In recent years many papers were devoted to the problem of approximate integration of system of differential equation by spline functions. The theory of spline functions presents a special interest and advantage in obtaining numerical solutions of differential equations.

The splines functions of even degree are defined in a similar manner with that for odd degree spline functions, but using the derivative-interpolating conditions. These spline functions preserve all the remarkable extremal and convergence properties of the odd degree splines, and are very suitable for the numerical solutions of the differential equation problems, especially for the delay differential equations with initial conditions.

In this paper we consider a spline approximation method for the numerical solution of a Lotka-Volterra system with delay. The purpose of the present study is to extend the results of [1], [2], [3], [5] from the delay differential equations to the delay

differential system. In the same manner we shall develop some theory and algorithms for the numerical solutions of a class of delay Lotka-Volterra system.

Let $\Delta_{n}$ be the following partition of the real axis

and let $m,n$ be two given natural numbers, satisfying the conditions $n\geq 1,m\leq n+1$. One denotes by $I_{k}$ the following subintervals

Definition 1. [3] For the couple ( $m,\Delta_{n}$ ) a function $s:\mathbb{R}\rightarrow\mathbb{R}$ is called a natural spline function of even degree $2m$ if the following conditions are satisfied:

where $\mathcal{P}_{k}$ represents the set of algebraic polynomials of degree $\leq k$.
We denote by $\mathcal{S}_{2m}\left(\Delta_{n}\right)$ the linear space of natural polynomial splines of even degree $2m$ with the simple knots $t_{1},\ldots,t_{n}$.

We now show that $\mathcal{S}_{2m}\left(\Delta_{m}\right)$ is a finite dimensional linear space of functions and we give a basis of it.

Theorem 1. [3] Any element $s\in\mathcal{S}_{2m}\left(\Delta_{n}\right)$ has the following representation

where the real coefficients $\left(A_{i}\right)_{0}^{m}$ are arbitrary, and the coefficients $\left(a_{k}\right)_{1}^{n}$ satisfy the conditions

Remark 1. [3] If $n+1=m$, then $a_{k}=0,k=\overline{1,n}$.

Theorem 2. [3] Suppose that $n+1\geq m$, and let $f:\left[t_{1},t_{n}\right]\rightarrow\mathbb{R}$ be a given function such that $f^{\prime}\left(t_{k}\right)=y_{k}^{\prime},k=\overline{1,n}$, and $f\left(t_{1}\right)=y_{1}$, where $y_{k}^{\prime},k=\overline{1,n}$, and $y_{1}$ are given real numbers. Then there exists a unique spline function $s_{f}\in\mathcal{S}_{2m}\left(\Delta_{n}\right)$, such that the following derivative-interpolating conditions

hold.
Corollary 1. [3] There exists a unique set of $n+1$ fundamental natural polynomial spline functions $S_{k}\in\mathcal{S}_{2m}\left(\Delta_{n}\right),k=\overline{1,n}$, and $s_{0}\in\mathcal{S}_{2m}\left(\Delta_{n}\right)$ satisfying the conditions:

It is clear that the functions $\left\{s_{0},S_{k},k=\overline{1,n}\right\}$, form a basis of the linear space $\mathcal{S}_{2m}\left(\Delta_{n}\right)$, and for $s_{f}$ we obtain the representation

But because $s_{0}(t)=1$, it follows that

Let us introduce the following sets of functions

Theorem 3. [3] (Minimal norm property). If $s\in\mathcal{S}_{2m}\left(\Delta_{n}\right)\cap\underset{\begin{subarray}{c}0,f\\ 0,f\end{subarray}}{m+1}\left(\Delta_{n}\right)$, then

holds, $\|\cdot\|_{2}$ being the usual $L_{2}$-norm.
For any function $f\in W_{2}^{m+1}\left(\Delta_{n}\right)$, we have the following corollaries.
Corollary 2. [3] $\left\|f^{(m+1)}\right\|_{2}^{2}=\left\|s_{f}^{(m+1)}\right\|_{2}^{2}+\left\|f^{(m+1)}-s_{f}^{(m+1)}\right\|_{2}^{2}$.
Corollary 3. [3] $\left\|s_{f}^{(m+1)}\right\|_{2}\leq\left\|f^{(m+1)}\right\|_{2}$.
Corollary 4. [3] $\left\|f^{(m+1)}-s_{f}^{(m+1)}\right\|_{2}\leq\left\|f^{(m+1)}\right\|_{2}$.
Remark 2. [3] If $\widetilde{s}:=s_{f}+p_{m}$, where $p_{m}\in\mathcal{P}_{m}$, it follows $\left\|\widetilde{s}^{(m+1)}\right\|_{2}\leq\left\|f^{(m+1)}\right\|_{2}$.
Theorem 4. [3] (Best approximation property). If $f\in W_{2}^{m+1}\left(\Delta_{n}\right)$ and $s_{f}\in\mathcal{S}_{2m}\left(\Delta_{n}\right)$ is the derivative-interpolating spline function of even degree, then, for any $s\in\mathcal{S}_{2m}\left(\Delta_{n}\right)$ the relation

holds.
Remark 3. [3] If $s_{f}-s\in\mathcal{P}_{m}$ then

1. 3.

