The generalized Euler-Maclaurin formula for the numerical solution of Abel-type integral equations.

Transcription

1 The generalized Euler-Maclaurin formula for the numerical solution of Abel-type integral equations. Johannes Tausch Abstract An extension of the Euler-Maclaurin formula to singular integrals was introduced by Navot []. In this paper this result is applied to derive a quadrature rule for integral equations of the Abel type. We present a stability and convergence analysis and numerical results that are in good agreement with the theory. The method is particularly useful when combined with fast methods for evaluating integral operators. Introduction The generalized Abel equation is a weakly singular Volterra integral equation of the first kind which usually appears in the form t (t τ α k(t, τg(τ dτ = f(t. ( Here t T, < α < and the kernel k(, is smooth and satisfies k(t, t =. The following solvability result was given by Atkinson []. Theorem. If f(t = t α β f(t with f C [, T ] and β < then ( has a unique solution g which is of the form g(t = t β g(t and g C [, T ]. Numerical methods for ( are usually based on collocation or product integration methods. This subject has been studied extensively and is reviewed in the monographs by Linz [7] and Brunner []. A more recent approach to discretizing ( is the use of the convolution quadrature rules []. The aforementioned discretization methods lead to a recurrence formula for the numerical approximations g n of g(t n which are of the form h α n w n,j k(t n, t j g j = f(t n, n =,,,... ( where h is the stepsize, t n = nh and w n,j are quadrature weights.

2 The present paper examines another discretization method of ( which is based upon the generalization of the Euler-Maclaurin expansion to integrals with algebraic singularities. The formula was first discovered by Navot [] and further generalized in a series of papers, see e.g., [6,, 5]. We will use this result to obtain a discretization of ( in the form n (t n t j α k(t n, t j g j + h α n h j= j=[n p] + r (p n j k(t n, t j g j = f(t n, (3 n =,,,.... The first sum, or history part, is the usual composite trapezoidal rule. The second sum contains only a very small number of terms and can be considered as a correction to account for the singularity of the integrand at t = τ. The parameter p controls the order of the method; we will show that for p = the order is O(h α and the scheme is stable for α (,. When p = and p = the order is O(h 3 α and O(h 4 α, respectively, but the interval of α s for which the method is stable is reduced. For larger values of p the important case α = is no longer stable, and therefore the resulting schemes are probably only of limited interest for solving first-kind equations. A complication we encounter is that the remainder in the generalized Euler- Maclaurin expansion depends in a subtle way on the interval length. To ensure that we obtain uniform bounds in t we must initially restrict the discussion to the case that the solution has sufficiently many vanishing derivatives at t =. To handle the general case, including singularities, we will discuss a correction of the right hand side to maintain the convergence rates. This procedure is similar to Lubich s approach to overcome a similar issue for convolutional quadrature schemes []. We note however, that other techniques have been discussed to handle singularities, see, e.g. [5]. The difference between the two recurrence formulas is that ( contains quadrature weights that depend on n and j, whereas the history part in (3 does not. This simplifies matters significantly if fast summation methods are used to compute the history part in the recurrence formula efficiently. Fast methods are based on separating the t n and t j variables. In (3 this can accomplished fairly easily, for instance, by Taylor expanding the kernel in (. On the other hand, recurrence ( involves in addition a separation of the n and j-index of the weights, which does not seem to be practical. It is not our goal to discuss fast methods here. Instead, we refer to the paper [4] where the p = method has been used as a time discretization for integral formulation of the heat equation. It should be noted that the idea to use the generalized Euler- Maclaurin expansion to obtain discretizations of singular Volterra integral equations is not new, although it has received only very limited attention in the past. It appears that the first paper to use this result for Volterra equations of the second kind is Tao and Yong [3]. Later, the equation of the first kind was treated by transformation to an equation of the second kind, see [8] and [9].

3 The Generalized Euler-Maclaurin Expansion The result by Navot is a generalization of the Euler-Maclaurin formula for integrals with an algebraic singularity at one endpoint of the interval of integration []. Later, the formula was generalized and its derivation was simplified in various ways. Probably the most efficient tool is the Mellin transform, which first used by Verlinden and Haegemans [5]. The methodology in their paper applies for integrals over a semiinfinite interval, but can be easily extended to integrals over [, ] using neutralizer functions []. We will briefly recall this technique and then discuss arbitrary length intervals. Basic expansion for the semiinfinte interval. function ϕ is given by Φ(z = and the inverse transform is given by x z ϕ(x dx The Mellin transform of a ϕ(x = πi c+i c i x z Φ(z dz, where c R is chosen such that Φ is analytic for Re(z c. Suppose ϕ(x = x α ϕ(x where ϕ C [, decays superalgebraically, that is, lim x xk ϕ(x =, for every k. From the inverse transform it follows that h ϕ(jh = h πi c+i c i (jh z Φ(z dz = πi c+i c i ζ(zφ(zh z dz, (4 where h > is the stepsize, c > and ζ( is the Riemann zeta function. Integration by parts leads to Φ(z = ω s (z α where ω s ( is defined by ω s (z = x z α+s ϕ (s+ (x dx, s =,,... (5 ( s+ z... (z + s. (6 The integral in (5 is an analytic function for Re(z > (s + α. Furthermore, Res(ζ(z, z = =, so the integrand in (4 has residuals Res ( ζ(zφ(zh z, z = = Φ( = 3 ϕ(x dx

4 and Res ( ζ(zφ(zh z, z = α s = ϕ(s ( ζ(α sh s+ α s! for s =,,... Thus we may move the integration contour in (4 into the left plane and obtain q ϕ (s ( h ϕ(jh = ϕ(x dx + ζ(α sh s+ α + R (q (h (7 s! s= for some integer q. This is the generalized Euler-Maclaurin expansion. The remainder is given by R (q (h = πi c +i c i where α q < c < α q. The remainder R (q can be estimated by c +i ζ(zφ(zh z+ dz h c π c i ζ(zφ(zh z+ dz, (8 ζ(c + iy Φ(c + iy dy From (5 it follows that Φ(c + iy decays superalgebraically as y ±. On the other hand, it is known that for a fixed real part, the zeta function increases at most like a polynomial, see [5]. Therefore the integral on the right hand side converges and the remainder is O(h c. Expansion for the interval [, t]. We now derive a similar expansion for integrals of the type that appear in (. Of course, the form of the generalized Euler-Maclaurin expansion for a finite interval is well known. However, in Volterra integral equations, the length is time dependent and enters the constants in the estimates. We will discuss this dependence in the following. It turns out that we must assume that the solution is of the form g(τ = τ γ g(τ where g is smooth in [, T ] and γ is sufficiently large to ensure that the estimates are independent of time. Later, it will become clear that γ = p + α, where p is the order of the method, defined in (3. By Theorem. this is equivalent to limiting the discussion to right hand sides of the form f(t = t α+γ f(t. Section 5 will describe a modified method that maintains its convergence properties for general, even singular solutions. To simplify notations, we will write ϕ(t, τ = k(t, τg(τ and ϕ(t, τ = τ γ ϕ(t, τ where ϕ C [, T ] and γ > is a not yet specified parameter. Furthermore, we introduce a neutralizer function χ(x with the properties χ C (R, χ(x =, x 3, χ(x =, x 3 (9 4

5 and, in addition, An example of such a function is χ(x = { ( + tanh x 3 χ( x = χ(x. ( } 3 x, 3 < x < 3. Using these definitions the integrand in ( can be written as the sum of two functions, where (t τ α ϕ(t, τ = τ γ ϕ (t, τ + (t τ α ϕ (t, t τ ( x ϕ (t, x = χ (t x α ϕ(t, x, ( t x ϕ (t, x = χ (t x γ ϕ(t, t x t Furthermore, we set ϕ (t, x = x γ ϕ (t, x and ϕ (t, x = x α ϕ (t, x. All these functions can be trivially extended to the semiinfinite interval x. With these definitions it follows that and t (t τ α ϕ(t, τ dτ = ϕ (t, x dx + ϕ (t, x dx n (t n t j α ϕ(t n, t j = ϕ (t n, t j + ϕ (t n, t j. We can apply the generalized Euler-Maclaurin formula to both sums and combine the expansions. The definitions of the functions ϕ and ϕ imply that s ϕ x s (t, = s ϕ τ s (t, t( s, If γ > q the derivatives up to order q of ϕ (t, x vanish for x =. Thus the function will not contribute to the residuals and will only appear in the remainder. (t n t j α ϕ(t n, t j = n h + s= where the remainder is given by R (q E (t, h = πi tn (t n τ α ϕ(t n, τ dτ q ( s s ϕ s! τ s (t n, t n ζ(α sh s+ α + R (q E (t n, h( c +i c i ( ζ(z Φ (t, z + Φ (t, z h z+ dz ( and Φ (t, z and Φ (t, z are the Mellin transforms of ϕ (t, x and ϕ (t, x with respect to the x-variable. 5

6 Lemma. If g(t = t γ g(t with g C [, T ], then there is a smooth function I E (t, h such that the remainder is given by R (q E (t, h = tγ+c α h c I E (t, h, q + α < c < q + α. Proof. As in (5 we obtain from integration by parts Φ (t, z = ω s (z α ( s+ x z α+s ϕ (t, x dx. x Changing variables x x/t and sorting out the powers of t shows that where I (s (t, z = 3 Likewise, we find for Φ that where I (s (t, z = 3 Φ (t, z = t z α+γ ω s (z αi (s (t, z ( s+ x z α+s χ(x( x γ ϕ(t, t( x dx. x Φ (t, z = t z α+γ ω s (z + γi (s (t, z, ( s+ x z+γ+s χ(x( x α ϕ(t, tx dx. x By the properties of parameter dependent integrals the functions I (s (t, z and (t, z are smooth in t and in z. Furthermore, they are bounded in t [, T ] I (s and Im(z, when Re(z is fixed. Hence it follows from ( that R (q E (t, h = α h πi tγ+c c c +i ( ζ(z ω s (z + γi (s (t, z + ω s(z αi (s (t, z (th Im(z dz c i The zeta function increases like a polynomial on the integration contour, whereas ω s decreases like z s. Thus we can choose s large enough such that the integral converges absolutely with bounds independent of t and h. Then the integral is a smooth function of all parameters. Quadrature Rule. To derive a quadrature rule we retain the first p terms in expansion ( and replace derivatives by finite difference approximations. The weights r (p j are determined such that p s= ζ(α s ( s s! s ϕ τ s (t, th α+s = h α p j= ϕ(t, t hjr (p j + R (p D (t, h. 6

7 Here we set ϕ = if the second argument is negative. If γ > p + then this extension is p+-times continously differentiable. Thus we can apply the Taylor expansion to obtain a linear system for the r (p j p j s r (p j = ζ(α s, s =,..., p. (3 j= Note that the weights are independent of t. We list the weights for the important cases p =,, [ r ( r ( r ( r ( r ( r ( = ζ(α, ] = [ ] [ ζ(α ζ(α = 3 ], ζ(α ζ(α ζ(α In view of Lemma 3.5 below, it is necessary to express the remainder as the product of certain powers of h and t and a function that is sufficiently regular on the triangle T := {(t, h : h t T }. (4 Lemma. If g(t = t p+ α g(t with g C [, T ], the remainder R (p D (t, h is of the form R (p D (t, h = hp+ α t α I (p D (t, h (5 where I (p D C ( T. Proof. Using the integral form of the remainder in the Taylor expansion, we get R (p ( p p jh D (t, h = h α r (p j τ p ϕp+ (t, t h + τ dτ p! τ p+ j= Since ϕ(t, τ = τ p+ α ϕ(t, τ there is a smooth ˆϕ such that τ p+ ϕ(t, τ = τ+ α ˆϕ(t, τ, where τ + is the positive part of τ. Hence R (p ( p p jh D (t, h = h α r (p j (t jh + τ + α p! τ p ˆϕ(t, τ dτ j= Changing variables τ hτ leads to where R (p D (t, h = hp+ α α ( p t p! I j (t, h = j p j= r (p j I j (t, h ( ht α (j τ τ p ˆϕ(t, hτ dτ, which is a smooth and bounded function when t h

8 3 Stability Analysis We first discuss the stability of discretizations of the p = rule for the constant kernel k(t, τ =. In this case the discretization rule leads to the semicirculant matrix a ζ(αh α A = ζ(αh α a a a a a ( whose coefficients are {, n =, a n = n α ζ(α, n. (7 The characteristic function associated with a semicirculant matrix with entries a n is given by a(z = a n z n. n= The stability of recurrence (3 hinges on the properties of the inverse matrix which is another semicirculant matrix whose coefficients are the expansion coefficients of a (z a (z = A n z n. (8 n= Here and in the following we will use the convention that capitalized coefficients denote the expansion coefficients of the reciprocal function. A frequently used tool is the following result which goes back to Hardy [4]. Lemma 3. If c =, c n >, n >, n c n = and then C n, n > c n+ c n c n c n, n > (9 and C n =. ( From the behavior of the zeta function in the interval (,, shown in Figure, it follows that the sequence a n in (7 is indeed positive. Furthermore, (9 is fulfilled when ζ(α α which implies that α must be in the interval (ᾱ, with ᾱ = In this case the inverse is a bounded operator and we have n= A n <, n >, and A =, when α (ᾱ,. ( Unfortunately, there is no information for α (, ᾱ. Furthermore, in the subsequent analysis it is necessary to have asymptotic estimates of the A n s. More information on the A n s for all α can be obtained from Eggermont s analysis of the product integration method for the Abel equation. We summarize the development in the appendix of the paper [3]. 8

9 Lemma 3. Let c n be a sequence with c =, and set { g n = c, n =, (c n c n, n >. If the g n satisfy g n >, ( g n g n < g n+ g n <, n > (3 and, in addition, if there is s R and β [, such that then y c n sy β, y, (4 n= C n sin ( βπ sπ. (5 nβ+ Our main stability result depends on Lemma 3.. Note, however, no information on the sign of the C n s is given, thus the conclusions from Lemma 3. will be still important later on. Theorem 3.3 The coefficients A n in (8 satisfy A n π sin(( απ( αζ(α n α, and A n =. (6 Proof. We first establish that the coefficients a n of (7 satisfy the conditions of Lemma 3.. For this sequence, the g n s are given by { + ζ(α g n =, n =, ζ(α ((n α n α (7, n, The positiveness of the g n s follows easily from the fact that ζ(α increases from to in the interval (,, see Figure. To verify the second hypothesis of (3, consider the case n = first, which is equivalent to n= α ζ(α < α 3 α. (8 α This must be verified by computing function values. The validity of this inequality is illustrated in Figure. When n > the validity of (3 for the g n s defined in (7 follows from elementary calculus. From (7 it also follows that y a n ζ(α( α y α n= hence the exponent in (4 is β = α which implies the asymptotic estimate for the A n s. 9

10 Since n A n converges, it follows that a (z is continuous on the closed unit disk, and by the Abel limit theorem it follows that n= A n = lim x a (x = Figure : Graph of the function ζ( (left and the ratios in (3 when n = (right, bottom and n = 3 (right, top. The following two lemmas will be necessary for the stability analysis. Lemma 3.4 Let Φ C [, T ], t j = hj, then Proof. Write A n j a j Φ(t j = δ n, Φ(t n + O(h (9 j= A n j a j Φ(t j = j= j= n A n j a j Φ(t n A n j a j (Φ(t n Φ(t j The first term is the convolution product of {a j } j and its inverse and accounts for the first term in the right hand side of (9. The first term in the second sum is O(h. To estimate the remainder of the second term, note that it follows j=

11 from the generalized Euler-Maclaurin formula that n h (t n t j +α t α j ( tn = O (t n τ +α τ α dτ = O (t n. Hence, using Φ(t n Φ(t j C(t n t j and Theorem 3.3 n n A n j (Φ(t n Φ(t j a j Ch (t n t j +α t α j = O(h. This proves the assertion. Lemma 3.5 Let Φ C ( T, then for any λ > A n j t λ j Φ(t j, h = O ( t λ +α n h α (3 Proof. For simplicity of notations, we omit the second argument from Φ. Setting Ψ(t = t λ Φ(t we have t λ n Φ(t n t λ j Φ(t j tn = Ψ(t n Ψ(t j = Ψ (τ dτ (t n t j max Ψ (τ. t j τ t n It follows from the chain rule that if j n, Because of j A j =, t λ n Φ(t n t λ j Φ(t j { C(tn t j t λ n, λ, C(t n t j t λ j, λ <. t j A n j t λ j Φ(t j = = n A n j t λ n Φ(t ( n A n j t λ n Φ(t n t λ j Φ(t j j=n n A j t λ n Φ(t ( n A n j t λ n Φ(t n t λ j Φ(t j.(3 For λ we estimate using Theorem 3.3 A n j t λ j Φ(t j C t λ n j=n j +α + C t λ n C ( t λ n n +α + t λ n hn α Ctn λ +α h α (n j +α n h

12 For the case < λ < note that n h (t n t j +α t λ j Hence, beginning with (3 A n j t λ j Φ(t j C t λ n ( tn = O (t n τ +α τ λ dτ j=n+ Ct λ +α n h α. j +α + C h α Thus the assertion has been shown in both cases. n = O ( tn λ +α, (t n t j +α t λ Higher Order Methods. In the case p > and a constant kernel k(t, τ = discretization (3 leads to the system ˆr (p j = Ag + ˆR (p g = h +α ζ(α f where A is defined in (6 and ˆR (p is a semicirculant matrix with coefficients r(p ζ(α, j =, To determine stability we factor r(p j ζ(α, j p,, j > p. A + ˆR (p = A(I + A ˆR(p j (3 and investigate whether the second factor has a continuous inverse. To that end, set S (p = A ˆR(p and find the values of α and p for which the coefficients of S (p satisfy S (p = s (p <. (33 n= If (33 is satisfied then I +A ˆR(p has a continuous inverse in l. The following lemma gives a necessary condition for stability. Lemma 3.6 For every p there is an α p (, such that condition (33 is satisfied for α (α p,. Proof. If p = then S ( = and there is nothing to show, and hence we may set α =. If p > and α > ᾱ we have from ( that A =. Hence n

13 S (p = A ˆR(p ˆR (p. Furthermore, it follows from (3 and (3 that ˆr (p = p m (p s ( s ζ(α s (34 ζ(α s= where m (p s are the columns of [M (p ], and M (p s,j = js. Since the zeta function is only singular when α =, it follows that for any p fixed r (p as α. Hence there is an α p < for which S (p <. Using the technique of the proof we compute values of α and α that guarantee stability. When p = we have [ ˆr ( ˆr ( When α > ᾱ we can conclude that ] = ζ(α [ ζ(α ζ(α S ( A ˆR ( ζ(α = 4 =: s (α (35 ζ(α As shown in Figure, s (α is always less than unity, but (35 is valid only if α (ᾱ,, hence we have α = ᾱ = ] Figure : Graph of the functions s (bottom and s (top. When p = we have ˆr ( ˆr ( = 3 ˆr ( ζ(α [ ζ(α ζ(α Figure shows the graph of s (α := ζ(α ˆr(. From the graph it is evident that α (α, with α = is necessary for stability. ] 3

14 4 Discretization Error The quadrature error consists of two parts, the error introduced by truncating the Euler-Maclaurin expansion, and the error introduced by replacing the derivatives in the leading terms of the expansion by finite differences. Setting q = p + in (, the m-th time step the quadrature error is d m := ( p+ p+ ϕ (p +! τ p+ (t m, t m ζ(α p h p+ α + R (p+ E (t m, h + R (p D (t m, h. (36 Subtracting equations ( and (3 shows that the discretization errors e j = g(t j g j and the quadrature error d m are related by p (t m t j α k(t m, t j e j + h α m h j= Equation (37 is equivalent to m a m j k(t m, t j e j + j= m j=m p j= r (p j k(t m, t m j e m j = d m. (37 ˆr (p m j k(t m, t j e j = hα ζ(α d m. (38 where the coefficients ˆr (p j are defined in (3. To estimate the error we convolve both sides of the equation with the sequence {A n } n. For the first term in the left hand side, we calculate ϕ ( n = = = m= A n m j= m=j j= m= m j= a m j k(t m, t j e j A n m a m j k(t m, t j e j n j A n j m a m k(t m+j, t j e j Lemma 3.4 applied to the inner sum with Φ(τ = k(τ + t j, t j shows that there are coefficients κ ( n,j = O( such that ϕ ( n = e n + h j= κ ( n,j e j (39 Proceeding in a similar manner shows that the second term satisfies ϕ ( n = n p j= m= A n j mˆr (p m k(t m+j, t j e j. 4

15 In the above sum one has k(t m+j, t j = k(t j, t j + O(h = + O(h. Thus there are coefficients κ ( n,j = O( such that n ϕ ( n = s (p n j e j + h κ ( n,j e j, (4 j= where s (p j are the coefficients of S (p = A ˆR(p. In matrix notation, (39 and (4 can be combined as ( I + S (p e + hke = h α A d (4 j= where K is a lower triangular matrix with entries K n,j = κ ( n,j + κ( n,j which are bounded independently of h. We continue with an estimate for the right hand side. Lemma 4. If γ = + p α the coefficients of ẽ = h α A d are bounded by ẽ n = O(h p+ α Proof. The coefficients of ẽ = h α A d are given by ẽ n = h α n A n j d j In view of (36 we have to analyze the effect of the convolution on three parts, all of which are in a form that fit the hypotheses of Lemma 3.5. For the first part note that the p+-st derivative of ϕ is of the form t α I(t, where I( is smooth, hence h α n A n j tj α I(t j h p+ α = O ( h p+ α t = O ( h p+ α The estimate of the second part follows from Lemma.. In that lemma, we may set, for instance, c = p, then h α n ( A n j R (p+ E (t j, h = O h c t γ+c n = = O ( h p+ t γ p n = O ( h p+ α t γ p +α n ( α h = O ( h p+ α t n n α. Hence this part is O ( h p+ α multiplied by a decaying term. For the third part of (36 use Lemma. h α n A n j R (p D (t j, h = h p+ A n j tj α I (p D (t j, h = O(h p+ α. 5

16 Adding the three parts together completes the proof. If α and p are such that I + S (p is bounded independently of n, then (4 is equivalent to e + h (I + S (p ( Ke = I + S (p ẽ. (4 Here, the matrix has entries bounded independently of h and the right hand side has the same asymptotic estimate as ẽ. Our final result follows directly from the discrete Gronwall lemma. Theorem 4. If α p < α < and γ = p + α then the discretization error satisfies the asymptotic estimate e n = O(h p+ α. 5 The general case In Section we have introduced a quadrature rule which has the desired order only if the integrand is O(τ p+ α as τ. We will write this rule in the form tn (t n τ α ϕ(t n, τ dτ = ϕ(t n, t j w n j + O(h p+ α where the weights are w j = h α r (p, j =, j α + r (p p j, j p, j α, j > p. We will show how to modify this rule if ϕ is replaced by τ β ϕ(t, τ, where β < and ϕ is a smooth function in both variables, that has no conditions on the behavior near τ =. The modification depends on the Taylor expansion of ϕ at τ = q s ϕ ϕ q (t, τ = s! τ s (t, τ s (43 s= which will be subtracted from ϕ to obtain a function of order τ q+ β. The quadrature rule is applied to the difference, the rest is integrated analytically, as shown in the following calculation tn = = (t n τ α τ β ϕ(t n, τ dτ tn (t n τ α τ β( ϕ(t n, τ ϕ q (t n, τ dτ + ( t β j ϕ(t n, t j ϕ q (t n, t j w n j + t β j ϕ(t n, t j w n j + c n. tn tn (t n τ α τ β ϕ q (t n, τ dτ (t n τ α τ β ϕ q (t n, τ dτ 6

17 This is the original quadrature rule with a correction term. Since only polynomials appear in this term, the integral can be evaluated in closed form tn c n = (t n τ α τ β ϕ p (t n, τ dτ t β j ϕ p (t n, t j w n j Here, E α,β s = = q s ϕ s! τ s (t n, q s= s= s ϕ s! τ s (t n, tn Eα,β s denotes the Euler integral E α,β s = (t n τ α τ s β dτ t +s α β n ( x α x s β dx = t s β j w n j t s β j. Γ( αγ( + s β. Γ( + s α β w n j We now give a recipe to solve the Abel integral equation with the general hand side as it appears in Theorem... Compute the coefficients g i, i =,..., q in the expansion of the solution g(τ = g + g τ + g τ These follow directly from the expansion coefficients of the kernel and the right hand side. Set g q (t = (g + g τ g q τ q and ĝ q (t = g(t τ β g q (t.. Subtract τ β g q (t in ( and solve the equivalent Abel equation t (t τ α k(t, τĝ(τ dτ = f(t t (t τ α k(t, ττ β g q (τ dτ. with ĝ(τ = O(τ +q β as the new unknown. To apply rule (3 q must satisfy + q β p + α, hence the order of the Taylor expansion in (43 is the smallest integer q with q p + + β α. 3. In every time step, compute the correction factors c n and evaluate the integral on the right hand side with the corrected quadrature rule. The additional error introduced in this procedure comes from replacing the integral on the right hand side with the corrected quadrature rule. The analysis of this error parallels the discussion of Sections and 4 and is therefore omitted. 6 Numerical Example Our test problem is the integral equation t (t τ α exp(t τg(τ dτ = exp(tt α β 5 7 k= E α,β k t k

18 with known solution g(t = exp(t 5 k= tk β. We solve this equation in the interval [, ] with stepsize h = /N and β =.5. α =.5 α =.5 α =.85 N error order error order error order Table : Maximum errors and orders of convergence when p =. α =.5 α =.5 α =.85 N error order error order error order e e e e e e e Table : Maximum errors and orders of convergence when p =. Tables -3 display the convergence of the error for different values of α and p, all of which are in good agreement with the theoretical analysis. In particular, the method is unstable for small values of α when p, but otherwise the orderh p+ α convergence is clearly visible. When p = and α =.5 the method still converges at the expected rate even though this parameter combination is not in the range with guaranteed stability that is implied by Figure. In Figure 3 we further illustrate the stability of α =.5 for different p by displaying the ( coefficients of the semicirculant matrices A + ˆR. (p It appears that after some transient effects the entries have the same asymptotic behavior when p =, p =, and p =. For larger values of p the coefficients blow up and the method is unstable. Figure 4 shows the errors for α =.5 and p = as a function of time. Even though the analytical solution is singular at t =, the convergence happens at the expected rate and the error is small near the singularity. To put the results presented into the right perspective, it should be noted that it is possible to derive stable rules of even higher order based on convolutional quadrature. In fact, Lubich presents results for a forth-order scheme []. However, the point here was to introduce quadrature rules of the simpler form 8

19 Figure 3: Magnitude of the coefficients of p = and p =. ( A + ˆR (p for α =.5 and p =, Figure 4: Errors for α =.5, β =.5, p =, N =,,..., 64 as a function of t. 9

20 α =.5 α =.5 α =.85 N error order error order error order e e e e e e e e e e e e Table 3: Maximum errors and orders of convergence when p =. (3 which is advantageous, for instance, for the fast evaluation of time-dependent boundary integral operators. References [] K.E. Atkinson. An existence theorem for Abel integral equations. SIAM J. Math. Anal., 5:79 736, 974. [] H. Brunner. Collocation Methods for Volterra Integral and Related Functional Differential Equations. Cambridge, 4. [3] P.P.B. Eggermont. A new analysis of the trapezoidal - discretization method for the numerical solution of Abel-type integral equations. J. Integral Equations, 3:37 33, 98. [4] G.H. Hardy. Divergent series. Oxford, Clarendon Press, 949. [5] M. Huang and Y. Xu. Superconvergence of the iterated hybrid collocation method for weakly singular Volterra integral equations. J. Integral Equations Appl., 8(:83 6, 6. [6] J.N.Lyness and B.W.Ninham. Numerical quadrature and asymptotic expansions. Math. Comp., (98:6 78, April 967. [7] P. Linz. Analytical and Numerical Methods for Volterra Equations. Studies in Applied Mathematics. SIAM, 985. [8] Y. Liu and L. Tao. High-accuracy combination algorthim and a posteriori error estimation for soling the first kind Abel integral equations. Appl. Math. Comput., 78:44 45, 6. [9] Y. Liu and L. Tao. Mechanical quadrature rules and extrapolation for solving first kind Abel integral equations. J. Comput. Appl. Math., :3 33, 7. [] C. Lubich. Fractional linear multistep methods for Abel-Volterra integral equations of the first kind. IMA J. Numer. Anal., 7:97 6, 987.

21 [] G. Monegato and J.N. Lyness. The Euler-Maclaurin expansion and finitepart integrals. Numer. Math., 8(:73 9, 998. [] I. Navot. An extension of the Euler-Maclaurin summation formula to functions with a branch singularity. J. Math. and Phys., 4:7 76, 96. [3] L. Tao and H. Yong. A generalization of discrete Gronwall inequality and its application to weakly singular Volterra integral equation of the second kind. J. Math. Anal. Appl., 8:56 6, 3. [4] J. Tausch. A fast method for solving the heat equation by layer potentials. J. Comput. Phys., 4: , 7. [5] P. Verlinden and A. Haegemans. An error expansion for cubature with an integrand with homogeneous boundary singularities. Numer. Math., 65:383 46, 993.