HIGH ORDER PRODUCT INTEGRATION METHODS FOR A ...

COMMUNICATIONS ON PURE AND APPLIED ANALYSIS Volume 3, Number 2, June 2004

Website: http://AIMsciences.org pp. 217–235

HIGH ORDER PRODUCT INTEGRATION METHODS FOR A VOLTERRA INTEGRAL EQUATION WITH LOGARITHMIC SINGULAR KERNEL

T. Diogo Centro de Matem´ atica e Aplica¸c˜ oes, Instituto Superior T´ ecnico Av. Rovisco Pais, 1049-001 Lisboa, Portugal

N.B. Franco Laborat´ orio Interdisciplinar de Computa¸c˜ ao Cient´ıfica Faculdades COC, 14096-160 Ribeir˜ ao Preto - SP, Brazil

P. Lima Centro de Matem´ atica e Aplica¸c˜ oes, Instituto Superior T´ ecnico Av. Rovisco Pais, 1049-001 Lisboa, Portugal

(Communicated by Tao Tang)

Abstract. This work is concerned with the construction and analysis of high order product integration methods for a class of Volterra integral equations with logarithmic singular kernel. Sufficient conditions for the methods to be convergent are derived and it is shown that optimal convergence orders are attained if the exact solution is sufficiently smooth. The case of non-smooth solutions is dealt with by making suitable transformations so that the new equation possesses smooth solutions. Two particular methods are considered and their convergence proved. A sample of numerical examples is included.

1. Introduction. We consider the Volterra integral equation of the second kind Z t F (t) + P (t, s)F (s)ds = H(t), t ∈ [0, T ], (1.1) 0

where 1 1 1 P (t, s) := √ p π ln(t/s) s

(1.2)

and H(t) is a given function. The above equation arises in some heat conduction problems with mixed-type boundary conditions [1]. As an example, consider ∂2u 1 ∂u = 2 , ∂x2 a ∂t

0 ≤ x ≤ l,

(1.3)

2000 Mathematics Subject Classification. 65R20. Key words and phrases. Volterra integral equations, weakly singular kernel, product integration methods, convergence.

217

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T. DIOGO, N.B. FRANCO AND P. LIMA

with the conditions u(x, −∞) = 0, (1.4) ∂u (0, t) − u(0, t) = φ1 (t), (1.5) ∂x ∂u − (l, t) − u(l, t) = φ2 (t). (1.6) ∂x The solution u(t, x) can be expressed in terms of single layer potentials (see [1], [23]) as follows   −x2 −(x − l)2 Z t a 2 2   (t − τ )−1/2 ρ1 (τ ) e 4a (t − τ ) + ρ2 (τ ) e 4a (t − τ )  dτ (1.7) u(x, t) = √ 2 π −∞ Above ρ1 (τ ), ρ2 (τ ) are such that u(x, t) satisfies conditions (1.4)-(1.6). By imposing those conditions, the system of two integral equations is obtained   −l2   Z u a l 1 1 1 2 4a ln(u/x)   √ q −p e ψ2 dx x π 0 ln(u/x) x 2a2 ln3 (u/x)       Z u 1 1 a 1 1 1 p −√ ψ1 dx − ψ1 = H1 (1.8) x u u π 0 ln(u/x) x −l2   1 1 2   q e 4a ln(u/x) ψ1 dx −p x ln(u/x) x 0 2a2 ln3 (u/x)       Z u a 1 1 1 1 1 p −√ ψ2 dx − ψ2 = H2 (1.9) x u u π 0 ln(u/x) x a √ π

Z

u





l

1

where ψk (s) := ρk (− ln s), hk (s) := 2φk (− ln s), k = 1, 2. If l is large compared to a, then we may consider the system       Z u 1 a 1 1 1 1 p ψ1 dx − ψ1 = H1 (1.10) −√ x u u π 0 ln(u/x) x       Z u a 1 1 1 1 1 p −√ ψ2 dx − ψ2 = H2 (1.11) x x u u π 0 ln(u/x) We note that the above equations are independent and may be treated separately, each of them being of the form of (1.1). A more complex problem leading to a system of equations of the type of (1.1) was considered by Bartoshevich [1]. In [2] he developed a method for its solution which involved expansions in terms of Watson’s operators. Sub-Sizonenko [24] provided an analytic L2 solution for (1.1) and other expressions valid in weighted Lp spaces were derived by Rooney [22] and Lamb ([13], [14]) but none is useful if a tabulated solution is required. Rt We note that 0 P (t, s)ds is divergent. Following [14] we use the transformations y(t) = t−µ F (t),

f (t) = t−µ H(t),

where µ > 0 is a constant. From (1.1) we obtain

(1.12)

HIGH ORDER PRODUCT INTEGRATION METHODS

219

t

Z y(t) +

q(t, s)y(s)ds = f (t),

t ∈ [0, T ],

(1.13)

0

with

 s µ 1 1 1 . q(t, s) := √ p s π ln(t/s) t

In [6] this equation was transformed into the more tractable equation Z t p(t, s)y(s)ds = g(t), t ∈ [0, T ], y(t) −

(1.14)

(1.15)

0

with

 s µ 1 . (1.16) t s For a given non-negative integer m, let C m [0, T ] denote the normed space of the real functions φ whose derivatives of order up to m are continuous on [0, T ], with kφkm := max max φ(j) (t) . p(t, s) :=

0≤j≤m t∈[0,T ]

The results concerning existence and uniqueness of solution for the above equations, published in [6], [10], can be summarized as follows. (i) If µ > 1 in (1.14) and the function f in (1.13) belongs to C m [0, T ] then the problem (1.13)-(1.14) possesses a unique solution y ∈ C m [0, T ]. Similarly, if If µ > 1 in (1.16) and the function g in (1.15) belongs to C m [0, T ] then (1.15)-(1.16) possesses a unique solution y ∈ C m [0, T ]. Moreover, if g is given by Z t g(t) = f (t) − q(t, s)f (s)ds 0

then the solution of the problem (1.13)-(1.14) is also the solution of (1.15)-(1.16). (ii) If 0 < µ ≤ 1 and f ∈ C 1 [0, T ] (with f (0) = 0 if µ = 1), then (1.13)-(1.14) has a family of solutions in C[0, T ] of which only one has C 1 continuity. Similar conclusions are valid for (1.15)-(1.16) if g ∈ C 1 [0, T ] (with g(0) = 0 if µ = 1). It should be noted that (i) is in contrast with the smoothness properties of weakly singular equations possessing a singularity in the kernel of the form k(t, s) = (t − s)−α , 0 < α < 1 ( Abel type equations). For those equations, a smooth forcing function leads to a solution which has typically unbounded derivatives at t = 0 (see [3]). Furthermore, unlike the function k, the functions q and p do Rt Rt not satisfy 0 q(t, s)ds → 0 as t → 0 and 0 p(t, s)ds → 0 as t → 0. In fact √ these integrals are constant and equal to 1/ µ and 1/µ, respectively. Finally, all the iterated kernels associated with p (we note that q is the iterated kernel of second order) are unbounded. The ideas of iterated and discrete iterated kernels were used by Dixon and McKee [8] to obtain generalised weakly singular Gronwall inequalities (see also [3]). These are an important tool for proving convergence of discretization methods for weakly singular integral equations but it is usually required the existence of at least one bounded iterated kernel. Owing to the above properties of the kernels p and q, special techniques are needed to prove convergence of discretisation methods for the above equations. The numerical study of (1.13)(1.14) and (1.15)-(1.16), in the case of µ > 1, has been the subject of several papers. Tang et al [25] applied two low order product integration methods to equation (1.13)(1.14). They obtained approximations to y(t) of orders one and two with the Euler and Trapezoidal methods, respectively. Diogo et al [6] considered a fourth order

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T. DIOGO, N.B. FRANCO AND P. LIMA

Hermite-type collocation method for the equivalent equation (1.15)-(1.16) and Lima and Diogo [15] have developed an extrapolation method, based on Euler’s method. In order to deal with unbounded solutions a variable transformation was used so that the solution of the new equation was smooth. Euler’s method was then applied and first order of convergence was proved. Moreover, under certain conditions, the error was shown to allow an asymptotic error expansion in integer powers of the stepsize which permitted the use of Richardson’s extrapolation. In the present work, we are concerned with the construction and analysis of higher order product integration methods based on generalized Newton-Cotes formulae for the solution of (1.15)-(1.16), in the case µ > 1. A convergence analysis is given which requires the exact solution to be sufficiently smooth and the case of non-smooth solutions is dealt with by using the technique considered in [15]. An appropriate transformation of the dependent variable is made so that the new equation possesses smooth solutions and the convergence results can be applied. The idea of solution regularization transformations was used in [19], [20], [7] with Abel-type equations and, recently, in [18], for both continuous and Abel-type kernels. Product integration methods based on generalized Newton-Cotes formulae for Abel-type equations were studied in [16], [12] and [5]. For a comprehensive study of numerical methods for Volterra integral equations we refer to [3]. Among more recent works we refer to [18] and the references therein. For other discretisation methods for Abel equations which deal with smooth solutions see for instance [9] and [4]. This paper is organised as follows. In Section 2 a class of product integration methods based on the use of a main interpolatory quadrature rule together with several end rules is introduced. The consistency order of these methods is analysed in Section 3 by using the summation formula for functions with algebraic singularities given in [17]. In general, if the main rule is based on n + 1-points then consistency of order n + 1 can be achieved if the solution y(t) is in C n+1 [0, T ]. However, if µ is sufficiently high, these continuity requirements can be relaxed, by considering an appropriate transformation of the equation as referred to above. Finally, a sufficient condition for the product integration methods to be convergent is derived in Section 4 which requires the sum of the modulus of the weights to be uniformly bounded by some constant less than one. In particular, this is shown to hold if the weights are non-negative. In Section 4 two particular methods corresponding to the values n = 1 and n = 2 are considered and their convergence is proved. The paper concludes with a sample of numerical examples illustrating the theoretical results obtained. 2. High Order Product Integration Methods. In order to construct numerical methods for (1.15)-(1.16), let us define the grid {tj = jh,

0 ≤ j ≤ N;

Nh = T}

and consider the discretised form of (1.15) Z ti y(ti ) − p(ti , s)y(s)ds = g(ti ),

0 ≤ i ≤ N.

(2.1)

0

To approximate the integral in (2.1), some (n + 1)-point interpolatory formula is used repeatedly, followed by an end rule or a series of end rules when necessary. The coefficients of the resulting quadrature are calculated analytically.

HIGH ORDER PRODUCT INTEGRATION METHODS

221

Suppose first that i is a multiple of n, say i = nr. We can rewrite (2.1) as r−1 Z tn(j+1)  µ X s 1 y(ti ) − y(s) ds = g(ti ). (2.2) t s i j=0 tnj Approximating y(s), s ∈ (tj , tj+n ) by a polynomial of degree n, yields Z tj+n  µ Z tj+n  µ n X s s 1 1 y(s)ds ' lk (s)ds, y(tj+k ) ti s ti s tj tj

(2.3)

k=0

where the lk are the Lagrange polynomials of degree n given by n Y s − tj+i 0 ≤ k ≤ n. lk (s) = t − tj+i j+k i=0 i6=k

By making the transformation s = tj + vh, we have Z tj+n  µ Z n s 1 (tj + vh)µ−1 lk (tj + vh)dv lk (s)ds = h ti s (ti )µ tj 0 Z n 1 = µ (j + v)µ−1 ρk (v)dv, i 0

(2.4)

where ρk is a polynomial of degree n, ρk (v) := lk (tj + vh) =

n Y v−i . k −i i=0

(2.5)

i6=k

Let us define

Z bγ (j) :=

n

(j + v)µ−1 ργ (v)dv,

0 ≤ γ ≤ n.

(2.6)

0

Substituting (2.4) into (2.3) and Z tj+n  µ s t i tj

using (2.6), we obtain n 1 X 1 y(s)ds ' µ y(tj+k )bk (j). s i

(2.7)

k=0

Combining (2.7) and (2.2), we obtain for the case i = nr the following approximate equation r−1 n 1 XX ynr − ynj+k bk (nj) = g(tnr ), (2.8) µ (nr) j=0 k=0

where the yi are approximate values of y(ti ). When i is not a multiple of n, say i = nr + ν, 1 ≤ ν ≤ n − 1, it is convenient to rewrite (2.1) in the form Z ti−n−ν Z ti y(ti ) − p(ti , s)y(s)ds − p(ti , s)y(s)ds = g(ti ). (2.9) 0

ti−n−ν

The integral over [0, ti−n−ν ] is approximated by the main repeated formula. In order to approximate the integral over [ti−n−ν , ti ], end rules based on polynomial interpolation of degrees (n + 1), (n + 2), · · · , (2n − 1) are used. Let us define Z n+ν dνk (i) := (i − n − ν + v)µ−1 ρ˜k (v)dv, 0 ≤ k ≤ n + ν. (2.10) 0

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T. DIOGO, N.B. FRANCO AND P. LIMA

where ρ˜j (v) :=

n+ν Y k=0

v−k , j−k

0 ≤ j ≤ n + ν.

(2.11)

k6=j

We obtain the following approximate equation for the case when i = nr + ν   r−2 X n n+ν X X 1  ynr+ν − ynj+k bk (nj) + ynr−n+k dνk (nr + ν) = g(tnr+ν ). (nr + ν)µ j=0 k=0

k=0

(2.12) Combining (2.8) and (2.12), we obtain the discretisation method Φh y = 0

(2.13)

with  [Φh y]i =

yi − y˜i , Pi yi − j=0 ωij yj − gi ,

0 ≤ i ≤ n − 1, n ≤ i ≤ N,

(2.14)

where y˜i , 0 ≤ i ≤ n − 1, are given starting values. We can write (2.13)-(2.14) in matrix form as Φh y = (I − AN )y − g, (2.15) with Here AN

y = (y0 , y1 , · · · , yN )T and g = (˜ y0 , · · · , y˜n−1 , gn , · · · , gN )T . ˜ N DN , with =A   1 1 1 , · · · , , DN = diag 0, · · · , 0, µ , n (n + 1)µ Nµ

˜ N being integrals (or sums of integrals) of the types (2.6) and the elements of A (2.10). We note that, since µ > 1, all the integrals involved can be calculated analytically. 3. Convergence. Let δ(h, ti ) denote the local consistency error, defined by Z ti  µ i X s 1 δ(h, ti ) := y(s)ds − ωij y(tj ). (3.1) ti s 0 j=0 We require the following definition of consistency. Definition 3.1. The discretisation (2.13)-(2.14) is said to be consistent of order p with (1.15)-(1.16) at t = ti , i ≥ n, if there exists a constant C, independent of h, such that for h ∈ (0, h0 ), h0 > 0, we have |δ(h, ti )| ≤ Chp .

(3.2)

In order to investigate the consistency order of the discretisation (2.13)-(2.14), let us first suppose that i is a multiple of n. It follows, with i = nr, that ! Z ti  µ r−1 Z tn(j+1)  µ n X s 1 s 1 X δ(h, ti ) = y(s)ds − lk (s)y(tnj+k ) ds ti s ti s 0 j=0 tnj k=0 ! r−1 Z tn(j+1)  µ n X X s 1 = y(s) − lk (s)y(tnj+k ) ds. (3.3) ti s j=0 tnj k=0

HIGH ORDER PRODUCT INTEGRATION METHODS

223

Assume that y ∈ C m [0, T ], 0 < m ≤ n+1. We have from the Lagrange interpolation theory n X 1 (m) y(s) − y (tnj + nθj h)τ (s), (3.4) lk (s)y(tnj+k ) = m! k=0 Qm−1 for some θj ∈ (0, 1) and where τ (s) := k=0 (s − tnj+k ). Then we can obtain the following bound |y(s) −

n X

lk (s)y(tnj+k )| ≤

k=0

where τ ∗ (v) =

hm max |y (m) (s)| max |τ ∗ (nv)|, m! s∈[0,T ] v∈[0,1]

(3.5)

Qm−1

(v − k), which substituted into (3.3) gives Z ti  µ s 1 m ds = O (hm ) . |δ(h, ti )| ≤ Ch t s i 0

k=0

(3.6)

If i is of the form i = nr + ν, 1 ≤ ν ≤ n − 1, we apply the main rule repeatedely on [0, ti−n−ν ] and use an end rule of degree n + ν on [ti−n−ν , ti ]. Then δ(h, ti ) = δ 1 (h, ti ) + En+ν (h, ti ),

(3.7)

with δ 1 (h, ti ) = O(hm ) and where the second term is the error associated with the end rule. We now prove that this contribution will not affect the global consistency error. If y ∈ C q [0, T ], 1 ≤ q ≤ n + ν + 1, then  µ Z 1 ti s 1 (q) En+ν (h, ti ) = y (θi )λ(s)ds, (3.8) q! ti−n−ν ti s Qq−1 with λ(s) := k=0 (s − ti−n−ν+k ) and where θi ∈ (ti−n−ν , ti ). Making the transformation s = ti−n−ν + vh, gives Z n+ν (i − n − ν + v)µ−1 hq max |y (q) (s)| max |λ∗ (v)| dv, |En+ν (h, ti )| ≤ q! s∈[0,T ] iµ v∈[0,n+ν] 0 Qq−1 where λ∗ (v) = k=0 (v − k). That is, En+ν (h, ti ) = O(hq ) and it is clear that this error will not affect the global consistency error. We have thus proved Theorem 3.2. If y ∈ C m [0, T ], 0 < m ≤ n + 1, then δ(h, ti ) = O (hm ). In particular, if y ∈ C n+1 [0, T ] then the discretisation (2.13)-(2.14) is consistent of order n + 1. Let us now assume that y ∈ C n+1 [0, T ] ∩ C n+2 (0, T ). Using (3.4) with m = n + 1 and making the transformation s = tnj + nvh, we obtain from (3.3) that Z 1 r−1 X (tnj + nvh)µ−1 (n+1) 1 nh (y (tnj + nvh) δ(h, ti ) = (n + 1)! j=0 ti µ 0

=

+ O(h))hn+1 τ ∗ (nv)dv Z 1 r−1 X hn+1 (tnj + nvh)µ−1 (n+1) τ ∗ (nv) nh y (tnj + nvh)dv (n + 1)! 0 ti µ j=0 + O(hn+2 ).

(3.9)

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T. DIOGO, N.B. FRANCO AND P. LIMA

The following Lemma will allow us to analyse the sum nh

r−1 X (tnj + nvh)µ−1 (n+1) y (tnj + nvh). ti µ j=0

(3.10)

Lemma 3.3. If y ∈ C n+2 [0, T ] and µ > 1 is not an integer then, for each s ∈ [0, 1], h

r−1 X

(tl + sh)µ−1 y (n+1) (tl + sh)

l=0 tr

Z

tµ−1 y (n+1) (t)dt + tr µ h γ(0, 1 − s)

= 0

y (n+1) (tr ) + o(h) tr

(3.11)

where γ(a, s) is the periodic generalized zeta function defined by (3.13). Proof. Defining g(u) := y (n+1) (u tr ), we may write Z tr Z 1 tµ−1 y (n+1) (t)dt = tr µ uµ−1 g(u)du. 0

(3.12)

0

We now make use of the generalized Euler-Maclaurin summation formula for func¯ r := tions with algebraic singularities given by Lyness and Ninham [17]. Letting h h/tr = 1/r, we have, for each s ∈ [0, 1], Z tr tµ−1 y (n+1) (t)dt 0

= tr

µ



¯r h

r−1 X

¯ r )µ−1 g((l + s)h ¯r) − h ¯ r γ(0, 1 − s)g(1) ((l + s)h

l=0


d µ−1 2 ¯ ¯ t g(t) t=1 + O(hr ) γ(1 − µ, s)g(0) + hr γ(−1, 1 − s) dt  X  r−1 ¯r ¯ r )µ−1 g((l + s)h ¯r) − h ¯ r γ(0, 1 − s)g(1) + o(hq ) . h ((l + s)h r

¯µ −h r = tr µ

l=0

Here q = max(µ + 1, 3) and γ(a, s) is the periodic generalized zeta function defined by γ(a, s) := ζ(a, s¯),

s − s¯ = integer,

0 < s¯ ≤ 1,

(3.13)

where ζ(a, s) is the generalized Riemann zeta function ζ(a, s) =

∞ X k=0

1 , (s + k)a

∞ 2a! X sin[2πsk − πa/2] ζ(−a, s) = (2π)a+1 k a+1

(Re a > 0).

k=1

From above the desired result follows. When i = nr, using (3.11) in (3.9) gives Z ti µ−1 Z 1 hn+1 t y (n+1) (ti ) (n+1) δ(h, ti ) = τ ∗ (nv) (t)dt + h γ(0, 1 − v) µ y (n + 1)! 0 ti ti 0 
+ o (h) dv + O hn+2 . (3.14)

HIGH ORDER PRODUCT INTEGRATION METHODS

Theorem 3.4. Suppose that n is even. If y ∈ C n+2 [0, T ] and y δ(h, ti ) = O(hn+2 ).

225 (n+1)

(0) = 0 then

Proof. Suppose that µ > 1 is not an integer. We shall use the fact that if n is even then Z 1 τ ∗ (nv)dv = 0. (3.15) 0

This, together with the condition y (n+1) (0) = 0, gives Z ti µ−1  Z 1
t hn+1 ∗ (n+1) δ(h, ti ) = τ (nv) (t) + o (h) dv + O hn+2 µ y (n + 1)! 0 ti 0

(3.16)

which shows that δ(h, ti ) = O(hn+2 ) if i = nr. When i = nr + ν, 1 ≤ ν ≤ n − 1, the error will be given by (3.7). As noted before, the contribution of the end rule will not affect the global result. To estimate the first term, an application of Lemma 3.3 to (3.10), with tµi replaced by ((nr + ν)h)µ , yields δ 1 (h, ti ) = O(hn+2 ). We finish the proof by noting that, if µ is an integer, then the expansion (3.11) reduces to the usual form in terms of the Bernoulli polynomials. The conclusions about the consistency order remain valid. 3.1. Non-smooth solutions. We have seen that in order to obtain consistency of order at least n+1 the function y(t) was required to be at least in the class C n+1 [0, T ] (cf. Theorems 3.2 and 3.4). We now show that these continuity requirements can be relaxed if µ is sufficiently high. Following the approach used in [15], we can transform the original equation into a new equation, so that the two equations are equivalent away from the origin. Moreover, the corresponding solution of the new equation will be smooth. Let β > 0 be such that µ − β > 1. Multiplying both sides of (1.15) by tβ and defining y(t) := tβ y(t) β

g(t) := t g(t)

(3.17) (3.18)

we obtain the equation y(t) −

Z t  µ−β s 1 y(s)ds = g(t), t s 0

(3.19)

which is equivalent to (1.15) for t > 0. If we apply the scheme (2.13)-(2.14) to this equation, we obtain approximations y k of y(tk ) = tβk y(tk ), k ≥ n. Then we take as an approximation to y(tk ) the value y k /tβk . The idea is to chose β such that tβ y(t) will have the required continuity so that Theorems 3.2 and 3.4 can be applied. In particular, suppose that y(t) = tα f (t), with f (t) ∈ C n+1 [0, T ]. If we choose β such that 0 ≤ β < µ − 1 and α + β is an integer, then tβ y(t) ∈ C n+1 [0, T ] and consistency of order n + 1 follows from Theorem 3.2. 3.2. Two convergence results. The following definition will be needed. Definition 3.5. The starting values are said to be convergent of order p if there exists a constant C2 , independent of h, such that |y(ti ) − yi | ≤ C2 hp ,

0 ≤ i ≤ n − 1.

(3.20)

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T. DIOGO, N.B. FRANCO AND P. LIMA

Theorem 3.6. Let µ > 1 in (1.16). If the exact solution y of (1.15)-(1.16) is such that the discretisation (2.13)-(2.14) is consistent of order p (p ≤ n + 2), the starting values are convergent of order p and the weights ωij satisfy the condition i−1 X j=0

|ωij | ≤ β < 1, 1 − |ωii |

i = n, n + 1, . . . .

(3.21)

then the scheme (2.13)-(2.14) is convergent of order p. Proof. Let us define ei := y(ti ) − yi . We have Z ti i X ei = p(ti , s)y(s)ds − ωij yj

=

0

j=0

i X

Z

ti

p(ti , s)y(s)ds −

ωij (y(tj ) − yj ) + 0

j=0

i X

ωij y(tj ),

(3.22)

j=0

which gives, after using (3.1) and taking modulus, |ei | (1 − |ωii |) ≤

i−1 X

||ωij ||ej | + |δ(h, ti )|.

(3.23)

j=0

Let us now prove that |ωii | ≤ 1/µ. If i is a multiple of n, say, i = nr, then from (2.8) and (2.6) Z n bn (n(r − 1)) (nr − n + v)µ−1 = ρn (v)dv, ωii = (nr)µ (nr)µ 0

(3.24)

(3.25)

which gives Z |ωii | ≤ 0

n

rµ − (r − 1)µ 1 (nr − n + v)µ−1 dv = ≤ . µ µ (nr) µr µ

If i = nr + ν, 1 ≤ ν ≤ n − 1, then from (2.12) and (2.10) Z n+ν dνn+ν (nr + ν) ωii = = (nr − n + v)µ−1 ρ˜n+ν (v)dv (nr + ν)µ 0

(3.26)

(3.27)

and so Z |ωii | ≤ 0

n+ν

(nr − n + v)µ−1 (nr + ν)µ − (nr − n)µ 1 dv = ≤ . (nr + ν)µ µ(nr + ν)µ µ

(3.28)

Then we obtain, for some constant C independent of h, |ei | ≤

i−1 X j=0

|ωij | |ej | + Chp , 1 − |ωii |

(3.29)

where we have used (3.2). Finally, an application of a Gronwall lemma ([16]) yields the desired result. Theorem 3.7. Let µ > 1 in (1.16). If the exact solution y of (1.15)-(1.16) is such that the discretisation (2.13)-(2.14) is consistent of order p (p ≤ n + 2), the starting values are convergent of order p and the weights ωij are non-negative, then the scheme (2.13)-(2.14) is convergent of order p.

HIGH ORDER PRODUCT INTEGRATION METHODS

227

Proof. We can show that (3.21) holds with β = 1/µ. We begin by proving that i X

ωij =

j=0

1 , µ

i = n, n + 1, . . .

(3.30)

Let us examine the two cases i = nr and i = nr + ν, 1 ≤ ν ≤ n − 1, separately. When i = nr, we obtain, from (2.8) nr X

ωij

=

j=0

n X k=0

=

=

r−1 1 X bk (nl) (nr)µ l=0

1 (nr)µ

r−1 Z n X l=0

(nl + v)µ−1

0

n X

ρk (v)dv

k=0

r−1 1 X1 [(nl + n)µ − (nl)µ ], (nr)µ µ

(3.31)

l=0

where we have used (2.5) and the fact that n X

ρk (v) =

n X

lk (tnl + vh) = 1,

0 ≤ v ≤ n.

k=0

k=0

It follows from (3.31) that nr X

1 , µ

ωij =

j=0

(3.32)

since r−1 X

[(nl + n)µ − (nl)µ ] = (nr)µ ,

(3.33)

l=0

In a similar way, when i = nr + ν, 1 ≤ ν ≤ n − 1, we obtain from (2.12) ! nr+ν n+ν r−2 X n X X X 1 ν ωij = dk (nr + ν) . bk (nl) + (nr + ν)µ j=0 l=0 k=0

(3.34)

k=0

Using (2.10), we obtain nr+ν X

n+ν

ωij

=

j=0

X (nr − n)µ + µ µ(nr + ν)

k=0

= =

1 (nr + ν)µ

Z

n+ν

(n(r − 1) + v)ν−1 ρ˜k (v)dv

0

(nr − n)µ + µ

Z

n+ν µ−1

(n(r − 1) + v) 0

n+ν X

! ρ˜k (v)dv

k=0

(nr + ν)µ 1 = . µ(nr + ν)µ µ

(3.35)

We then have i−1 X j=0

1 − ωii ωij 1 µ = ≤ , 1 − ωii 1 − ωii µ

i = n, n + 1, . . . .

and an application of Theorem 3.6 completes this proof. 4. Two Product Integration Methods.

(3.36)

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T. DIOGO, N.B. FRANCO AND P. LIMA

4.1. The product trapezoidal method. In the case when n = 1 the integral in (2.1) is approximated by the product trapezoidal rule applied repeatedly over [0, ti ]. We then have the following approximate equation (cf. (2.8)) yi −

i−1 1 X (yj b0 (j) + yj+1 b1 (j)) = g(ti ), iµ j=0

1 ≤ i ≤ N,

(4.1)

with 1

Z b0 (j) =

(j + v)µ−1 (1 − v)dv,

(4.2)

(j + v)µ−1 vdv.

