.15) and (3.17) end the proof of the first assertion by the theorem.
(ii) By (2.1) and (2.11) it follows that the function $\varepsilon:=u-\hat{u}$ can be written for every $n=0,1,\ldots,N-1$, thus:

where:

If we now substract the discretized collocation equation (2.10) from the exact collocation equation (2.6) and we use relations (2.9) and (3.18), we are led to:

where $r_{n,i}:=\left(E_{n,i}^{(1)}\left[u_{i}\right],\ldots,E_{n,i}^{(m)}\left[u_{i}\right]\right)$, and the matrices $\hat{D}_{n},\hat{F}_{n},\mathscr{E}_{n}$ and the vectors $\hat{q}_{n,i}$ have the same sizes as $D_{n},F_{n},E_{n}$ and $q_{n,i}$ from (3.10), the differences between them consisting in the fact that the integrals from (3.9) are replaced with quadrature formulas of the form (2.8).

The above expression has the same structure as (3.10). From the smoothness hypothesis and from the assumptions on the order of the quadrature formulas (3.3) and (3.4) we have: $\left\|r_{n,n}\right\|_{1}=O\left(h_{n}^{r_{0}}\right)$ and $\left\|r_{n,i}\right\|_{1}=O\left(h_{i}^{r_{1}}\right)$. Thus, repeating the reasoning from the proof of the assertion (i), it easily results that relation (3.5) is true.

Now by (3.2) and (3.5) it results:

for all $k=0,1,\ldots,s$, with $s=\min\left\{s^{\prime},m+d+1\right\}$.
COROLLARY 3.2 Let the assumptions of Theorem 3.1 hold. If the quadrature formulas (2.8) are of interpolatory type, with $\mu_{0}=\mu_{1}=m+d$, then the approximation $\hat{u}\in S_{m+d}^{(d)}\left(Z_{N}\right)$ defined by the discretized collocation equation (2.10) leads to an error $\hat{e}(t)$ satisfying:

for $k=0,\ldots m+d$, and for every choice of the collocation parameters $\left\{c_{j}\right\}_{j=\overline{1,m}}$ with $0<c_{1}<\ldots<c_{m}\leq 1$.

In many papers (see [1], [2], [3]) the quadrature formulas used have $\mu_{0}=\mu_{1}=m$, $d_{j}=c_{j}$ and $d_{j,l}=c_{j}c_{l}(j,l=1,\ldots,m)$. The possibility of employing some quadrature formulas of the this type in our method would lead to some simplifications. These simplifications are useful when they do not spoil the convergence order given by Theorem 3.1 (i), namely $s=m+d+1$. An answer to this problem is given in the following corollary.

COROLLARY 3.3. If in VIDE (2.5), $p\in C^{m+d}(I),q\in C^{m+d}(I)$ and $K\in C^{m+d}(S)$ and if $m\geq d$, then there exists the set of collocation parameters $\left\{c_{j}\right\}_{j=1,n}$ such that for the approximation $\hat{u}\in S_{m+d}^{(d)}\left(Z_{N}\right)$ given by the discrete collocation equations (2.10) in which $\mu_{0}=\mu_{1}=m,d_{j}=c_{j}$ and $d_{j,l}=c_{j}c_{l}$ we have:

for $k=0,\ldots,m+d($ as $h\cdot 0$ with $Nh\leq\gamma T)$.
Proof. If $\mu_{0}=\mu_{1}=m$ and $m\geq d$ then we choose the set of collocation parameters $\left\{c_{j}\right\}_{j=\overline{1,m}}$ to be formed by the $m$ Gauss points for $(0,1)$.

Remark 3.4. (i) The results of the above theorems for $d=0$ and $m\geq 1$ are similar to the results given in [3], while for $d=n-1$ and $m=1(n\in N)$ they are similar to the result from [6].
(ii) The extension of the above arguments to nonlinear VIDE (1.1) is straightforward: in the error equations (3.10) and (3.19), the roles of $p_{n,j}$ and of $k_{n,j}\left(t_{i}+\tau h_{i}\right)$ are taken, respectively, by $\partial f\left(t_{n}+c_{j}h_{n},z_{n,j}\right)/\partial y$ and $\partial K\left(t_{n}+c_{j}h_{n},t_{i}+\tau h_{i},z_{i}(\tau)\right)/\partial y$, with $z_{n,j}$ and $z_{i}(\tau)$ denoting suitable intermediate values arising in the application of the Mean-Value Theorem (see [3], [5]).

The notion of local superconvergence is used when on a set of interior points $Z_{N}$ (or $\bar{Z}_{N}$ ), the approximate solution has a convergence order greater than the global convergence order. From Theorem 3.1 we notice that the only conditions imposed on the collocation parameters $\left\{c_{j}\right\}_{j=\overline{1,m}}$ are that they must be distinct and
they must belong to $(0,1]$. The local superconvergence on $\bar{Z}_{N}$ is closely connected with the choice of the collocation parameters (see [3], [4], [5]) and with the relation between their number and the number of the coefficients of the approximate solution determined from the smooth conditions.

We will give the following theorem concerning the aspects presented above:

(I) the given functions $p,g$ and $K$ from VIDE (2.5) are $m+p$ times continuously differentiable on their respective domains $I$ and $S$ (where $d+1<p\leq m$ );
(II) $m\geq d+2$;
(III) the collocation parameters $\left\{c_{j}\right\}_{j=1,m}$, with $0<c_{1}<\ldots<c_{m}\leq 1$ are chosen such that:

Then, for all quasi-uniform mesh sequences $\left\{\Pi_{N}\right\}$ with sufficiently small $h>0$, we have:
(i) if $u\in S_{m+d}^{(d)}\left(Z_{N}\right)$ is the approximate solution defined by the exact collocation equation (2.6) and $y$ is the exact solution of VIDE (2.5) then:

(ii) if the quadrature formulas (2.8) satisfy (3.3) and $\hat{u}\in S_{m+d}^{(d)}\left(Z_{N}\right)$ is the approximate solution defined by the discretized collocation equation (2.10) then:

where $\alpha=\min\left\{m+p,s^{\prime}\right\}$;
(iii) if $m\geq d+2$ and the collocation parameters $\left\{c_{j}\right\}_{j=1,m}$, are chosen such that relation (4.1) holds and $c_{m}=1$, then:

and

where $\alpha=\min\left\{m+p,s^{\prime}\right\}$.

Proof. (i) The exact collocation equation (2.6) can be written in the form: