COMPARATIVE STUDY OF NUMERICAL METHODS FOR A CLASS OF INTEGRAL EQUATIONS WITH WEAKLY SINGULAR KERNEL Teresa Diogo, Pedro Lima
Abstract
where µ > 0 and g is a given function. The above equation can arise in some heat conduction problems with mixed-type boundary conditions [10] and has been studied analytically by several authors. If µ > 1 then (1.1) has a unique solution in C m [0, T ] if g belongs to C m [0, T ]. For the numerical solution of (1.1), certain classes of product integration methods were studied in [5], [6] and [10]. More recently, our investigation has been focused on the case when 0 < µ ≤ 1. Although in this case (1.1) has a family of solutions in C[0, T ], Euler’s method was shown to converge to a certain particular solution [11]. In order to improve the accuracy of the Euler approximations, extrapolation procedures were employed and the results compared with the ones obtained by the trapezium method. The case when g is not sufficiently smooth is dealt with by using an appropriate regularization technique. We also investigate the use of graded meshes to recover the optimal convergence rates.
In this work we consider a certain class of Volterra integral equations possessing an infinite set of solutions. We investigate the application of Euler’s method together with the use of extrapolation procedures. Several examples are considered illustrating our results as well as the use of graded meshes and the application of higher order methods. keywords Volterra integral equations; weakly singular kernel; Euler’s method
1
Introduction
This work is concerned with the following Volterra integral equation Lµ u(t) = g(t), Lµ u(t) := u(t) −
t ∈ [0, T ], Z
t 0
sµ−1 u(s)ds, tµ
(1.1) (1.2)
2
0
Departamento de Matem´ atica, Instituto Superior T´ ecnico, Av. Rovisco Pais, 1049-001 Lisbon, Portugal Email:
[email protected],
[email protected]
Existence and uniqueness
Lemma 2.1. [5] If µ > 1 and the function g in (1.1) belongs to C m [0, T ] then (1.1) has a unique solution u ∈ C m [0, T ].
Work supported by Funda¸ca ˜o para a Ciˆ encia e Tecnologia, Centro de Matem´ atica Aplicada/ I.S.T. (Lisbon, Portugal)
1
A further result was proved in [8] which includes the case when 0 < µ ≤ 1.
Setting t = tk in (1.1) gives
Lemma 2.2. (a) If 0 < µ ≤ 1 and g ∈ C 1 [0, T ] (with g(0) = 0 if µ = 1) then (1.1) has a family of solutions u ∈ C[0, T ] given by the formula
u(tk ) = t−µ k
Z
tk
sµ−1 u(s)ds + g(tk ) (3.1) 0
Defining, for 0 ≤ i ≤ N − 1,
u(t) = c0 t1−µ + g(t) + γ + Z t sµ−2 (g(s) − g(0))ds, (2.1) t1−µ
Di :=
Z
ti+1
sµ−1 ds = hµ ti
(i + 1)µ − iµ , (3.2) µ
0
we obtain the following algorithm for the approximate values uhk of u(tk )
where γ :=
1 g(0), µ−1 0,
if µ < 1, (2.2)
uhk
if µ = 1,
1≤k≤N
We have the following convergence result. Theorem 3.1. Consider equation (1.1) with g ∈ C 1 [0, T ] and 0 < µ ≤ 1. Then the approximate solution uh obtained by Euler’s method converges to the particular solution of (1.1) which is in C 1 [0, T ].
0
We note that (2.3) can be obtained from (2.1) with c0 = 0. Indeed, it follows from (2.1) that
Remark 3.1. It should be noted that the case when g ∈ / C 1 [0, T ] can be dealt with using the regularization techniques developed in [10]. If there exists α ∈ IR, such that g(t) = tα g¯(t), where g¯ ∈ C 1 [0, T ], then equation (1.1) can be transformed into the following auxiliary equation
(2.4)
and this limit is zero when µ > 1.
3
Di uhi = g(tk )
In the case 0 < µ < 1 we take as initial value uh0 = u(0) = µ/(µ − 1) g(0). If µ = 1 then we require g(0) = 0 and set uh0 = 0.
(b) If µ > 1 and g ∈ C m [0, T ], m ≥ 0, then the unique solution u ∈ C m [0, T ] of (1.1) is Z t u(t) = g(t) + t1−µ sµ−2 g(s)ds. (2.3)
t→0+
k−1 X i=0
and c0 is an arbitrary constant. Out of the family of solutions there is one particular solution u ∈ C 1 [0, T ]. Such a solution is unique and can be obtained from (2.1) by taking c0 = 0.
c0 = lim tµ−1 u(t),
−
t−µ k
Numerical solution
Lµ+α v(t) = g¯(t), Here we consider the application of Euler’s method to equation (1.1).