   The numerical solutions of Lotka-Volterra system with delay by spline functions of even degree

   Report issue for preceding element

Let us consider the following delay differential system with a constant delay $\omega>0$

with initial conditions

and we suppose that $f^{u}:D\subset\mathbb{R}^{4}\rightarrow\mathbb{R}$, satisfies all the conditions assuring the existence and uniqueness of the solutions $y^{u}:[a,b]\rightarrow\mathbb{R}$ of the problem (3.1)+(3.2).

We propose an algorithm to approximate the solutions $y^{u}$ of the problem (3.1)+(3.2) by spline functions of even degree $s^{u}\in\mathcal{S}_{2m}\left(\Delta_{n}\right)$, where $\Delta_{n}$ is a partition of $[a,b]$ and $m,n$ are two integers satisfying the conditions $n\geq 1$ and $m\leq n+1$.

For $t\in[a,a+\omega]$, the problem (3.1)+(3.2) reduces to the following usual initial value problems:

Theorem 5. If $y^{u}$ are the exact solutions of the problem (3.1)+(3.2), then, there exists some unique spline functions $s_{y^{u}}\in\mathcal{S}_{2m}\left(\Delta_{n}\right)$ such that:

The assertion of this theorem is a direct consequence of Theorem 2 by substituting $t_{1}$ by $a$ and $f$ by $y^{u}$.

Denoting $y_{k}^{u}:=y^{u}\left(t_{k}\right),\bar{y}_{k}^{u}:=y^{u}\left(t_{k}-\omega\right),k=\overline{1,n},u=1,2$, we have

Corollary 5. If the functions $\left\{s_{0},S_{k},k=\overline{1,n}\right\}$ are the fundamental spline functions in $\mathcal{S}_{2m}\left(\Delta_{n}\right)$, then we can write

where $y_{k}^{1},y_{k}^{2},k=\overline{2,n}$, are unknown, and

We shall call the function $s_{y^{u}}(t)$, the approximating solution of the problem $(3.1)+(3.2)$ and it can be written as follows

For simplicity, in writing (3.5), let us use the following index sets:

Thus, we can write (3.5) in the form

where the values $y_{k}^{u},k=\overline{2,n}$, and $\bar{y}_{k}^{u},k=\overline{1,p},u=1,2$ are unknown.
Before giving an algorithm to determine these values, we shall give the following estimation error and convergence theorem.

Theorem 6. [3]If $y^{u}\in W_{2}^{m+1}[a,b],u=1,2$ are the exact solutions of the problem (3.1) $+(3.2)$ and $s_{y^{u}}$ is the spline approximating solution for $y^{u}$, the following estimations hold:

for $k=1,2,\ldots,m$ where $\left\|\Delta_{n}\right\|:=\max_{i=2,n}\left\{t_{i}-t_{i-1}\right\},u=1,2$.
Corollary 6. [3]If $y^{u}\in W_{2}^{m+1}[a,b]$, we have

Corollary 7. [3] $\lim_{\left\|\Delta_{n}\right\|\rightarrow 0}\left\|y^{u(k)}-s_{y^{u}}^{(k)}\right\|_{\infty}=0,k=\overline{1,m},u=1,2$.

For any $t\in[a,b]$, we suppose that $y^{u}(t)\approx s_{y^{u}}(t),u=1,2$.
If we denote, as usual, $e^{u}(t):=y^{u}(t)-s_{y^{u}}(t),t\in[a,b]$, we have

or

If we denote

then we have $y_{i}^{u}=w_{i}^{u}+e_{i}^{u},\bar{y}_{i}^{u}=\bar{w}_{i}^{u}+\bar{e}_{i}^{u}$, where

In what follows, we suppose that in (3.1)+(3.2) the functions

are continuous. Thus,

where

We can write the system (4.1) in the form

where

supposing that

on $D$. Obviously, $E_{i}^{u}\rightarrow 0$ and $\bar{E}_{i}^{u}\rightarrow 0$ for $\left\|\Delta_{n}\right\|\rightarrow 0,u=1,2$.

Now, we have to solve the following nonlinear system:

Let us denote:

and