HIT Summer Seminar: 5-16 July 2010
Seven lectures on
Theory and numerical solution of
Volterra functional integral
equations
Hermann Brunner
Department of Mathematics and Statistics
Memorial University of Newfoundland
St. John’s, NL
Canada
Department of Mathematics
Hong Kong Baptist University
Hong Kong SAR
P.R. China
1
Topics of lectures
• Lecture 1: Theory of linear Volterra integral
equations
• Lecture 2: Nonlinear Volterra integral equations and applications
• Lecture 3: Basic elements of collocation methods
• Lecture 4: Collocation methods for VIEs with
smooth solutions
• Lecture 5: Collocation methods for Volterra
integral equations with singular kernels
• Lecture 6: Collocation methods for VIEs with
delay functions
• Lecture 7: Additional topics / suggestions
for future research
2
Lecture I:
Theory of linear Volterra integral equations
A linear Volterra integral equation (VIE) of the
second kind is a functional equation of the
form
u(t) = g(t) +
Z t
0
K(t, s)u(s) ds,
t ∈ I := [0, T].
Here, g(t) and K(t, s) are given functions, and
u(t) is an unknown function.
The function K(t, s) is called the kernel of the
VIE.
A linear VIE of the first kind is given by
Z t
0
K(t, s)u(s) ds = g(t),
t ∈ I.
Here, the unknown function occurs only under
the integral sign.
A linear VIE of the third kind has the form
r(t)u(t)) = g(t) +
Z t
0
K(t, s)u(s) ds,
t ∈ I,
where the given function r(t) = 0 at some points
(or on a subinterval) of [0, T] .
(We shall see later that such VIEs are related to so-called
integral-algebraic Volterra equations.)
3
Linear Volterra integral operators:
Notation:
I := [0, T], D := {(t, s) : 0 ≤ s ≤ t ≤ T}
• The classical Volterra integral operator
V : C(I) → C(I) is defined by
Z t
(V u)(t) :=
0
K(t, s)u(s) ds,
t ∈ I,
with K ∈ C(D) .
• The weakly singular Volterra integral operator Vα : C(I) → C(I) has the form
(Vαu)(t) :=
Z t
0
(t − s)−αK(t, s)u(s) ds,
0 < α < 1,
with algebraic singularity (t − s)−α , and K ∈ C(D),
K(t, t) 6= 0 (t ∈ I) .
The weakly singular Volterra integral operator corresponding to a logarithmic singularity,
V1 : C(I) → C(I) is given by
(V1u)(t) :=
Z t
0
log(t − s)K(t, s)u(s) ds ,
with K ∈ C(D), K(t, t) 6= 0 (t ∈ I) .
4
Volterra integro-differential equations: (VIDEs)
A functional differential equation of the form
u0(t) = a(t)u(t) + b(t) + (Vαu)(t),
0 ≤ α ≤ 1,
is called a linear Volterra integro-differential equation. It is complemented by an initial condition:
u(0) = u0 , where u0 is a given number. The
(continuous) functions a, b and K are given.
More general VIDEs ( k ≥ 1 ):
kX
−1
u(k)(t) = a0(t)u(t) +
aj(t)u(j)(t) + b(t)
j=1
k
X
+
(ν)
(Vα u(ν))(t) ,
ν=0
(ν)
with Volterra integral operators Vα
by
(ν)
(Vα u(ν))(t) :=
Z t
0
defined
(t − s)−αKν (t, s)u(ν)(s) ds
(0 ≤ α < 1).
5
Some history:
Vito VOLTERRA (1860-1940) was a very famous Italian mathematician. His papers on integral equations (which are now called Volterra
integral equations) appeared in 1896, and they
– together with the papers of the equally famous Swedish mathematician Ivar Fredholm
– also mark the beginning of Functional Analysis.
Ivar FREDHOLM (1866-1927) wrote his celebrated papers on what are now known as Fredholm integral equations in 1900 and 1903.
,→ For biographies of famous mathematicians, see
www-history.mcs.st-and.ac.uk
6
Ordinary differential equations and VIEs
• The first-order initial-value problem
u0(t) = a(t)u(t) + b(t), t ∈ I := [0, T]; u(0) = u0,
is equivalent to the second-kind Volterra integral equation
u(t) = u0 +
|
Z t
b(s) ds +
0{z
=g(t)
Z t
}
0
a
(s) u(s) ds .
| {z }
=K(t,s)
Here, the kernel K(t, s) does not depend on t !
• The second-order initial-value problem
u00(t) = a(t)u(t) + b(t), u(0) = u0, u0(0) = v0,
is equivalent to a second-kind Volterra integral
equation whose kernel K(t, s) now does depend on t :
u(t) = g(t) +
Z t
0
K
(t, s) u(s) ds,
| {z }
t ∈ I,
where
g(t) := u0 + v0t +
Z t
0
(t − s)b(s) ds
and
K(t, s) := (t − s)a(s) ds .
7
But: A VIE of the second-kind,
u(t) = g(t) +
Z t
0
K(t, s)u(s) ds ,
is in general not equivalent to an initial-value
problem for an ordinary differential equation,
since
d
dt
Z t
0
!
K(t, s)u(s) ds
= K(t, t)u(t) +
Z t
∂ K(t, s)
∂t
0
u(s) ds ,
where, in general,
∂ K(t, s)
6≡ 0 .
∂t
Thus,
u0(t) = g0(t) + K(t, t)u(t) +
=: a(t)u(t) + b(t) +
Z t
∂ K(t, s)
|0
Z t
0
∂ t {z
u(s) ds
}
H(t, s)u(s) ds ,
with u(0) = g(0) . This is an initial-value problem for a Volterra integro-differential equation.
8
Remark: Fredholm integral equations
In a Fredholm integral equation the limits of integration are fixed numbers (given by the endpoints of the interval of integration):
u(t) = g(t) +
Z T
0
K(t, s)u(s) ds,
t ∈ [0, T] .
Fredholm integral equations are related to boundaryvalue problems for differential equations.
Example: The boundary-value problem
u00(t) = a(t)u(t) + b(t), u(0) = A, u(T) = B,
is equivalent to the Fredholm integral equation
u(t) = g(t) +
Z T
G(t, {z
s)a(s)} u(s) ds, t ∈ [0, T] ,
0 |
=K(t,s)
with
(B − A)t
+
g(t) := A +
T
Z
T
G(t, s)b(s) ds
0
and
G(t, s) :=
− Ts (T − t)
− Tt (T − s)
if
if
s≤t
t ≤ s.
9
Basic Volterra theory
In 1896 Vito Volterra published the first of his
fundamental papers on integral equations. It
contains the following fundamental result (which
may be viewed as marking the beginning of Functional
Analysis).
Theorem 1.1:
Assume that the kernel K(t, s) of the linear
Volterra integral equation
u(t) = g(t) +
Z t
0
K(t, s)u(s) ds,
t ∈ I := [0, T],
is continuous on D := {(t, s) : 0 ≤ s ≤ t ≤ T} .
Then for any function g(t) that is continuous
on I (that is, g ∈ C(I)), the VIE possesses a
unique solution u ∈ C(I) . This solution can be
written in the form
u(t) = g(t) +
Z t
0
R(t, s)g(s) ds, t ∈ I ,
for some R ∈ C(D) . The function R = R(t, s)
is called the resolvent kernel of the given kernel K(t, s) .
10
Remark:
,→ Recall: The (unique) solution of the VIE
u(t) = g(t) +
Z t
0
K(t, s)u(s) ds,
t ∈ I := [0, T],
with g ∈ C(I), K ∈ C(D) is given by
u(t) = g(t) +
Z t
0
R(t, s)g(s) ds, t ∈ I ,
where R = R(t, s) is the resolvent kernel of
the given kernel K(t, s) .
If we define the integral operator R : C(I) → C(I)
by
(Rg)(t) :=
Z t
0
R(t, s)g(s) ds, t ∈ I ,
and if we write the VIE in operator form,
u = g + V u,
or
(I − V)u = g
(where I denotes the identity operator), then we
have the following relationship:
(I − V)u = g
⇒
u = (I + R)g .
By Theorem 1.1 this implies that the inverse
operator (I − V)−1 always exists, and hence
(by uniqueness of R(t, s))
(I − V)−1 = I + R .
11
Proof of Volterra’s Theorem for
u(t) = g(t) +
Z t
K(t, s)u(s) ds,
0
t ∈ I.
(1)
Let u0(t) := g(t) and define an infinite sequence of functions {uk(t)}k≥1 by
uk(t) := g(t) +
Z t
0
K(t, s)uk−1(s) ds, t ∈ I
(Picard iteration).
Thus:
u1(t) = g(t) +
Z t
0
K(t, s)g(s) ds .
We can show (using mathematical induction) that
for any k ≥ 1 ,
uk(t) = g(t) +
Z t X
k
Kj(t, s) g(s) ds, t ∈ I ,
0 j=1
|
{z
}
where the so-called iterated kernels Kj(t, s) of
K(t, s) in (1) are defined by K1(t, s) := K(t, s)
and
Kj(t, s) :=
,→
Z t
s
K(t, v)Kj−1(v, s) dv (j ≥ 2).
lim uk(t) = ?
k→∞
12
Does the limit exist? ⇒ Yes, since K(t, s) is
continuous (and thus |K(t, s)| ≤ M, (t, s) ∈ D)
for some constant M. The infinite series
∞
X
Kj(t, s) is called the Neumann series, and
j=1
k
X
R(t, s) := lim
k→∞
Kj(t, s)
((t, s) ∈ D)
j=1
is the resolvent kernel of the given kernel K(t, s) .
It is continuous on D (since all iterated kernels
Kj(t, s) are continuous and the convergence is uniform).
Therefore:
lim uk(t) = z(t),
k→∞
t ∈ I,
for some continuous function z(t) .
Exercise 1.1:
(a) Show that z(t) is a solution (in C(I) ) of the VIE
Z t
u(t) = g(t) +
K(t, s)u(s) ds, t ∈ [0, T] .
0
(b) Show that this is the only solution: z(t) = u(t) .
Note: If u and w are two solutions, then
Z t
|u(t) − w(t)| ≤
|K(t, s)||u(s) − z(s)| ds, t ∈ I .
0
,→ Gronwall inequality / comparison theorem !
13
Corollary 1.2:
Assume that the given functions in
u(t) = g(t) +
Z t
0
K(t, s)u(s) ds, t ∈ I ,
satisfy g ∈ Cd(I), K ∈ Cd(D) for some d ≥ 1 .
Then the regularity of the solution of this VIE
is described by u ∈ Cd(I) . In other words, the
solution u inherits the regularity of the data g
and K .
(Proof: Show that the iterated kernels satisfy Kj ∈ Cd(D)
for all j ≥ 1 . Then use the uniform convergence of the
Neumann series to obtain that R ∈ Cd (D).)
We shall see below that the regularity result
of Corollary 1.2 does not remain valid for the
weakly singular VIE
u(t) = g(t) +
Z t
0
(t − s)−αK(t, s)u(s) ds, t ∈ I ,
with 0 < α < 1 : if g ∈ Cd(I), K ∈ Cd(D) (d ≥
1) then u ∈ C(I) but u 6∈ C1(I) .
14
VIEs with convolution kernels: K(t, s) = k(t − s)
Corollary 1.3:
Assume that k ∈ C(I).
g ∈ C(I) the VIE
u(t) = g(t) +
Z t
0
Then for any given
k(t − s)u(s) ds, t ∈ I,
possesses a unique solution given by
u(t) = g(t) +
Z t
0
r(t − s)g(s) ds, t ∈ I :
the resolvent kernel R(t, s) of K(t, s) = k(t − s)
has also convolution form: R(t, s) = r(t − s) .
Note:
Linear VIEs with convolution kernels can of course
(theoretically) be solved by Laplace transform
techniques.
(See also:
Gripenberg, Londen & Staffans (1990): Chapter 1.)
15
Gronwall’s Lemma and comparison theorems
Lemma 1.4: (Generalized Gronwall lemma)
Let I := [0, T] and assume that for given a, b ∈ C(I),
with b(t) ≥ 0 (t ∈ I) and a(t) non-decreasing
on I, the function z ∈ C(I) satisfies the integral inequality
z(t) ≤ a(t) +
Z t
0
b(s)z(s) ds, t ∈ I .
Then
z(t) ≤ a(t) exp
Z t
0
!
b(s) ds
for all t ∈ I .
If a(t) = α = constant and b(t) = β = constant (> 0 ), then we obtain Gronwall’s original
lemma (1919): if
z(t) ≤ α + β
Z t
0
z(s) ds, t ∈ I ,
then
z(t) ≤ α exp(β t), t ∈ I .
16
Two comparison theorems: (Beesack (1969, 1975))
Theorem 1.5:
Assume:
(i) g ∈ C(I), K ∈ C(D) ;
(ii) g(t) ≥ 0 (t ∈ I), K(t, s) ≥ 0 ((t, s) ∈ D) ;
(iii) R(t, s) is the resolvent kernel of K(t, s) .
If z ∈ C(I) satisfies the inequality
z(t) ≤ g(t) +
Z t
0
K(t, s)z(s) ds, t ∈ I ,
then
z(t) ≤ g(t) +
Z t
0
R(t, s)g(s) ds, t ∈ I .
Theorem 1.6:
Assume that the given functions gi(t) and Ki(t, s)
(i = 1, 2 ) satisfy:
(i) gi ∈ C(I), |g1(t)| ≤ g2(t) (t ∈ I) ;
(ii) Ki ∈ C(D), |K1(t, s| ≤ K2(t, s) ((t, s) ∈ D) .
Then the solutions of the two VIEs
ui(t) = gi(t) +
Z t
0
Ki(t, s)ui(s) ds, t ∈ I
(i = 1, 2) are related by
|u1(t)| ≤ u2(t) + |g1(t) − g2(t)|, t ∈ I .
17
Remark: Abstract theory of Volterra integral equations
Since the late 1960s the theory of Volterra integral equations in abstract settings (e.g. in Banach spaces) has
received increasing attention. Here are some of the key
books and papers (see also References / Lecture I):
• A. Friedman & M. Shinbrot, Volterra integral equations in Banach spaces, Trans. Amer. Math. Soc., 126
(1967), 131–179.
• R.K. Miller & G.R. Sell, Volterra Integral Equations
and Topological Dynamics, Memoirs Amer. Math. Soc.,
No. 102, American Mathematical Society, Providence,
R.I., 1970.
• R.C. Grimmer, Resolvent operators for integral equations in a Banach space, Trans. Amer. Math. Soc., 273
(1982), 333–349.
• R.C. Grimmer & A.J. Pritchard, Analytic resolvent
operators for integral equations in Banach space, J. Differential Equations, 50 (1983), 234–259.
• O. Diekmann & S.A. van Gils, Invariant manifolds
for Volterra integral equations of convolution type, J.
Differential Equations, 54 (1984), 139-180.
• J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser Verlag, Basel-Boston, 1993.
• M. Väth, Abstract Volterra equations of the second
kind, J. Integral Equations Appl., 10 (1998), 319–362.
• M. Väth, Volterra and Integral Equations of Vector
Functions, Marcel Dekker, New York, 1999.
18
Fredholm integral equations
It follows from Theorem 1.1 that for every constant λ the linear Volterra integral equation of
the second kind,
u(t) = g(t) + λ
Z t
0
K(t, s)u(s) ds, t ∈ [0, T] ,
with continuous g and K, has a unique continuous solution.
This is in general not true for a linear Fredholm
integral equation of the second kind,
u(t) = g(t) + λ
Z T
0
K(t, s)u(s) ds, t ∈ [0, T] .
(2)
,→ A (real or complex) value of µ for which
(F φ)(t) :=
Z T
0
K(t, s)φ(s) ds = µφ(t), t ∈ [0, T]
possesses a continuous solution φ(t) 6≡ 0 is called
an eigenvalue of the Fredholm integral operator
F . The corresponding solution φ(t) is called
an eigenfunction of F .
Exercise 1.2: Let K(t, s) = A(t)B(s) where A, B ∈ C(I)
are given (real-valued) functions. Show that there may
exist λ ∈ IR so that for given g ∈ C(I) the Fredholm integral equation (2) has more than one solution in C(I) .
Is it possible that (2) has no solution?
19
Volterra-Fredholm integral equations
An integral equation of the form
Z tZ
u(t, x) = g(t, x) + λ
0
Ω
K(t, s, x, ξ)u(s, ξ) dξ ds,
with t ∈ I := [0, T], x ∈ Ω := [a, b] , is called a
Volterra-Fredholm (or: mixed) integral equation.
In contrast to Fredholm integral equations, it
has a unique solution u ∈ C(I × Ω) for all (real
or complex) parameters λ, whenever g ∈ C(I × Ω)
and K ∈ C(D × Ω2) :
Theorem 1.7: (Diekmann (1978), Kauthen (1989))
Under the above conditions on g and K the
above Volterra-Fredholm integral equation possesses a unique solution u ∈ C(I × Ω) .
This solution is given by
u(t, x) = g(t, x) +
Z tZ
0
Ω
R(t, s, x, ξ)g(s, ξ) dξ ds ,
where the resolvent kernel R(t, s, x, ξ) is a continuous function on D × Ω2 .
(Proof: ,→ Exercise !)
20
Volterra integral equation of the first kind:
The starting point of Volterra’s first paper of
1896 was the first-kind integral equation
Z t
0
H(t, s)u(s) ds = f (t),
t ∈ I := [0, T] .
Assume that the kernel H(t, s) is continuous
and has a continuous partial derivative ∂ H(t, s)/∂ t ,
and that f (t) has a continuous derivative and
satisfies f (0) = 0 . Then (differentiate both sides
with respect to t):
H(t, t)u(t) +
Z t
∂ H(t, s)
u(s) ds = f 0(t) .
∂t
If H(t, t) 6= 0 for all t ∈ I , then (divide by H(t, t))
we obtain a VIE of the second kind:
0
Z t
−1 ∂ H(t, s)
f 0(t)
u(t) =
+
u(s) ds .
H(t, t)
∂t }
0 H(t, t)
| {z }
|
{z
=g(t)
=K(t,s)
⇒ Under the above conditions on f and the
kernel H the first-kind VIE has a unique continuous solution u(t) on the interval [0, T]
(Volterra, 1896), because g(t) and K(t, s) are
continuous functions.
21
Exercise 1.3:
(a) Is the condition H(t, t) 6= 0 for all t ∈ [0, T] necessary
for the existence of a unique solution of
Z t
H(t, s)u(s) ds = f (t) ?
0
(b) Consider the first-kind Volterra integral equation
Z t
(t − s)k−1
u(s) ds = f (t), t ∈ [0, T] ,
(
k
−
1
)!
0 |
{z }
=H(t,s)
where k is an integer with k ≥ 1, and f (t) has continuous derivatives of at least order k .
Does this VIE possess a unique (continuous) solution ?
(c) Does the VIE
Z t
(2t − 3s)u(s) ds = t2, t ∈ [0, T] ,
0
possess a unique (continuous) solution on [0, T] ?
22
• VIEs with weakly singular kernels
In his second paper of 1896, Volterra studied
VIEs with discontinuous (unbounded) kernels,
u(t) = g(t) +
Z t
0
(| t − s)−α
K(t, s)} u(s) ds, 0 < α < 1,
{z
=:Kα (t,s)
where K(t, s) is continuous on D and satisfies K(t, t) 6= 0 (t ∈ I) . The kernel Kα(t, s) is
an example of a weakly singular kernel: it is
unbounded when s = t but its integral over any
bounded interval [0, T] is finite. (Such a kernel is
called an integrable kernel.)
A second-kind VIE with a different kind of weakly
singular kernel (not studied by Volterra) is
u(t) = g(t) +
Z t
0
log(t − s)K(t, s)u(s) ds, t ∈ I .
Its kernel has a logarithmic singularity (which
is also integrable).
Remark: VIEs with weakly singular kernels
(t − s)−αK(t, s) (0 < α < 1) are often called
Abel integral equations. (The Norwegian mathematician Niels Henrik Abel (1802-1829) was the first to
study first-kind integral equations with such kernels.)
23
A special case: K(t, s) = λ, 0 < α < 1 :
u(t) = u0 + λ
Z t
0
(t − s)−αu(s) ds, t ∈ I .
(3)
Definition: (Mittag-Leffler function)
Let β > 0 and z ∈ C. The function
∞
X
zj
Eβ (z) :=
Γ(1 + jβ)
j=0
is called the Mittag-Leffler function.
Remark: The Swedish mathematician Gösta MittagLeffler (1846-1927) introduced this function (which can
also be defined for complex β with Re(β)> 0) in 1903.
Examples:
•
E1(z) = ez .
√
•
E2(z) = cosh( z) .
Theorem 1.8:
For every α ∈ (0, 1) the VIE (3) possesses a
unique continuous solution given by
u(t) = E1−α(λΓ(1 − α)t1−α)u0 .
(This result is due to Hille and Tamarkin (1930).)
Exercise 1.4: Prove Theorem 1.8 by using Picard iteration. Show that u ∈ C(I) \ C1(I) for all u0 6= 0.
24
Theorem 1.9 below generalizes the result of
Theorem 1.8. We use the notation
Kα(t, s) := (t − s)−αK(t, s)
(0 < α < 1) .
Theorem 1.9: Let 0 < α < 1 and assume that
g ∈ Cd(I), K ∈ Cd(D) for some d ≥ 0 .
(a) If d = 0 the VIE
u(t) = g(t) +
Z t
0
Kα(t, s)u(s) ds, t ∈ I,
possesses a unique solution u ∈ C(I) . This solution has the representation
u(t) = g(t) +
Z t
0
Rα(t, s)g(s) ds, t ∈ I ,
where the resolvent kernel Rα(t, s) of the kernel Kα(t, s) has the form
Rα(t, s) = (t − s)−αQα(t, s) .
Here, Qα(t, s) is continuous on D .
(b) If d ≥ 1 every nontrivial solution has the
property that u 6∈ C0(I) : as t → 0+ the solution behaves like
u0(t) ∼ Ct−α .
25
Proof of Theorem 1.9:
In analogy to the proof of Theorem 1.1 ( α = 0 ) we use
Picard iteration: setting u0(t) := g(t) we define an infinite sequence of functions {uk(t)}k≥1 by
Z t
uk(t) := g(t) +
(t − s)−α K(t, s)uk−1(s) ds, t ∈ I .
0
Here, the resulting iterated kernels {Kα,j(t, s)} of
Kα(t, s) := (t − s)−αK(t, s) are defined by Kα,1(t, s) := Kα(t, s)
and
Z t
Kα,j(t, s) :=
Kα(t, v)Kα,j−1(v, s) dv (j ≥ 2; (t, s) ∈ D).
s
For example,
Z
Kα,2(t, s) =
t
(t − v)−α (v − s)−α K(t, v)K(v, s) dv .
s
To establish uniqueness we the following result.
Lemma 1.10: (Generalized Gronwall lemma)
Assume that
(a) g ∈ C(I), g(t) ≥ 0 (t ∈ I) and g is non-decreasing
on I .
(b) The function z ∈ C(I) satisfies the inequality
Z t
z(t) ≤ g(t) + λ
(t − s)−α z(s) ds, t ∈ I ,
0
for some λ > 0 and α ∈ (0, 1) .
Then
z(t) ≤ E1−α(λΓ(1 − α)t1−α)g(t), t ∈ I .
( ,→ Proof: Exercise 1.6.)
26
Exercise 1.6:
(a) Prove the generalized Gronwall lemma (Lemma 1.10)
for
Z t
z(t) ≤ g(t) + λ
(t − s)−α z(s) ds, t ∈ I ,
0
with λ > 0 and 0 < α < 1 .
(b) State and prove the analogue of Lemma 1.10 for the
integral inequality
Z t
z(t) ≤ g(t) + λ
log(t − s)z(s) ds, t ∈ I .
0
Exercise 1.7:
Analyze the regularity of the solution of the VIE
Z t
u(t) = tβ + λ
(t − s)−α u(s) ds, t ∈ I ,
0
when β > 0, β 6∈ IN and 0 < α < 1 .
27
• Linear Volterra integro-differential equations (VIDEs).
(Recall: I := [0, T], D := {(t, s) : 0 ≤ s ≤ t ≤ T}.)
Theorem 1.11: (Grossman & Miller (1970))
If a ∈ C(I) and K ∈ C(D), then for any b ∈ C(I)
and any u0 the VIDE
u0(t) = a(t)u(t) + b(t) +
Z t
K(t, s)u(s) ds, t ∈ I,
0
∈ C1(I) satisfying
has a unique solution u
This solution is given by
u(t) = r(t, 0)u0 +
Z t
0
r(t, s)b(s) ds,
u(0) = u0.
t ∈ I,
where the (differential) resolvent kernel r(t, s)
depends on a and K (but not on b ).
Moreover, smooth data imply smooth solutions:
a, g ∈ Cd(I)
and
K ∈ Cd(D) ⇒ u ∈ Cd+1(I)
for any d ≥ 1.
(Proof: Application of Volterra’s 1896 theorem: integrate both sides of the VIDE, to obtain a VIE of the
second kind; then use Theorem 1.1. ,→ Exercise 1.8.)
