THE STABILITY OF COLLOCATION METHODS FOR HIGHER-ORDER VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS

The numerical stability of the polynomial spline collocation method for general Volterra integro-differential equation is being considered. The convergence and stability of the new method are given and the efficiency of the new method is illustrated by examples. We also proved the conjecture suggested by Danciu in 1997 on the stability of the polynomial spline collocation method for the higher-order integro-differential equations.


Introduction
In this paper, we analyze the stability properties of the polynomial spline collocation method for the approximate solution of general Volterra integro-differential equation.Consider the linear pth-order Volterra integro-differential equation of the form y (p) (t) = q(t) + p−1 j=0 p j (t)y ( j) (t) + p−1 j=0 t 0 k j (t,s)y ( j) (s)ds, t ∈ I := [0,T], y (i) (0) = y (i)  0 , i = 0,1,..., p − 1. (1.1) Here, the functions q, p j : I → R and k j : D → R ( j = 0,1,..., p − 1) (with D := {(t, s) : 0 ≤ s ≤ t ≤ T}) are assumed to be (at least) continuous on their respective domains.For more detail of these equations and many other interesting methods for the approximated solution, stability procedures and applications are available in earlier literatures [1,2,3,4,5,6,7,10,11,12,13,14,15,18].The above equation is usually known as basis test equation and is suggested by Brunner and Lambert [4].Since then it, has been widely used for analyzing the stability properties [3,4,5,6,7,8,9,18] of various methods.Volterra integro-differential equation (1.1) will be solved numerically using polynomial spline spaces.To describe the polynomial spline spaces, let N : 0 = t 0 < t 1 < ••• < t N = T be the mesh for the interval I, and set 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) is approximated by an element u in the polynomial spline space n t n , for j = 0,1,...,d, t n ∈ Z N . (1. 3) It is 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 .
According to Micula [16] and Miculá and Micula [17], an element u ∈ S (d) m+d (Z N ) for all n = 0,1,...,N − 1 and t ∈ σ n has the following form: where From (1.4), we see that the element u ∈ S (d) m+d (Z N ) is well defined provided the coefficients {a n,r } r=1,...,m for all n = 0,1,...,N − 1 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 ) is determined by imposing the condition that u satisfies the initial-value problem (1.1) on X(N) and the initial conditions, that is, Here, we assume that the mesh sequence { N } is uniform, that is, h n = h, for all n = 0,1,...,N − 1.

Numerical stability
In order to discuss numerical stability, we study the behavior of the method as applied to the pth-order test Volterra integro-differential equation where α j , ν are constants and q : I → R is sufficiently smooth.We refer to a polynomial spline collocation method in the space S (d) m+d (Z N ), as an (m,d, p)-method, where p is the order of the integro-differential equation.
Definition 2.1.An (m,d, p)-method is said to be stable if all solutions {u(t n )} remain bounded as n → ∞, h → 0, while hN remains fixed.
From (1.4), we observe that the first d + 1 coefficients of u ∈ S (d)  m+d (Z N ) are determined by the smooth conditions, and the exact collocation equation (1.7) can be used to determine the last m coefficients.For the convenience, we introduce the following notations: β n := β n,r r=1,...,m , with β n,r := a n,r h d+r (n = 0,1,...,N − 1). ( Using (2.2) and t := t n + τh ∈ σ n in (1.4), we obtain the following: By direct differentiation of (2.3) and using the smooth conditions of the approximation u ∈ S (d)  m+d (Z N ), we get a relationship between vector η n+1 and vectors η n and β n as follows: where A is the (d + 1) × (d + 1) upper triangular matrix, and B is the (d + 1) × m matrix, whose elements are given by a j,r := For d ≥ p, apply the collocation method to test (2.1) and use the representation (2.3) to obtain the following collocation equation: where V is the m × m-matrix, W is the m × (d + 1)-matrix, and R n is the m-vector, whose elements are given by ) (2.9) We state the following result for pth-order VIDEs which describes a stability criterion for the collocation spline method.The proof of this theorem is similar to the proof given by Danciu [9] for first-order VIDEs.
Theorem 2.2.An (m,d, p)-method is stable if and only if all eigenvalues of matrix M := A + BV −1 W are in the unit disk, and all eigenvalues with |λ| = 1 belong to a 1 × 1 Jordan block, where the matrices A and B are defined in (2.5).
Remark 2.3.The dimension of the matrix M is dim(d + 1).Moreover, let M 0 be the matrix M with h = 0, and let λ (0) and λ be the eigenvalues of M 0 and M, respectively, then it follows that the matrix M 0 has (2.10)

Applications
In this section, we will investigate some special cases.
(I) For the case d = p, the approximation space is S (p) m+p (Z N ).From the above theorem and Remark 2.3, we have the following theorem.Theorem 3.1.For every choice of the collocation parameters {c j } j=1,...,m , an (m, p, p)-method is stable for all m ≥ 1.
Edris Rawashdeh et al. 3079 (II) For the case m = 1, this choice of m corresponds to a classical spline function, that is, the approximate solution u ∈ S (d)  1+d (Z N ).By Remark 2.3, M 0 is the matrix M with h = 0, and λ (0) and λ are the respective eigenvalues of M 0 and M, and we have is the collocation parameter, then, for m = 1 and d ≥ p, the trace of the matrix M 0 can be computed by the following formula: Proof.Let V 0 and W 0 be the matrices V and W with h = 0, respectively.Then, for m = 1, we have from (2.7) and (2.8) that V 0 is a 1 × 1-matrix and W 0 is a 1 × (d + 1)-matrix, whose elements are given by Now, from the definition of the matrices A and B as in (2.5) (note that the diagonal entry of the matrix A is one), we have (3.4)However, by the binomial expansion, we have the following identity: Hence, For the stability of the spline collocation method S (d) 1+d (Z N ) (m = 1), we have the following theorem.
If d = p + 2 and c 1 ∈ (0,1], then, from (3.2), we have (3.13) Edris Rawashdeh et al. 3081 Hence, Thus, from Theorem 2.2, a (1,d, p)-method is unstable for any choice of the collocation parameter c 1 ∈ (0,1] when d ≥ p + 2. (III) For the case m = 2, we can prove the following theorem.The proof is similar to the proof given in [9] for first-order integro-differential equation (p = 1).Theorem 3.4.Let 0 < c 1 < c 2 ≤ 1 be the collocation parameters, then (i) (2, p, p)-method is stable for every choice of the collocation parameters, (IV) For the case d = p + 1, the approximation u ∈ S (p+1) m+p+1 (Z N ) and the dimension of the matrix M 0 are p + 2, whose p+1 = 1 are its first p + 1-eigenvalues.To compute the p + 2-eigenvalue, we need the following results.But, first we introduce the following notations: Proof.We will prove the lemma by induction on the dimension of the matrix, starting with 2 × 2-matrices.For the 2 × 2-matrices, the result is clearly true.For m × m-matrices (m > 2), we define where a, b are constants to be determined.By the induction hypothesis, we obtain On the other hand, from the definition of f and by the induction hypothesis, we have