(4.3)

0 1

Z b1 (j) = 0

By examining the integrals (4.2) and (4.3), we immediately conclude that the weights of the trapezoidal method (4.1) are positive, for any value of µ > 1. Then convergence follows from Theorem 3.7. 4.2. The product Simpson’s method. We consider the case when n = 2. To approximate the integral in (2.1), the product Simpson’s rule is used repeatedly over [0, ti ] if i is even. When i is odd, we use the product Simpson’s rule over [0, ti−3 ] and the product three-eights rule will be used over [ti−3 , ti ]. In this case, (2.8) and (2.12) take the form, respectively, y2r −

r−1 2 1 XX y2j+k bk (2j) = g(t2r ) (2r)µ j=0

(4.4)

k=0

and y2r+1 −

1 (2r + 1)µ

  r−2 X 2 3 X X  y2j+k bk (2j) + y2r−2+k dk (2r + 1) = g(t2r+1 ), j=0 k=0

k=0

(4.5) with 1 b0 (j) = 2

2

Z

(v − 1)(v − 2)(j + v)µ−1 dv,

0

Z

2

v(v − 2)(j + v)µ−1 dv,

b1 (j) = − 0

1 b2 (j) = 2

Z

2

v(v − 1)(j + v)µ−1 dv,

(4.6)

0

and d0 (i)

=

d1 (i)

=

d2 (i)

=

d3 (i)

=

Z 1 3 − (v + i − 3)µ−1 (v − 1)(v − 2)(v − 3)dv, 6 0 Z 1 3 (v + i − 3)µ−1 v(v − 2)(v − 3)dv, 2 0 Z 1 3 − (v + i − 3)µ−1 v(v − 1)(v − 3)dv, 2 0 Z 1 3 (v + i − 3)µ−1 v(v − 1)(v − 2)dv. 6 0

(4.7)

HIGH ORDER PRODUCT INTEGRATION METHODS

229

˜ N has the form Here the matrix A               

0 0 0 b0 (0) b1 (0) b2 (0) d0 (3) d1 (3) d2 (3) d3 (3) b0 (0) b1 (0) b2 (0) + b0 (2) b1 (2) b2 (2) b0 (0) b1 (0) b2 (0) + d0 (5) d1 (5) d2 (5) d3 (5) b0 (0) b1 (0) b2 (0) + b0 (2) b1 (2) b2 (2) + b0 (4) b1 (4) b2 (4) b0 (0) b1 (0) b2 (0) + b0 (2) b1 (2) b2 (2) + d0 (7) d1 (7) d2 (7) d3 (7) .. .



..

              .

We note that, unlike in the Trapezoidal method, the condition ωij ≥ 0 does not hold for all i, j and all values of µ > 1. Therefore, we cannot apply Theorem 3.7 for all values of µ. However, we have the following result. Theorem 4.1. Let 1 < µ ≤ 2. Consider the integrals bk (2r), k = 0, 1, 2, r ≥ 0; and dl (2r + 1), l = 0, 1, 2, 3, r ≥ 1. We have b0 (2r) = 0 if r = 0 and µ = 2 and in the remaining cases all the integrals bk (2r) and dl (2r + 1) are positive. As a consequence, the weights ωij , i = 1, 2, . . . ; j = 0, 1, . . . of algorithm (4.4)-(4.5) are non-negative. Proof. We only give here the proof for the integrals bk (2r), k = 0, 1, 2, as the dl (2r + 1), l = 0, 1, 2, 3 can be studied using similar arguments. Consider b0 (2r) and note that Z 1 Z 2 (v − 1)(v − 2)dv = 5/6 and (v − 1)(v − 2)dv = −1/6. (4.8) 0

1

On the other hand, the function gr (v) := (2r+v)µ−1 is a positive increasing function on [0, 2]. Then, by the mean value theorem for integrals, Z 2 (v − 1)(v − 2)(2r + v)µ−1 dv 0

Z

1

Z

2

(v − 1)(v − 2)dv + gr (v2 )

= gr (v1 ) 0

(v − 1)(v − 2)dv 1

= 5/6 gr (v1 ) − 1/6 gr (v2 ),

(4.9)

with v1 ∈ (0, 1) and v2 ∈ (1, 2). We may then write 2(2r)µ b0 (2r)

=

5/6(2r + v1 )µ−1 − 1/6 (2r + v2 )µ−1

> 5/6(2r)µ−1 − 1/6 (2r + 2)µ−1 =

5/6 2µ−1 rµ−1 − 1/6 2µ−1 (r + 1)µ−1

=

2µ−1 [5/6 rµ−1 − 1/6 (r + 1)µ−1 ]

=

2µ−1 [5/6 rµ−1 − 1/6 rµ−1 (1 + 1/r)µ−1 ]

> 2µ−1 rµ−1 [5/6 − (1/6) 2]

(4.10)

and the above quantity is positive for all r ≥ 1. If r = 0, we easily obtain that b0 (0) = (−µ + 2)/2, which is zero if µ = 2 and positive if 1 < µ < 2. Next, b1 (2r) is easily seen to be positive for all values of µ. We finally consider b2 (2r). Let gr (v) := (2r + v)µ−1 . Using the mean value theorem for integrals and

230

T. DIOGO, N.B. FRANCO AND P. LIMA

the fact that Z

1

Z v(v − 1)dv = −1/6

2

v(v − 1)dv = 5/3,

and

(4.11)

1

0

we obtain (2r)µ b2 (2r) = −1/6gr (v1 ) + 5/3gr (v2 ), with v1 ∈ (0, 1) and v2 ∈ (1, 2). As gr (v1 ) < gr (v2 ), then it follows that b2 (2r) > 0. Assuming consistency, it follows from Theorem 3.7 that Simpson’s method is convergent if 1 < µ ≤ 2. Convergence for the values µ > 2 will be established with the help of Theorem 3.6 and this will require the following result. Theorem 4.2. Let µ > 1. Then the weights ωij of algorithm (4.4)-(4.5) satisfy  λ2   , i = 2r,  i  µ X (a) (4.12) |ωij | ≤   λ3 j=0   , i = 2r + 1, r ≥ 1 µ µ (b) 0 < ωii ≤ 2 , i = 2, 3, . . . (4.13) µ + 3µ + 2 Above λ2 = 1.25 and λ3 ' 1.63113 are the Lebesgue constants associated with the Lagrange polynomials of second and third degrees, respectively (see e.g. [21]). Proof. (a) We examine the two cases i = 2r and i = 2r + 1, separately. From (4.4), we see that 2r 2 r−1 X X 1 X ωij = bk (2l). (4.14) (2r)µ j=0 k=0

l=0

Moreover 2r X

|ωij |

≤

j=0

r−1 Z 2 X 1 X 2 µ−1 (2l + v) |ρk (v)|dv (2r)µ 0 l=0

≤

≤

1 (2r)µ 1 (2r)µ

l=0

2

l=0

λ2 := max

v∈[0,2] 2r X

(2l + v)µ−1 λ2 dv

0

r−1 X λ2

where

It follows that

k=0

r−1 Z X

µ

2 X

[(2l + 2)µ − (2l)µ ]

|ρk (v)| = 1.25

k=0

|ωij | ≤

j=0

since

r−1 X

(4.15)

1.25 , µ

(4.16)

[(2l + 2)µ − (2l)µ ] = (2r)µ .