28
• VIDEs with weakly singular kernels
Theorem 1.12:
Consider the VIDE with weakly singular kernel,
u0(t) = a(t)u(t) + b(t) +
Z t
0
(t − s)−αK(t, s)u(s) ds ,
with initial condition u(0) = u0 and 0 < α < 1 .
(a) If a, b ∈ C(I) and K ∈ C(D) , then this equation possesses a unique solution u ∈ C1(I) satisfying u(0) = u0 .
(b) If a, b ∈ Cd(I) and K ∈ Cd(D) (for any
d ≥ 1 ), then
u ∈ C(I) ∩ Cd+1(0, T],
with
u00(t) ∼ Ct−α at t = 0+ .
(Brunner (1983), Lubich (1983), B., Pedas & Vainikko
(2001))
Exercise 1.9:
Prove Theorem 1.12.
Exercise 1.10:
Determine the solution of the VIDE
Z t
u0(t) = g(t) + λ
(t − s)−α u(s) ds, 0 < α < 1 ,
0
satisfying u(0) = u0 .
29
• Non-compact Volterra integral operators
It follows from Theorems 1.1 and 1.9 (Volterra,
1896) that for any K ∈ C(D) and any λ 6= 0
the homogeneous VIE
(Vαu)(t) :=
Z t
0
(t − s)−αK(t, s)u(s) ds = λu(t)
(with 0 ≤ α < 1, t ∈ [0, T]) has only the trivial
solution u(t) ≡ 0 .
(Note that Vα : C(I) → C(I) is a compact integral operator.)
This is in general not true if Vα is replaced by
(Ap,αu)(t) :=
Z t
0
(tp − sp)−αK(t, s)u(s) ds
with p > 1, α ∈ (0, 1) .
Example: The generalized Abel integral operator (from C[0, 1] → C[0, 1]),
(A2,1/2u)(t) =
Z t
0
(t2 − s2)−1/2u(s) ds ,
is not compact ⇒ there exist uncountably
many values λβ ∈ (0, π/2] so that for any β ≥ 0,
A2,1/2(tβ ) ds = λβ tβ , t ∈ [0, 1].
(Atkinson (1976); see also: G. Vainikko, Cordial Volterra
integral equations, Numer. Funct. Anal. Optim., 30
(2009), 1145-1172.)
30
Basic references:
• V. Volterra, Sulla inversione degli integrali definite (in
Italian) [On the invertibility of definite integrals], Atti R.
Accad. Sci. Torino, 31 (1896), 311-323.
• V. Volterra, Theory of Functionals and of Integral
and Integro-Differential Equations, Dover Publications,
New York, 1959.
• R.K. Miller, Nonlinear Volterra Integral Equations,
W.A. Benjamin, Menlo Park, CA, 1971.
• G. Gripenberg, S.-O. Londen & O. Staffans, Volterra
Integral and Functional Equations, Cambridge University
Press, Cambridge, 1990.
[Most comprehensive monograph on linear and nonlinear
VIEs]
• C. Corduneanu, Integral Equations and Applications,
Cambridge University Press, Cambridge, 1991.
• H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations, Cambridge University Press, Cambridge, 2004. (Chapters 2
and 6)
(,→ See also the handout ”References: Lecture I ” for
additional papers and books on the theory and applications of Volterra integral equations.)
31
HIT Summer Seminar: 5-16 July 2010
Lecture 2 of
Theory and numerical solution of
Volterra functional integral
equations
Hermann Brunner
Department of Mathematics and Statistics
Memorial University of Newfoundland
St. John’s, NL
Canada
Department of Mathematics
Hong Kong Baptist University
Hong Kong SAR
P.R. China
32
Lecture 2:
Nonlinear Volterra integral equations and
applications
• General nonlinear Volterra integral equation:
u(t) = g(t) +
Z t
0
(t − s)−αk(t, s, u(s)) ds
(0 ≤ α < 1) .
• Volterra-Hammerstein integral equation:
u(t) = g(t) +
Z t
0
(t − s)−αK(t, s)G(s, u(s)) ds
(0 ≤ α < 1) ).
• Implicit Volterra integral equation:
F(u(t)) = g(t) +
Z t
0
k(t, s, u(s)) ds (0 ≤ α < 1) .
Example: G(s, u) = up :
u(t) = g(t) +
Z t
0
k(t − s)up(s) ds, p > 1.
33
Volterra-Hammerstein integral equations:
Most nonlinear Volterra integral equations of
the second kind arising in applications are of
the form
u(t) = g(t) +
Z t
0
K(t, s)G(s, u(s)) ds,
t ∈ [0, T] .
Note that here the nonlinearity G(s, u(s)) does
not depend on t.
Example 1:
The first-order nonlinear differential equation
u0(t) = F(t, u(t)),
t ∈ [0, T]; u(0) = u0,
is equivalent to the nonlinear VIE
u(t) = u0 +
Z t
0
F(s, u(s)) ds,
t ∈ [0, T] :
here, K(t, s) = 1 for all (t, s) ∈ D .
Example 2:
The second-order differential equation
u00(t) = F(t, u(t)), u(0) = u0, u0(0) = v0,
is equivalent to the nonlinear VIE
u(t) = u0 + v0t +
Z t
0
(t − s)F(s, u(s)) ds :
here, we have K(t, s) = t − s .
34
Nonlinear VIEs:
The basic existence theorem
Similar to the existence theory for nonlinear ordinary differential equations (ODEs) the solution of a nonlinear VIE may not be unique, or it
exists only on some subinterval [0, t̄) with t̄ < T .
Recall that when proving the (local) existence
and uniqueness of a solution to the initial-value
problem for an ODE,
u0(t) = f (t, u(t)), t ≥ 0;
u(0) = u0 ,
one applies fixed-point iteration (Picard iteration:
recall Lecture I) to the equivalent nonlinear VIE
u(t) = u0 +
Z t
0
f (s, u(s)) ds, t ≥ 0,
The general nonlinear VIE
u(t) = g(t) +
Z t
0
k(t, s, u(s)) ds, t ≥ 0 ,
can be treated in the same way, and thus Theorem 2.2 below is a generalization of the classical
local existence theorem for ODEs.
,→ We first consider two special cases.
35
• Reduction of certain Volterra-Hammerstein
integral equations to (systems of) ODEs:
u(t) = g(t) +
Z t X
r
0 j=1
Aj(t)bj(s, u(s)) ds ,
(4)
where Aj = Aj(t) and bj = bj(t, z) are continuous functions.
(A kernel of the form
k(t, s, z) =
r
X
Aj(t)bj(s, z)
j=1
is sometimes referred to as a degenerate kernel (or a
separable kernel).
Since the nonlinear VIE (4) is equivalent to a
system of nonlinear ODEs (see Exercise 2.1 below), the local existence and uniqueness of its
solution can be established by using ODE theory.
Exercise 2.1:
Show that the above nonlinear VIE (4) is equivalent to
a system of r nonlinear ODEs.
36
• Finite-time blow-up of VIE solutions:
,→ Example: The semi-linear ODE,
u0(t) = λu(t) + εup(t) (t ≥ 0), u(0) = u0 ,
is equivalent to the semi-linear VIE
u(t) = u0 +
Z t
0
(λu(s) + εup(s))ds, t ≥ 0 . (5)
Assume that p > 1, λ < 0, ε > 0, u0 > 0 .
Theorem 2.1: (cf. Brunner (2004))
(a) There exists a finite Tb > 0 such that
lim u(t) = +∞
(6)
t→T−
b
if and only if u0 in (5) is sufficiently large:
u0 > (−λ/ε)1/(p−1) .
(b) If (6) holds, then this blow-up time Tb is
given by

Tb =

1
λ
ln 1 + p−1  .
λ(p − 1)
εu
0
Proof: Since the above ODE is a Bernoulli differential
equation, its exact solution is easily found; it is
1/(p−1)
1
u(t) =
.
u01−pe−λ(p−1)t − (ε/λ)[1 − e−λ(p−1)t]
If there exists a finite t = Tb > 0 so that the denominator becomes zero, then the solution exists only in the
interval [0, Tb ) .
37
• Existence of solutions for general nonlinear
VIEs:
Notation: D := {(t, s) : 0 ≤ s ≤ t ≤ T} and
ΩB := {(t, s, z) : (t, s) ∈ D, z ∈ IR, |z − g(t)| ≤ B} ,
MB := max{|k(t, s, z)| : (t, s, z) ∈ ΩB} ,
for given B > 0 .