(3.3)
where u(t) = tα v(t). For the use of variable transformation methods with Abel-type equations see for example [7] and the references therein.
Let Xh := {ti = ih, 0 ≤ i ≤ N } be an uniform grid with stepsize h := T /N . 2
4
5
Error expansions
In [10], we proved that if µ > 1 and g ∈ C m [0, T ] is such that g 0 (0) = · · · = g (m−1) (0) = 0, m ≥ 1, the numerical solution uh , obtained by Euler’s method, allows an asymptotic error expansion in integer powers of h h
h
(r u − u )k =
m−1 X
5.1
Numerical examples Euler’s method and convergence acceleration
We have considered the numerical solution of (1.1), with g(t) of the form g(t) := tα g¯(t),
q
m
Aq (tk )h + O (h ) , (4.1)
t ∈ [0, 1]
(5.1)
where g¯(t) satisfies the conditions of corollary 4.1. Letting g¯(t) := (1 + t), then if µ + α > 0 and µ + α 6= 1, the solution of (1.1) is given by
q=1
where Aq ∈ C m−q [0, T ] do not depend on h. The above expansion was the basis for applying the Richardson’s extrapolation method in order to improve the numerical results.
u(t) := tα u ¯(t) µ+α µ+α+1 = tα (5.2) . +t µ+α−1 µ+α
In [11] it has been shown that if g 0 (0) 6= 0 the asymptotic error expansions will contain terms of more general forms.
Examples 5.1, 5.2 correspond to the choices of µ = 1, α = −0.5 and µ = 0.5, α = −0.3, respectively. In example 5.3 we have taken µ = 1, α = 0 and g(t) = g¯(t) = t, giving u(t) = 2t. The numerical approximations were computed by applying Euler’s method to the auxiliary equation (3.3). The absolute errors in the approximate solutions uh to u are shown in figures 1-3 for the values 1 , n = 0, · · · , 5 of the stepsize h. hn = 21n 40 The following quantity has been used as an estimate of the convergence order h kr u − uh k∞ k := −log2 . (5.3) kr2h u − u2h k∞
Theorem 4.1 Let g ∈ C 2 [0, T ], g 0 (0) 6= 0 and 0 < µ ≤ 1. Then the numerical approximation uh to (1.1) satisfies the error estimate (rh u − uh )k = C1 (tk ) hµ + O(h), if 0 < µ < 1, (4.2) and (rh u − uh )k = C2 (tk ) h ln h + O(h), if µ = 1, (4.3) where C1 , C2 do not depend on h. Corollary 4.1 Let g(t) = tα g¯(t), with g¯ ∈ h C [0, T ] and g¯0 (0) 6= 0. Define uhi := tα i vi , i = h 1, . . . , N , where v is the approximate solution of (3.3). Then,
Extrapolation methods based on the midpoint rule have been studied in [9] for nonlinear Volterra equations with smooth kernels (see also [3] and the references therein). In the case µ > 1, an asymptotic error expansion was obtained in [10] for Euler’s method which contained only integer powers of the stepsize h. This allowed the use of the Richardson’s extrapolation algorithm to improve the numerical results. The examples below have produced expansions of more general types (cf.
2
(rh u − uh )k = C10 (tk ) hµ+α + O(h), if 0 < µ + α < 1, and (rh u − uh )k = C20 (tk ) h ln h + O(h), if µ + α = 1. 3
corollary 4.1). A natural way to accelerate the convergence of the numerical results is then to use the E-algorithm of Brezinski [1] which is a generalization of the Richardson’s extrapolation. It was designed under the assumption that we know an asymptotic expansion for a given sequence Sn :
Figure 1: Absolute errors in example 5.1 Error 0.5 0.4 0.3
Sn = S + a1 g1 (n) + a2 g2 (n) + . . . + ak gk (n), n = 0, 1, 2, . . . (5.4)
0.2 0.1
where gi (n) are predefined sequences, which satisfy the condition gi+1 (n) = o(gi (n)), as n → ∞, ai being real numbers that do not depend on n. The performance of this algorithm is illustrated by comparing the errors in the Euler’s approximations uhk ' u(tk ) using a stepsize h = 1/1280 with the errors of the extrapolated values. The results are shown in tables 1-3.
0
t > 0.
ti 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
(5.5)
An application of corollary 4.1 gives an asymptotic error expansion with the form (rh u − uh )i = C1 (ti ) h1/2 + O(h).
(5.6)
The absolute errors in the solution are displayed in figure 1 and the convergence order estimate given by (5.3) is k = 0.49, which confirms the theoretical prediction. In order to accelerate the convergence, the E-algorithm has been applied with the auxiliary sequences g1 (n) = 3/2 hn , g4 (n)
1/2 hn , g2 (n) = h2n , g5 (n) =
= hn , 5/2 hn .
0.4
0.6
0.8
1
Table 1: Absolute errors with h = 1/1280 and after extrapolation, in example 5.1
Example 5.1. Let α = −0.5, µ = 1. Then (5.2) takes the form u(t) = −t−1/2 + 3 t1/2 ,
0.2
u(ti ) −2.21359 −0.89443 −0.18257 0.31623 0.70711 1.03280 1.31475 1.56525 1.79196 2.00000
h = 1/1280 0.1056 0.1076 0.1086 0.1091 0.1095 0.1098 0.1100 0.1102 0.1103 0.1104
extrapol 0.120E − 4 0.336E − 5 0.162E − 5 0.219E − 6 0.698E − 7 0.241E − 6 0.617E − 6 0.530E − 6 0.429E − 6 0.713E − 6
This choice was based on the assumption that the asymptotic error expansion has the form (rh u − uh )i = C1 (ti ) h1/2 + C2 (ti ) h + C3 (ti ) h3/2 + C4 (ti ) h2 + C5 (ti ) h5/2 +
g3 (n) =
O(h3 ). The significant improvement in the error of the approximate values uhk of u(tk ) shown in table 1 supports the validity of the above conjecture. 4
x
Figure 2: Absolute errors in example 5.2
Table 2: Absolute errors with h = 1/1280 and after extrapolation, in example 5.2
Error 3 2.5 2 1.5 1 0.5 0
0.2
0.4
0.6
0.8
x
1
ti 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
u(ti ) 0.69834 1.53963 2.22431 2.83022 3.38565 3.90481 4.39610 4.86501 5.31538 5.75000
h=1/1280 0.4698 0.6668 0.8179 0.9452 1.0573 1.1586 1.2518 1.3385 1.4193 1.4969
extrapol 0.2607E − 4 0.7866E − 5 0.3941E − 5 0.4900E − 5 0.6160E − 6 0.2570E − 5 0.2563E − 5 0.8522E − 5 0.2847E − 5 0.7727E − 6
Example 5.2. Letting α = −0.3 and µ = 0.5 then the exact solution of (1.1) is u(t) = −0.25 t−0.3 + 6 t0.7 .
Example 5.3 If g(t) = t and µ = 1 then the unique solution of (1.1) is u(t) = 2t. In this case, the approximate solution uh , obtained by the Euler’s method, has an asymptotic error expansion with the form
(5.7)
From corollary 4.1, the approximate solution uh has an asymptotic error expansion with the form h
h
(r u − u )i = C1 (ti ) h
0.2
+ O(h).
(rh u − uh )i = C1 (ti ) h ln h + O(h). (5.10)
(5.8)
The graphics of the absolute errors are displayed in figure 3 .
The convergence order is very slow and we get very bad results with the Euler’s method (as shown in figure 2 and in 3rd column of table 2). In order to apply the E-algorithm, we have considered the auxiliary sequences: g1 (n) = h0.2 n , g2 (n) = 1.2 hn , g4 (n) = h2n g5 (n) = h2.2 n .
hn ,
The E-algorithm was applied based on the assumption that the following expansion is valid (rh u − uh )i = C1 (ti ) h lnh + C2 (ti ) h + C3 (ti ) h2 lnh + O(h2 )
g3 (n) =
Again we have assumed that the terms of the asymptotic error expansion may be extended to terms of orders higher than 2, with g2k−1 (n) = hα+µ+k−1 n g2k (n) = hkn , k = 1, 2, 3, . . .
and the results produced are shown in table 3.
5.2
The use of graded meshes
(5.9) Graded meshes were introduced in [2] to recover the optimal convergence rates of methods for Abel type equations with nonsmooth solutions. Here we have employed graded
The absolute errors in the 4th column of table 2 indicate that this assumption leads to a high performance of the E-algorithm. 5
Figure 4: Example 5.1 with graded meshes order 0.9
Figure 3: Absolute errors in example 5.3 Error 0.12
0.8
0.7
0.1
0.6
0.08 0.06
2
3
4
5
r
6
0.04
Figure 5: Example 5.2 with graded meshes
0.02 0
order
0.2
0.4
0.6
0.8
1
x 0.6 0.5 0.4 0.3
2
Table 3: Absolute errors with h = 1/1280 and after extrapolation, in example 5.3 ti 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
u(ti ) 0.2000 0.4000 0.6000 0.8000 1.0000 1.2000 1.4000 1.6000 1.8000 2.0000
h = 1/1280 0.424E − 2 0.478E − 2 0.510E − 2 0.532E − 2 0.550E − 2 0.564E − 2 0.576E − 2 0.587E − 2 0.596E − 2 0.604E − 2
3
4
5
6
r
meshes in the interval [0, 1], defined by r i , i = 0, · · · , N, (5.11) xi = N
extrapol 0.298E − 5 0.156E − 5 0.106E − 5 0.798E − 6 0.642E − 6 0.535E − 6 0.457E − 6 0.405E − 6 0.361E − 6 0.326E − 6
for several values of the grading exponent r. In the case r = 1 we have the uniform mesh. The performance of these meshes is shown in figures 4,5, for examples 5.1,5.2, respectively, indicating that first order convergence can be attained. Future work will include finding a way how to choose the appropriate value of r.
5.3
Other methods
We have applied product integration methods based on higher order quadrature methods to 6
(1.1), in the case 0 < µ ≤ 1, by using the regularization technique of remark 3.1. Here we consider the so-called product trapezoidal method, which results from using the product trapezium rule to approximate the integral on the right hand side of (3.1). We have performed several experiments with
Figure 6: Trapezoidal method with graded meshes in example 5.4 order
g(t) := tα (1 + t + t2 ) = tα g¯(t)
1.9
where g¯(t) is such that g¯00 (0) 6= 0, in analogy with the conditions of theorem 4.1. Then, with µ 6= 1, we have µ µ+1 α 2µ + 2 u(t) = t +t +t µ−1 µ µ+1 (5.12) Example 5.4: We set α = −0.3, µ = 0.5 in (5.12).
1.8 1.7 1.6 1.5 1.4 1.3
1.2
Figure 6 illustrates the performance of the trapezoidal method, using the meshes defined by (5.11). There, estimates for the convergence order are shown for several values of r. In the case r = 1 (uniform mesh), our numerical experiments seem to indicate that the order p is p = µ + α + 1, if µ + α < 1. This gives p = 1.2 for example 5.4. As in the case of Euler’s method, it should be possible to recover the optimal convergence order, 2, for an appropriate choice of r. For the sake of comparison we have also applied Euler’s method to this example in conjunction with extrapolation. In table 4 we have listed the absolute errors at ti = 0.5 obtained by both methods and also the error after using the E-algorithm. We see that an impressive improvement in accuracy can indeed be achieved by extrapolation.
6
1.4
1.6
1.8
Table 4: Errors at t = 0.5 for example 5.4 h 1/40 1/80 1/160 1/320 1/640 1/1280
Concluding remarks
We have introduced an integral equation possessing a singular kernel defined by p(t, s) = 7
Trapezoidal 0.308E − 01 0.134E − 01 0.586E − 02 0.255E − 02 0.111E − 02 0.485E − 03
Euler 2.103 1.839 1.605 1.398 1.218 1.060
Euler+extrap
0.275E − 5
r
sµ−1 /tµ with the following properties: p does Rt not satisfy 0 p(t, s)ds → 0 as t → 0+ and all the iterated kernels associated with p are unbounded. The case when the equation has an infinite set of solutions was considered. Both Euler and trapezoidal methods were applied to several examples. The results show, in particular, that the application of a low order method together with extrapolation procedures can produce highly accurate approximations, even in the cases of multiple or nonsmooth solutions. On the other hand, the use of graded meshes seems to be an alternative approach to recover the optimal convergence rates. We point out that the above properties of the kernel are in contrast to the ones of Abel-type kernels (t − s)−α , 0 < α < 1, and so the usual convergence arguments for weakly singular equations are not applicable here (see eg. [4]).
tion with weakly singular kernel, Appl. Num. Math. 9 (1992) 259–266. [6] T. Diogo, N. B. Franco and P. Lima, Analysis of product integration methods for a class of singular Volterra integral equations, Tendˆencias da Matem´atica Aplicada e Computacional, E.Andrade, G.Silva and A.Ranga (Eds.), 1 (2000) 373-387. [7] E. A. Galperin, E. J. Kansa, A. Makroglou, S.A. Nelson, Variable transformations in the numerical solution of second kind Volterra integral equations with continuous and weakly singular kernels; extensions to Fredholm integral equations, J. Comp. Appl. Math. 115 (2000) 193-211. [8] Weimin Han, Existence, uniqueness and smoothness results for second-kind Volterra equations with weakly singular kernels, J. Int. Eq. Appls. 6 (1994) 365384.
References
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[4] H. Brunner and P.J. van der Houwen, The Numerical Solution of Volterra Equations, North-Holland, Amsterdam, 1986. [5] T. Tang, S. McKee and T. Diogo, Product integration methods for an integral equa8