(4.17)

l=0

On the other hand, when i = 2r + 1, we have from (4.5) 2r+1 X j=0

1 ωij = (2r + 1)µ

r−2 X 2 X l=0 k=0

bk (2l) +

3 X k=0

! dk (2r + 1) .

(4.18)

HIGH ORDER PRODUCT INTEGRATION METHODS

231

Using (4.7), we have that 2r+1 X

|ωij |

≤

1 (2r + 1)µ

≤

1 (2r + 1)µ

j=0

! 3 Z X 3 (2r − 2)µ (2(r − 1) + v)µ−1 ρ˜k (v)dv λ2 + µ k=0 0 ! Z 3 3 X (2r − 2)µ λ2 + (2(r − 1) + v)µ−1 |˜ ρk (v)|dv µ 0 k=0

(2r − 2)µ λ2 λ3 ((2r + 1)µ − (2r − 2)µ ) ≤ + (2r + 1)µ µ µ(2r + 1)µ (2r − 2)µ (λ2 − λ3 ) λ3 = + (2r + 1)µ µ µ λ3 ≤ , µ where we have used 3 X λ3 := max |˜ ρk (v)| ' 1.63113. v∈[0,3]

(4.19)

k=0

(b) In the cases i = 2r and i = 2r + 1, r = 1, 2, . . ., ωii is given by (3.25) and (3.27), respectively, after setting n = 2 and ν = 1. From Theorem 4.1 it follows that the ωii are positive. If i is even, then Z 2 1 ωii = (i − 2 + v)µ−1 (v 2 − v) dv 2 iµ 0 = [2µ2 + 6µ + 4 + 2i2 − 6i − 3µi − (1 − 2/i)µ (4 − 2µ) − (1 − 2/i)µ (2i2 − 6i + µi)]/(2µ(µ + 1)(µ + 2)).

(4.20)

Considering the Taylor expansion for (1 − 2/i)µ , valid for i > 2, it can be seen that ωii will contain only negative powers of i. This implies that ωii is a decreasing sequence with respect to i and we easily obtain that µ , i = 2r, r ≥ 1 ωii ≤ ω22 = 2 µ + 3µ + 2 The case when i is odd can be treated in a similar way and we still have that ωii is decreasing with respect to i. Moreover, ω22 − ω33 = (5/2µ − 3/2)/((µ + 1)(µ + 2)(µ + 3)) > 0 since µ > 1. This completes the proof. We are now in a position to apply Theorem 3.6 in the case µ > 2. Indeed, using (4.12) and (4.13) it follows that i−1 X |ωij | 1.63113 µ2 + 3µ + 2 ≤ , 1 − ωii µ µ2 + 2µ + 2 j=0

i = 2, 3, . . . .

(4.21)

That is, condition (3.21) is satisfied with β = 1.63113 (µ2 +3µ+2)/(µ3 +2µ2 +2µ) < 1, since µ > 2. This completes the proof of convergence (of order p) of the product Simpson’s method for all values of µ > 1, assuming that it is consistent of order p and that the starting values are convergent of the same order. Remark 4.3. By similar arguments to the ones employed in the proof of Theorem 4.1, it can be shown that the weights of the discretisation method (2.13)-(2.14) in the cases n = 3, n = 4 are non-negative for the values 1 < µ ≤ 2. Therefore convergence follows from Theorem 3.7.

232

T. DIOGO, N.B. FRANCO AND P. LIMA

5. Numerical Results. In order to illustrate the theoretical results of the previous Sections we have considered the numerical solution of (1.15), with forcing function g(t) = tα (µ + α − 1)/(µ + α) and exact solution y(t) = tα , t ∈ [0, 2]. The following choices of α and µ have been considered. Example 5.1: µ = 1.5 and α = 4.5; Example 5.2: µ = 1.5 and α = 6.5; Example 5.3: µ = 1.3 and α = 3.0; Example 5.4: µ = 5.0 and α = −0.2. In Tables 5.1–5.6 the errors |ei | = |y(ti ) − yi | produced with several numerical methods of the type (2.13)-(2.14) are displayed for the cases n = 1, 2, 3, 4. Table 5.1 shows the results obtained with the product trapezoidal method applied to example 5.1. Here Theorem 3.2 can be used with m = n + 1 = 2 giving consistency of order two. Then Theorem 3.7 gives second order convergence and this is confirmed by the numerical results. Table 5.2 contains the errors obtained with the product Simpson’s method for example 5.1. In this case we have y (3) (0) = 0 and Theorem 3.4 can be applied with n = 2 to give consistency of order four. The results indicate that the convergence order is four and this is in agreement with the theoretical prediction. Table 5.1: The trapezoidal method example 5.1: µ = 1.5, α = 4.5

ti h = 0.05 h = 0.025 h = 0.0125 h = 0.00625

0.5 1.9D − 4 4.8D − 5 1.2D − 5 3.0D − 6

1.0 1.1D − 3 2.7D − 4 6.8D − 5 1.7D − 5

2.0 rate 6.2D − 3 1.5D − 3 2.0 3.9D − 4 2.0 9.7D − 5 2.0

Table 5.2: The Simpson’s method example 5.1: µ = 1.5, α = 4.5

ti 0.5 h = 0.05 3.1D − 6 h = 0.025 2.0D − 7 h = 0.0125 1.3D − 8 h = 0.00625 8.2D − 10

1.0 4.6D − 6 2.9D − 7 1.8D − 8 1.2D − 9

2.0 rate 6.6D − 6 4.2D − 7 4.0 2.6D − 8 4.0 1.6D − 9 4.0

Table 5.3: The three-eights method example 5.1 : µ = 1.5, α = 4.5

ti 0.5 h = 0.05 1.1D − 6 h = 0.025 1.0D − 7 h = 0.0125 9.1D − 9 h = 0.00625 6.5D − 10