Theorem 2.2: (Miller (1971))
Assume:
(a) g ∈ C(I), k ∈ C(ΩB) ;
(b) k satisfies the Lipschitz condition
|k(t, s, z) − k(t, s, z̃)| ≤ LB|z − z̃|
for all (t, s, z), (t, s, z̃) ∈ ΩB .
Then the nonlinear VIE
u(t) = g(t) +
Z t
0
k(t, s, u(s)) ds, t ≥ 0 ,
possesses a unique solution u ∈ C(I0) where
I0 := [0, δ0],
δ0 := min{T, B/MB} .
(A detailed proof can also be found in the book by Brunner (2004), Section 2.1.5.)
38
Exercise 2.2:
(a) Let p > 1 and u0 > 0. Does the solution of the
nonlinear VIE
Z t
u(t) = u0 +
(t − s)up (s) ds, t ≥ 0,
0
blow up in finite time?
(b) (hard!) Answer (a) for the VIE with weakly singular
kernel,
Z t
u(t) = u0 +
(t − s)−α up (s) ds,
0
where 0 < α < 1 .
Exercise 2.3: Extend Theorem 2.2 to nonlinear VIEs
with weakly singular kernels:
Z t
u(t) =
(t − s)−α k(t, s, u(s)) ds, 0 < α < 1 ,
0
where k(t, s, u) = K(t, s)G(u) .
39
Volterra-Hammerstein integral equations (VHIEs):
u(t) = g(t) +
Z t
0
K(t, s)G(s, u(s)) ds, t ≥ 0 :
(7)
Setting z(t) := G(t, u(t)), this VHIE may be
written as the pair of equations
z(t) = G t, g(t) +
Z t
0
!
K(t, s)z(s) ds , t ≥ 0,
(8)
and
u(t) = g(t) +
Z t
0
K(t, s)z(s) ds, t ≥ 0 .
(9)
Thus, instead of analyzing the existence of a
solution u(t) for the VIE (7), we could prove
the existence of a solution z(t) of the ’implicit’
equation (8) and then use (9) to find u(t) .
Remark: Equations (8),(9) can also be used as the basis
for the numerical solution of the nonlinear VIE (7)
,→ Lecture 6.
40
Remarks:
• Non-standard VIEs:
u(t) = g(t) +
Z t
0
K(t, s)G(u(t), u(s)) ds, t ∈ I .
(Zhang Ran, Guan Qingguang & Zou Yongkui (2010))
• Auto-convolution VIEs:
u(t) = g(t) +
Z t
0
K(t, s)G(u(t − s), u(s)) ds .
(von Wolfersdorf & Janno (1995), Berg & von Wolfersdorf (2005))
Exercise 2.4: Does the auto-convolution VIE
Z t
u(t) = g(t) +
u(t − s)u(s) ds, t ∈ I,
0
possess a unique solution u ∈ C(I) for given g ∈ C(I) ?
41
Applications: VIEs as mathematical models
• The renewal equation
The renewal VIE,
u(t) = g(t) +
Z t
0
k(t − s)u(s) ds,
t≥0
(where the kernel K(t, s) = k(t − s) is a convolution kernel: it depends only on the difference
t − s of the variables t and s) arises in the
mathematical modelling of renewal processes.
Examples:
,→ Model with single commodity, a single investment policy that is continually renewed, and
a single depreciation policy.
,→ Model of age-structured population in which
individuals die and new individuals are added
(born).
Question:
lim u(t) = ?
t→∞
(Feller (1941), ... , Miller (1975), ... ;
Diekmann, Gyllenberg & Thieme (1991) [Abstract
framework])
42
• Population growth models (I):
(Brauer (1975), Brauer & Castillo-Chávez (2001) )
u(t) = g(t) +
Z t
0
P(t − s)G(u(s)) ds, t ≥ 0 :
Representation of the size of a population u = u(t)
whose growth rate depends only on the population size, and with a probability of death that
depends only on age.
Here, G(u) is the number of members added to the population (in unit time) when the population size is u. The
function P(t) represents the probability that a member
of the population survives to age t, and the function
g(t) represents the number of members who are already
present at time t = 0 and who are still alive at time
t > 0.
,→ Population with harvesting:
If harvesting is carried out at a constant time
rate, the resulting mathematical model is the
VIE
u(t) = g(t) +
Z t
0
P(t − s)G)u(s)) ds − Φ(t), t ≥ 0 ,
where Φ(t) represents the number of members
of the population harvested up to time t > 0
who would otherwise have survived to time t .
(Here, Φ(t) is non-negative, (piecewise) continuous, and
so that Φ(∞) := lim Φ(t) exists.)
t→∞
43
• Population growth models (II):
(Cooke & Yorke (1973), Cooke (1976), Smith (1977),
Torrejón (1990), ... )
,→ Mathematical models of single-species population growth with immigration and given
age distribution:
u(t) =
Z t
t−τ
P(t − s)G(u(s)) ds + g(t), t ≥ 0 ,
where
(i) g(t) : number of immigrants at time t;
(ii) u(t) : total number of individuals alive at
time t;
(iii) P(t) : Proportion of population surviving
to age t (probability of survival). P(t) is non-negative
and nonincreasing.
(iv) G(u(t)) : number of births per unit time
at time t (births are dependent on the density of the
population at time t);
(v) τ > 0 : Life span (every individual dies at age
τ ).
,→ VIEs of this type also arise as models of economic
growth and of the spreading of infections (epidemics)
(Cooke (1976))
,→ Variable life span: τ = τ (t) > 0 : see Torrejón (1990).
44
• Population growth models (III):
(Gripenberg (1981, 1983))
,→ Spread of infections not inducing permanent
immunity can be modelled by the non-standard
nonlinear VIE
u(t) = c · f (t) −
× g(t) +
Z t
0
Z t
0
!
a(t − s)u(s) ds
!
b(t − s)u(s) ds , t ≥ 0.
Here, u(t) is the rate at which individuals that
are susceptible to the disease have become infected up to time t, and c > 0 is a given constant.
45
• Population growth models (IV):
(Diekmann (1978))
,→ Geographical spread of infections
A simple mathematical model (ignoring the effects
due to births and migration) for the spread of some
infectious disease in time and space of a population living in a habitat Ω (a closed subset of IRn)
is given by
u(t, x) =
Z tZ
0
Ω
K(t − τ, x, ξ)G(u(τ, ξ)) dξ dτ
+f (t, x),
t ≥ 0, x ∈ Ω ,
where u(t, x) is related to the quotient of the
number S(t, x) of susceptibles at time t > 0 at
the location x ∈ Ω and the number S0(x) of susceptibles at time t = 0.
(,→ A detailed derivation of this Volterra-Fredholm
integral equation can be found in Diekmann (1978).)
46
• Model for explosion in diffusive medium
(Roberts, Lasseigne & Olmstead (1993), Roberts
(1998, 2008))
The nonlinear VIE
Z t
(1 + s)q[u(s) + 1]p
q
u(t) = γ
0
π(t − s)
ds
where γ, p, q are positive parameters, arises as
a mathematical model in steel production: formation of shear bands in steel, when subjected
to very high strain rates ⇒ huge rise in temperature u(t) .
,→ Behaviour of solution of model VIE:
(I) Finite-time blow-up:
lim u(t) = ∞
t→T−
b
for some Tb < ∞ ?
(II) Quenching (= rapid cooling) of solutions
(Roberts (2007)):
lim u(t) < ∞
t→T−
q
and
lim u0(t) = ±∞
t→T−
q
for some Tq < ∞ ?
47
• Optimal control problems involving VIEs
(Gripenberg (1983), ... )
Find w(t) on IR+ so that the functional
J[w] :=
Z ∞
0
w(s) ds
is minimized under the condition that the solution u(t) of the VIE
u(t) = w(t) +
Z t
0
k(t − s)G(u(s)) ds, t ∈ IR+,
(10)
satisfies
lim u(t) ≥ inf{β ∈ IR+ : G(β)
t→∞
Z ∞
0
k(t) dt > β} .
Application:
u(t) : flow of available resources at time t;
G(u(t)) : investments at time t (available resources are
determined by previous investments and exterior inputs
w(t) by the VIE (10)).
Z
,→ Problem: Minimize total inputs J[w] =
∞
w(s) ds
0
(returns on investments suffice for consumption and reinvestments).
(Theory/applications of optimal control problems involving VIEs: see, e.g., Corduneanu (1991))
48
VIE models: other areas of applications
• Inverse problems in viscoelasticity:
Identification (recovery) of the memory kernel
k(t − s) in the hyperbolic PDE
r(x)utt(t, x) = div (β(x)∇u(t, x))
−
Z t
0
k(t − s)div (β(x)∇u(s, x)) ds + g(t, x)
(with appropriate initial and boundary conditions, plus some additional condition).
⇒ The problem can be reduced to a Volterra
integral equation (of the first kind) for the unknown memory kernel k(t − s) .
(Janno & von Wolfersdorf (1997, 2001), ... )
• Inverse problems related to wave propagation:
(Geophysics, accoustics, electrodynamics, ... )
,→ Most inverse problems for such hyperbolic
PDEs can be reduced to Volterra (operator)
integral equations.
(See the book by Kabanikhin & Lorenzi (1999))
• Kernel identification of Volterra systems:
(Pattern recognition models, nerve networks, ... )
(Brenner, Jiang & Xu (2009))
49
• Boundary integral equations (single-layer
potential)
The boundary integral equation for the homogeneous diffusion equation (on bounded Ω ⊂ IR2
with smooth boundary Γ ), Dirichlet boundary
data g and vanishing initial data is a VolterraFredholm integral equation of the first kind (Hamina & Saranen (1994)):
Z tZ 1
0
0
E(x(θ) − x(ϕ), t − τ )u(ϕ, τ ) dϕ dτ = f (θ, t)
on IR×[0, T]. Here, x(θ) is a smooth 1-periodic
representation of Γ, f (θ, t) := gΓ(x(θ), t) and
(
E(x, t) :=
(4π t)−1 exp(−|x|2/(4t)),
0,
t>0
t ≤ 0.
,→ Convergence analysis of collocation method
(−1)
in Sm−1(Ih) (with m ≥ 2) for time-stepping in
the above Volterra-Fredholm integral equation
of the first kind?
50
• VIEs with power-law nonlinearity:
uβ (t) =
Z t
0
(t − s)−αK(t, s)u(s) ds, t ∈ I,
with β > 1, 0 ≤ α < 1, K(t, s) ≥ 0 .
,→ Existence on non-trivial solution u(t) ?
(Buckwar (1997, 2005))
,→ Analysis of collocation methods ?
Numerical approximation of non-trivial solution by collocation based on piecewise polynomials: Open problem.
(See also Lecture 7.)
51
• VFIEs with state-dependent delays:
Example:
Mathematical model of population whose life
span τ depends on the (unknown!) size of
the population (due to crowding effects) (Bélair
(1990)):
u(t) =
Z t
t−τ (u(t))
k(t − s)G(u(s)) ds,
t > 0,
with u(t) = φ(t) for t ≤ 0.
In the model by Bélair we have
k(t − s) ≡ b = const > 0 .
In more general models, the convolution kernel k(t − s)
is positive and non-increasing (,→ memory kernel).
Remark:
The numerical analysis (e.g.: convergence properties and
asymptotic behaviour of collocation solutions) and the
efficient computational solution remain to be studied.
(Current work: Brunner & Maset (2010+))
52
Basic references:
• R.K. Miller, Nonlinear Volterra Integral Equations,
W.A. Benjamin, Menlo Park, CA, 1971.
• R.K. Miller, A system of renewal equations, SIAM J.
Appl. Math., 29 (1975), 20-34.
• F. Brauer, Constant rate harvesting of populations
governed by Volterra integral equations, J. Math. Anal.
Appl., 56 (1976), 18-27.
• K.L. Cooke, An epidemic equation with immigration,
Math. Biosci., 29 (1976), 135-158.
• O. Diekmann, Thresholds and travelling waves for
the geographical spread of infection, J. Math. Biology,
6 (1978), 109-130.
• G. Gripenberg, S.-O. Londen & O. Staffans, Volterra
Integral and Functional Equations, Cambridge University
Press, Cambridge, 1990.
• C. Corduneanu, Integral Equations and Applications,
Cambridge University Press, Cambridge, 1991.
• C.A. Roberts, D.G. Lasseigne & W.E. Olmstead,
Volterra equations which model explosion in a diffusive
medium, J. Integral Equations Appl., 5 (1993), 531-546.
• F. Brauer & C. Castillo-Chávez, Mathematical Models in Population Biology and Epidemiology, SpringerVerlag, New York, 2001.
(,→ See also the handout ”References: Lecture II” for
additional papers and books on applications of Volterra
integral equations.)
53
HIT Summer Seminar: 5-16 July 2010
Lecture 3 of
Theory and numerical solution of
Volterra functional integral
equations
Hermann Brunner
Department of Mathematics and Statistics
Memorial University of Newfoundland
St. John’s, NL
Canada
Department of Mathematics
Hong Kong Baptist University
Hong Kong SAR
P.R. China
54
Lecture 3:
Basic elements of collocation methods
We want to solve the VIE
u(t) = g(t) +
Z t
0
K(t, s)u(s) ds
(11)
on the interval I := [0, T] . Let
Ih := {tn : 0 = t0 < t1 < · · · < tN = T}
a mesh (or: grid) on I , and define
en := (tn, tn+1],
hn := tn+1 − tn (0 ≤ n ≤ N −1),
and h := max{hn : 0 ≤ n ≤ N } (mesh diameter).
,→ To find: ’good’ approximation uh(t) to
the solution u(t) of the VIE (11) so that
• uh(t) is defined for all t ∈ I ;
• uh(t) can be easily computed on non-uniform
meshes Ih ;
• the approximation error satisfies
max{|u(t) − uh(t)| : t ∈ I} ≤ Chp
where p (the order of the numerical method)
is as large as possible.
55
Direct quadrature (DQ) methods for VIEs
u(t) = g(t) +
Z t
0
K(t, s)u(s) ds,
t ∈ I = [0, T] :
Let t = tn = nh (n = 1, . . . , N ) be a mesh point
(of a uniform mesh Ih) and approximate the integral by some (high-order) quadrature formula:
Z t
n
0
K(tn, s)u(s) ds ≈ h
n
X
wn,`K(tn, t`)u(t`) .
`=0
If we denote by un an approximation to the
(unknown) value u(tn), then we obtain a system of linear algebraic equations for {un} :
[1 − hwn,nK(tn, tn)]un = g(tn) − h
nX
−1
wn,`K(tn, t`)u`
`=0
( n = n0, . . . , N , for some n0 ≥ 1 and given weights wn,`
(0 ≤ n ≤ n0 )). (,→ Wolkenfelt (1982))
But:
DQ methods are in general not feasible methods:
,→ Disadvantages:
• Difficult to implement on non-uniform meshes;
• The approximations un are only defined at the mesh
points;
• Generation of quadrature weights {wn,` } so that the
DG method has high order and is stable ?
56
The collocation method
,→ Approximation of solution u(t) of VIE by
a piecewise polynomial uh(t) : For given mesh
Ih and given integer m ≥ 1 we define
(−1)
Sm
−1(Ih ) := {v : v|en ∈ Pm−1 (0 ≤ n ≤ N − 1)} ,
where Pm−1 = Pm−1(en) is the set of (real)
polynomials on en = (tn, tn+1] of degree ≤ m − 1 .
(−1)
Sm
−1(Ih ) is called the space of piecewise polynomials of degree less than or equal to m − 1.
⇒
(−1)
dim Sm−1(Ih) = Nm .
(12)
Example 1: m = 1
⇒ S(−1)
(Ih ) : piecewise constant functions.
0
(,→ such a function contains N unknown coefficients)
Example 2: m = 2
⇒ S(−1)
(Ih ) : piecewise linear functions.
1
(,→ such a function contains 2N unknown coefficients)
In general: By (12), an element uh ∈ S(−1)
m−1 (Ih ) contains
Nm unknown coefficients. ⇒ To determine these coefficients, choose Nm distinct points in the interval [0, T]
at which the approximate solution uh (t) must satisfy the
given VIE !
,→ These points are called the collocation points.
57
Collocation points and collocation equation:
Let 0 < c1 < · · · < cm ≤ 1 be given numbers
(collocation parameters). The set
Xh := {tn + cihn : i = 1, . . . m (0 ≤ n ≤ N − 1)}
is called the set of collocation points. In each
subinterval (tn, tn+1] there are m such points,
and so we have |Xh| = Nm .
(−1)
,→ Find uh ∈ Sm−1(Ih) so that it satisfies the
given VIE at the points Xh :
uh(t) = g(t) +
Z t
0
K(t, s)uh(s) ds, t ∈ Xh.
This function uh(t) is called the collocation
solution for the VIE
u(t) = g(t) +
Z t
0
K(t, s)u(s) ds, t ∈ [0, T].
After we have computed the collocation solution uh(t) we can define the iterated collocation solution uit
h (t) :
uit
h (t) := g(t) +
Z t
K(t, s)uh(s) ds, t ∈ [0, T].
0
This may be viewed as a post-processing of the collocation solution uh (t): the accuracy (order of convergence)
of uit
h (t) is often much better than that of uh (t) .
58
Remark: Different types of meshes on I = [0, T]
Ih := {tn : 0 = t0 < t1 < · · · < tN = T} (N ∈ IN).
• Quasi-uniform mesh Ih : there exists a constant γ < ∞ (independent of N ) so that
max(n) hn
min(n) hn
≤γ
for all
N ≥ 1.
( ⇒ Nh ≤ γ T )
• Graded mesh Ih :
n r
T (n = 0, 1, . . . , N ),
tn =
N
with grading exponent r > 1 .
If r = 1 then the mesh Ih is a uniform mesh.
(Prove that a graded mesh is not quasi-uniform !)
• Geometric mesh Ih :
tn = qN−nT
(n = 0, 1, . . . , N ),
where q ∈ (0, 1) .
59
General piecewise polynomial spaces:
Recall: For the given interval I := [0, T] the
mesh Ih is given by
Ih := {tn : 0 = t0 < t1 < · · · < tN = T} ,
with
en := [tn, tn+1], hn := tn+1 − tn (0 ≤ n ≤ N −1),
and h := max{hn : 0 ≤ n ≤ N − 1} .
For given integers r ≥ 1 and 0 ≤ d < r we define the general space of piecewise polynomials
of degree r by
S(rd)(Ih) := {v ∈ Cd(I) : v|en ∈ Pr (0 ≤ n ≤ N −1)} .
(13)
( d)
Thus, Sr (Ih) ⊂ Cd(I) with
(d)
dim Sr (Ih) = N(r − d) + (d + 1) .
As we shall see (e.g. in collocation methods for
ODEs and Volterra integro-differential equations (VIDEs)), an important special case of
(13) corresponds to r = m + d : the dimension
of this linear space is given by
( d)
dim Sm+d(Ih) = Nm + (d + 1) .
If the ODE or VIDE is of the form u(k)(t) = · · ·
with k ≥ 1 , then we shall choose
d = k − 1.
60
Collocation solutions for ODEs
(Recall: Mesh (or: grid) on I := [0, T]:
Ih := {tn : 0 = t0 < t1 < · · · < tN = T},
with en := [tn , tn+1], hn := tn+1 − tn ;
h := max {hn : 0 ≤ n ≤ N − 1} is called the mesh diameter.)
,→ Collocation space for first-order ODEs:
Definition: For given integer m ≥ 1 ,
S(m0)(Ih) := {v ∈ C(I) : v|en ∈ Pm (0 ≤ n ≤ N −1)}
denotes the space of globally continuous piecewise polynomials (with respect to the given
mesh Ih) of degree m.
⇒
(0)
dim Sm (Ih) = Nm + 1 .
(0)
,→ The collocation solution uh ∈ Sm (Ih) for
u0(t) = f (t, u(t)), t ∈ I, u(0) = u0,
is determined by the collocation equation
uh(t) = f (t, uh(t)), t ∈ Xh, uh(0) = u0,
where Xh is the set of collocation points:
Xh := {tn + cihn : 0 < c1 < · · · < cm ≤ 1 (0 ≤ n < N )}.
61
Collocation for ODEs:
Approximation of the solution of the initialvalue problem for
u0(t) = f (t, u(t)) (t ∈ I), with u(0) = u0,
(0)
by an element uh in the collocation space Sm (Ih) ,
satisfying the initial-condition uh(0) = u0 .
(0)
Since dim Sm (Ih) = Nm + 1 : ⇒ choose the
set of collocation points Xh given by
{tn + cihn : 0 < c1 < · · · < cm ≤ 1 (0 ≤ n ≤ N − 1)} ,
where the {ci} denote given distinct real numbers (the collocation parameters) in (0, 1] .
⇒ |Xh| = Nm .
,→ Questions:
• Computational form of the collocation equation
u0h(t) = f (t, uh(t)),
t ∈ Xh ?
• Optimal global order of convergence (on I):
ku − uh k∞ ≤ Chp : p ≤ ?
• Optimal local order of convergence (on Ih ):
∗
max{|u(t) − uh (t)| : t ∈ Ih } ≤ Chp : p∗ > p ?
• Do the above optimal orders remain true for VIEs ?
• Collocation in smoother piecewise polynomial spaces:
S(d)
m (Ih )
with
1≤d<m?
62
• Collocation equation u0h(t) = f (t, uh(t)), t ∈ Xh :
Let
Lj(v) :=
m
Y
v − ck
,
c − ck
k6=j j
v ∈ [0, 1]
(j = 1, . . . , m),
denote the Lagrange canonical polynomials with
respect to the collocation parameters {ci} .
Setting Yn,j := u0h(tn + cjhn) and
u0h(tn + vhn) =
m
X
Lj(v)Yn,j,
v ∈ (0, 1],
j=1
we obtain the local representation of the col(0)
location solution uh ∈ Sm (Ih) on the subinterval [tn, tn+1]:
uh(tn + vhn) = uh(tn) + hn
m
X
βj(v)Yn,j, v ∈ [0, 1],
j=1
Z v
with βj(v) :=
Lj(s) ds .
0
,→ Computation of {Yn,j} (0 ≤ n ≤ N − 1):

Yn,i = f

m
X

ai,jYn,j
tn + cihn, yn + hn
 (i = 1, . . . , m),
j=1
where yn := uh(tn) and ai,j := βj(ci).
63
Computational form of collocation equation:
,→ The pair of equations (for 0 ≤ n ≤ N − 1):
uh(tn + vhn) = uh(tn) + hn
m
X
βj(v)Yn,j, v ∈ [0, 1]
j=1
(local representation of the collocation solu(0)
tion uh ∈ Sm (Ih) on the subinterval [tn, tn+1]),
and


Yn,i = f
m
X

ai,jYn,j
 (i = 1, . . . , m)
tn + cihn, yn + hn
j=1
(collocation equations for t = tn + cihn )
represents an m-stage continuous implicit
Runge-Kutta method for solving the ODE
initial-value problem
u0(t) = f (t, u(t)),
t ∈ [0, T];
u(0) = u0.
For arbitrary {ci} (and u ∈ Cd(I) with d ≥ m + 1):
(j)
ku(j) − uh k∞ ≤ Chm
(j = 0, 1).
,→ Question:
Is a global order p > m possible for special
choice(s) of the collocation parameters {ci} ?
64
(0)
Convergence results: uh ∈ Sm (Ih)
Define
Jν :=
• If u
Z 1
0
sν
m
Y
(s − ci) ds
i=1
∈ Cm+2(I) and
(0 ≤ ν ≤ m − 1).
J0 = 0 :
ku − uhk∞ ≤ Chm+1.
• Let u ∈ Cm+κ+1(I) (1 ≤ κ ≤ m).
If Jν = 0, ν = 0, . . . , κ − 1, and Jκ 6= 0:
max{|u(t) − uh(t)| : t ∈ Ih} ≤ Chm+κ.
κ=m ⇒
points:
{ci} are the Gauss (-Legendre)
max{|u(t) − uh(t)| : t ∈ Ih} ≤ Ch2m,
but:
max{|u0(t) − u0h(t)| : t ∈ Ih} ≤ Chm.
,→ Why O(h2m)-convergence on Ih for uh
but not for u0h ?
,→ Other choices of collocation parameters {cj}
?
65
(0)
Illustration: uh ∈ Sm (Ih) for
u0(t) = a(t)u(t) + b(t), t ∈ I, u(0) = u0 .
,→ The collocation equation can be written as
u0h(t) = a(t)uh(t) + b(t) − δh(t), t ∈ I,
with δh(t) = 0 for all t ∈ Xh.
⇒ The collocation error eh(t) := u(t) − uh(t)
satisfies
e0h(t) = a(t)eh(t) + δh(t), t ∈ I, eh(0) = 0 .
Thus:
Z t
eh(t) =
0
r(t, s)δh(s) ds, t ∈ I ,
where r(t, s) := exp
Z t
s
!
a(z) dz .
(a) t = tn + vhn (v ∈ [0, 1]) :
eh(t) =
=
Z t
n
0
nX
−1
r(t, s)δh(s) ds +
h`
`=0
+hn
Z 1
|0
Z v
|0
Z t
tn
r(t, s)δh(s) ds
r(t, t` + sh`)δh(t` + sh`) ds
{z
}
r(t, tn + shn)δh(tn + shn) ds .
{z
}
66
(b) t = tn (1 ≤ n ≤ N ) :
eh(tn) =
nX
−1
`=0
h`
Z 1
|0
r(tn, t` + sh`)δh(t` + sh`) ds.
{z
}
,→ Connection with optimal m-point interpolatory quadrature (with quadrature abscissas chosen
to be the collocation points {t` + cjh` }) ?
Setting
fn(t` + sh`) := r(tn, t` + sh`)δh(t` + sh`) ,
we write
Z 1
fn(t` + sh`) ds =
0
m
X
wjfn(t` + cjh`) + En,` .
j=1
Here, En,` denotes the quadrature error, and we have
fn(t` + cjh`) = 0
since
δh (t` + cjh` ) = 0 .
Thus:
eh(tn) =
n−1
X
h`En,` (n = 1, . . . , N ).
`=0
67
Special sets {cj} of collocation parameters:
The optimal collocation parameters correspond
to special abscissas in m-point interpolatory quadrature formulas of the form
Q(f ) :=
Z 1
0
f (s) ds =
m
X
wm,jf (cj) +Em(f ) ,
j=1
|
{z
=:Qm(f )
}
with 0 ≤ c1 < · · · < cm ≤ 1 and quadrature weights
wm,j =
Z 1
0
Lj(s) ds
(j = 1, . . . , m).
Definition: The quadrature formula Qm(f ) has
degree of precision ≥ q if
E m (f ) = 0
for all
f ∈ Pq .
Define
Jν :=
Z 1
0
sν
m
Y
(s − ci) ds
(ν = 0, . . . , m − 1).
i=1
Lemma 3.1: (Optimal degree of precision)
(a) The degree of precision of an m-point interpolatory quadrature formula satisfies q ≥ m − 1 .
(b) The quadrature formula Qm(f ) has (exact)
degree of precision m + κ (0 ≤ κ ≤ m − 1) if
and only if
Jν = 0 for ν = 0, . . . , κ − 1
and
Jκ 6= 0 .
68
,→ Recall:
Degree of precision is m + κ (1 ≤ κ ≤ m) if,
for ν = 0, . . . , κ − 1,
Jν :=
Z 1
0
sν
m
Y
(s − ci) ds = 0 .
i=1
Example 1: Gauss (-Legendre) points (κ = m)
,→ The {ci} are the zeros of Pm(2s − 1) (shifted Legendre polynomial of degree m).
m = 1 : c1 = 1/2
m = 2 : c1 = ( 3 −
m = 3 : c1 = ( 5 −
√
√
3)/6, c2 = (3 +
√
3)/6 .
15)/10, c2 = 1/2, c3 = (5 +
√
15)/10 .
Example 2a: Radau I points (κ = m − 1; c1 = 0)
,→ The {ci} are the zeros of Pm(2s − 1) + Pm−1(2s − 1).
m = 2 : c1 = 0, c2 = 2/3
m = 3 : c1 = 0, c2 = (6 −
√
6)/10, c3 = (6 +
√
6)/10 .
Example 2b: Radau II points (κ = m − 1; cm = 1)
,→ The {ci} are the zeros of Pm(2s − 1) − Pm−1(2s − 1).
m = 2 : c1 = 1/3, c2 = 1
√
√
m = 3 : c1 = (4 − 6)/10, c2 = (4 + 6)/10, c3 = 1 .
Example 3: Lobatto points (κ = m − 2,
c1 = 0, cm = 1 (m ≥ 2))
,→ The {ci} are the zeros of s(s − 1)P0m−1(2s − 1) .
m = 3 : c1 = 0, c2 = 1/2, c3 = 1 .
69
Superconvergence on I and Ih
(0)
Let uh ∈ Sm (Ih) be the collocation solution
for the initial-value problem
u0(t) = f (t, u(t)), t ∈ I, u(0) = u0 ,
with respect to given collocation parameters
{ci : 0 < c1 < · · · < cm ≤ 1}.
Theorem 3.2:
(a) If u ∈ Cd(I) (d ≥ m + 2) and
J0 :=
Z 1 Y
m
(s − ci) ds = 0,
0 i=1
then
ku − uhk∞ ≤ Chm+1 .
(b) Let 1 ≤ κ ≤ m . If u ∈ Cd(I) (d ≥ m+κ+1)
and
Jν :=
Z 1
0
sν
m
Y
(s − ci) ds = 0,
ν = 0, . . . , κ − 1,
i=1
then
max |u(tn) − uh(tn)| ≤ Chm+κ .
1≤n≤N
70
Corollary 3.3:
(a) If the {ci} are the Gauss (-Legendre) points
in (0, 1) (note that cm < 1 ), then
max |u(tn) − uh(tn)| ≤ Ch2m .
1≤n≤N
(b) If the {ci} are the Radau I points in [0, 1)
or the Radau II points in (0, 1], then
max |u(tn) − uh(tn)| ≤ Ch2m−1 .
1≤n≤N
The resulting numerical ODE schemes are, respectively, the continuous m-stage implicit RungeKutta-Gauss method and the continuous mstage implicit Runge-Kutta-Radau I/II methods.
Remarks:
• The m-stage continuous implicit Runge-Kutta
methods of orders 2m and 2m − 1 are collo(0)
cation methods in Sm (Ih). (Guillou & Soulé
(1969); see also Butcher (1964, 1965))
• But: Not all continuous implicit (or explicit)
Runge-Kutta methods are collocation methods
,→ Framework of perturbed collocation methods to
include all Runge-Kutta methods for ODEs: Nørsett &
Wanner (1981).
71
Question:
Consider an m-stage implicit Runge-Kutta method
of order p ≥ m, and assume that the RungeKutta abscissas {ci} satisfy ci 6= cj (i 6= j) .
When is such a method a collocation method?
Theorem:
An implicit m-stage Runge-Kutta method of
order p ≥ m and distinct {ci} is equivalent to
(0)
a collocation method in Sm (Ih) if, and only
if,
m
X
ν
c
aijcjν−1 = i , ν = 1, . . . , m (i = 1, . . . , m).
ν
j=1
(The above condition is known as (order) condition C(m)
in the Runge-Kutta theory; see Hairer, Nørsett & Wanner, Solving Ordinary Differential Equations I, SpringerVerlag, 1993, p. 212.)
72
ODEs: Collocation in smoother piecewise polynomial spaces ?
(m−1)
• uh ∈ Sm
(Ih) (d = m − 1):
,→ uh is divergent (as h → 0) when m ≥ 4 !
(Loscalzo & Talbot (1967))
(2)
• uh ∈ S4 (Ih) , 0 < c1 < c2 = 1:
uh is divergent if
1 − c1
>1
c1
(or:
c1 < 1/2).
(2)
• uh ∈ Sm (Ih) (m ≥ 4) :
uh is divergent if the {ci} are the Radau II
points.
(Complete convergence / divergence analysis for ODEs:
Mülthei (1979); see also Brunner (BIT, 2004))
Observation:
The natural (and optimal) piecewise polynomial spaces for (first-order) ODEs are the spaces
S(m0)(Ih) with m ≥ 1.
For VIEs (Lecture 4) and VFIEs (Lecture 6: Volterra
functional integral equations), the natural colloca(−1)
tion spaces are Sm−1(Ih) .
73
Basic references:
• P.H.M. Wolkenfelt, The construction of reducible
quadrature rules for Volterra integral and integro-differential
equations, IMA J. Numer. Anal., 2 (1982), 131-152.
• S.P. Nørsett & G. Wanner, Perturbed collocation
and Runge-Kutta methods, Numer. Math., 38 (1981),
193-208.
• H. Brunner & P.J. van der Houwen, The Numerical
Solution of Volterra Equations, CWI Monographs, Vol.
3, North-Holland, Amsterdam, 1986.
• J.C. Butcher, The Numerical Analysis of Ordinary
Differential Equations: Runge-Kutta and General Linear
Methods, Wiley, Chichester, 1987.
• E. Hairer, S.P. Nørsett & G. Wanner, Solving
Differential Equations I: Nonstiff Problems (2nd ed.),
Springer-Verlag, Berlin, 1993.
• H. Brunner, On the divergence of collocation in
smooth piecewise polynomial spaces for Volterra integral
equations, BIT, 44 (2004), 631-650.
(,→ See also the handout ”References: Lecture III” for
additional papers and books on numerical quadrature,
collocation and Runge-Kutta methods for ODEs, and
quadrature methods for VIEs.)
74