1.0 2.3D − 6 2.1D − 7 1.5D − 8 1.1D − 9

2.0 rate 4.7D − 6 3.3D − 7 3.8 2.3D − 8 3.9 1.5D − 9 3.9

To obtain the values in Table 5.3, we implemented a method based on the repeated use of the product three-eights rule as the main rule (three eight’s method) which was applied to example 5.1. Here Theorem 3.2 can be applied with m = n + 1 = 4 and the results show the expected fourth order convergence (cf. Remark 4.3). A further method corresponding to n = 4 was also implemented and applied to example 5.2. Here y (5) (t) = constant×t1.5 , so that y (5) (0) = 0 and Theorem 3.4 can

HIGH ORDER PRODUCT INTEGRATION METHODS

233

be applied. This, together with Remark 4.3, gives convergence of order six which is confirmed by the numerical results in Table 5.4. Table 5.4: A method based on a five-point rule rule example 5.2 : µ = 1.5, α = 6.5 ti 0.5 1.0 2.0 rate h = 0.05 3.8D − 8 1.54D − 7 2.4D − 7 h = 0.025 1.7D − 9 2.6D − 9 4.0D − 9 5.9 h = 0.0125 2.9D − 11 4.5D − 11 6.7D − 11 5.9 h = 0.00625 4.9D − 13 7.4D − 13 1.1D − 13 6.0 Table 5.5: The Simpson’s method example 5.3: µ = 1.3, α = 3.0

ti h = 0.05 h = 0.025 h = 0.0125 h = 0.00625

0.5 3.8D − 6 4.1D − 7 4.3D − 8 4.5D − 9

1.0 3.3D − 6 3.5D − 7 3.6D − 8 3.7D − 9

2.0 rate 2.8D − 6 2.9D − 7 3.26 3.0D − 8 3.30 3.0D − 9 3.30

Table 5.6: The Simpson’s method example 5.4: µ = 5.0, α = −0.2, β = 3.6

ti h = 0.05 h = 0.025 h = 0.0125 h = 0.00625

0.5 4.1D − 6 3.5D − 7 2.8D − 8 2.2D − 9

1.0 3.6D − 6 3.0D − 7 2.4D − 8 1.8D − 9

2.0 rate 3.1D − 6 2.5D − 7 3.6 1.9D − 8 3.7 1.5D − 9 3.7

We have also applied Simpson’s method to example 5.3 for which y (3) (t) ≡ constant 6= 0. From (3.14) we can only conclude that the method is consistent of order three and we would also expect to get convergence of order three. The numerical results of Table 5.5 seem to indicate that the convergence order might be slightly higher, maybe 3.3. We note that this is just α + µ − 1. Further numerical results for other values of µ seem to support the conjecture that the order should be given by min(α + µ − 1, n + 2). In Table 5.6 the results obtained for example 5.4 by Simpson’s method are displayed. In order to deal with the unbounded solution y(t) = t−0.2 we considered the transformed equation (3.19) with β satisfying the conditions µ − β > 1 and tβ y(t) ∈ C 3 [0, T ]. As µ = 5 we chose β = 3.6 so that the solution of the new equation (3.19) was tβ y(t) = t3.4 . Theorem 3.2 can be applied to equation (3.19) giving third order of consistency and this implies the same order of convergence. Again the order seems to be α + µ − 1. In general, we conjecture that order q of a method is given by: q = min(α + µ − 1, n + 1) if n is odd; q = min(α + µ − 1, n + 2) if n is even. 6. Concluding remarks. A class of product integration methods based on n + 1point interpolatory rules has been introduced for the solution of (1.15)-(1.16) with µ > 1. A sufficient condition for convergence was derived which required the weights to satisfy condition (3.21) (cf. Theorem 3.6). In particular, this is satisfied with β = 1/µ if the weights are non-negative (cf. Theorem 3.7). This is true for the trapezoidal method (n=1), for all values of µ > 1. However, for a discretisation

234

T. DIOGO, N.B. FRANCO AND P. LIMA

method of the type (2.13)-(2.14) with n > 1, the sign of the weights will depend on µ. In the cases n = 2, n = 3, n = 4 it can be shown that the weights remain non-negative if 1 < µ ≤ 2 and convergence follows from Theorem 3.4. However, if µ > 2 the weights will vary in sign and so Theorem 3.7 is not applicable. In this case the numerical results indicate that the following bound is valid i−1 X j=0

|ωij | ≤ 1/µ, 1 − |ωii |

i = n, n + 1, . . . .

In order to show numerical evidence of (6.1), the quantity   i−1 X  |ωij | SM AX := max , i = n, n + 1, . . . , N   1 − |ωii | j=0

(6.1)

(6.2)

has been computed for several values of N and µ. The results displayed on Tables 6.1 and 6.2 clearly support the conjecture that Theorem 3.6 should be applicable with β = 1/µ. In the particular case of Simpson’s method (n = 2), with µ > 2, we were able to obtain (4.21)), which is less sharp than (6.1), and this permitted proving convergence for the values of µ > 2. The above conjecture will be dealt with in future work. Finally, we shall try to extend the results so far obtained to more general equations than (1.16), namely with kernels of the type K(t, s)p(t, s). Table 6.1: SM AX for Simpson’s method ν = 1/µ N ν = 0.666667 ν = 0.25 ν = 0.153845 40 0.638654 0.243702 0.146756 80 0.652719 0.246862 0.150308 160 0.659708 0.248434 0.152080 320 0.663191 0.249218 0.152964 640 0.666493 0.249609 0.153405 Table 6.2: SM AX for the three-eights method ν = 1/µ N ν = 0.666667 ν = 0.25 ν = 0.153845 40 0.664053 0.244131 0.147251 80 0.665290 0.246942 0.150436 160 0.666017 0.248534 0.152198 320 0.666323 0.249229 0.152979 640 0.666504 0.249635 0.153435

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Received June 2003; revised January 2004. E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected]