HIT Summer Seminar: 5-16 July 2010

Seven lectures on

Theory and numerical solution ofVolterra functional integral

equations

Hermann Brunner

Department of Mathematics and StatisticsMemorial University of Newfoundland

St. John’s, NL

Canada

Department of MathematicsHong 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 equa-

tions and applications

• Lecture 3: Basic elements of collocation meth-

ods

• 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) +∫ t

0K(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∫ t

0K(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) +∫ t

0K(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

(Vu)(t) :=∫ t

0K(t, s)u(s)ds, t ∈ I ,

with K ∈ C(D) .

• The weakly singular Volterra integral opera-

tor Vα : C(I) → C(I) has the form

(Vαu)(t) :=∫ 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 opera-

tor corresponding to a logarithmic singularity,

V1 : C(I) → C(I) is given by

(V1u)(t) :=∫ t

0log(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

u′(t) = a(t)u(t) + b(t) + (Vαu)(t), 0 ≤ α ≤ 1 ,

is called a linear Volterra integro-differential equa-

tion. 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 ):

u(k)(t) = a0(t)u(t) +k−1∑j=1

aj(t)u(j)(t) + b(t)

+k∑

ν=0

(V(ν)α u(ν))(t) ,

with Volterra integral operators V(ν)α defined

by

(V(ν)α u(ν))(t) :=

∫ t

0(t− s)−αKν(t, s)u

(ν)(s)ds

(0 ≤ α < 1).

5

Some history:

Vito VOLTERRA (1860-1940) was a very fa-

mous Italian mathematician. His papers on in-

tegral equations (which are now called Volterra

integral equations) appeared in 1896, and they

– together with the papers of the equally fa-

mous Swedish mathematician Ivar Fredholm

– also mark the beginning of Functional Anal-

ysis.

Ivar FREDHOLM (1866-1927) wrote his cele-

brated papers on what are now known as Fred-

holm 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

u′(t) = a(t)u(t) + b(t), t ∈ I := [0,T]; u(0) = u0,

is equivalent to the second-kind Volterra inte-

gral equation

u(t) = u0 +∫ t

0b(s)ds︸ ︷︷ ︸

=g(t)

+∫ t

0a(s)︸ ︷︷ ︸

=K(t,s)

u(s)ds .

Here, the kernel K(t, s) does not depend on t !

• The second-order initial-value problem

u′′(t) = a(t)u(t) + b(t), u(0) = u0, u′(0) = v0,

is equivalent to a second-kind Volterra integral

equation whose kernel K(t, s) now does de-

pend on t :

u(t) = g(t) +∫ t

0K(t, s)︸ ︷︷ ︸u(s)ds, t ∈ I ,

where

g(t) := u0 + v0t +∫ 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) +∫ t

0K(t, s)u(s)ds ,

is in general not equivalent to an initial-value

problem for an ordinary differential equation,

since

d

dt

(∫ t

0K(t, s)u(s)ds

)

= K(t, t)u(t) +∫ t

0

∂K(t, s)

∂tu(s)ds ,

where, in general,

∂K(t, s)

∂t6≡ 0 .

Thus,

u′(t) = g′(t) + K(t, t)u(t) +∫ t

0

∂K(t, s)

∂tu(s)ds︸ ︷︷ ︸

=: a(t)u(t) + b(t) +∫ t

0H(t, s)u(s)ds ,

with u(0) = g(0) . This is an initial-value prob-

lem for a Volterra integro-differential equation.

8

Remark: Fredholm integral equations

In a Fredholm integral equation the limits of in-

tegration are fixed numbers (given by the end-

points of the interval of integration):

u(t) = g(t) +∫ T

0K(t, s)u(s)ds, t ∈ [0,T] .

Fredholm integral equations are related to boundary-

value problems for differential equations.

Example: The boundary-value problem

u′′(t) = a(t)u(t) + b(t), u(0) = A, u(T) = B,

is equivalent to the Fredholm integral equation

u(t) = g(t) +∫ T

0G(t, s)a(s)︸ ︷︷ ︸

=K(t,s)

u(s)ds, t ∈ [0,T] ,

with

g(t) := A +(B−A)t

T+

∫ T

0

G(t, s)b(s)ds

and

G(t, s) :=

− s

T(T− t) if s ≤ t

− tT(T− s) if 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) +∫ t

0K(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) +∫ t

0R(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 ker-

nel K(t, s) .

10

Remark:

→ Recall: The (unique) solution of the VIE

u(t) = g(t) +∫ t

0K(t, s)u(s)ds, t ∈ I := [0,T],

with g ∈ C(I), K ∈ C(D) is given by

u(t) = g(t) +∫ t

0R(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) :=∫ t

0R(t, s)g(s)ds, t ∈ I ,

and if we write the VIE in operator form,

u = g + Vu, 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) +∫ t

0K(t, s)u(s)ds, t ∈ I . (1)

Let u0(t) := g(t) and define an infinite se-

quence of functions uk(t)k≥1 by

uk(t) := g(t) +∫ t

0K(t, s)uk−1(s)ds, t ∈ I

(Picard iteration).

Thus:

u1(t) = g(t) +∫ t

0K(t, s)g(s)ds .

We can show (using mathematical induction) that

for any k ≥ 1 ,

uk(t) = g(t) +∫ t

0

k∑j=1

Kj(t, s)︸ ︷︷ ︸g(s)ds, t ∈ I ,

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) :=∫ t

sK(t,v)Kj−1(v, s)dv (j ≥ 2).

→ limk→∞

uk(t) = ?

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∞∑j=1

Kj(t, s) is called the Neumann series, and

R(t, s) := limk→∞

k∑j=1

Kj(t, s) ((t, s) ∈ D)

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:

limk→∞

uk(t) = z(t), 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

u(t) = g(t) +

∫ t

0

K(t, s)u(s)ds, t ∈ [0,T] .

(b) Show that this is the only solution: z(t) = u(t) .

Note: If u and w are two solutions, then

|u(t)−w(t)| ≤∫ t

0

|K(t, s)||u(s)− z(s)|ds, t ∈ I .

→ Gronwall inequality / comparison theorem !

13

Corollary 1.2:

Assume that the given functions in

u(t) = g(t) +∫ t

0K(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) +∫ 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). Then for any given

g ∈ C(I) the VIE

u(t) = g(t) +∫ t

0k(t− s)u(s)ds, t ∈ I,

possesses a unique solution given by

u(t) = g(t) +∫ t

0r(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 inte-

gral inequality

z(t) ≤ a(t) +∫ t

0b(s)z(s)ds, t ∈ I .

Then

z(t) ≤ a(t) exp

(∫ t

0b(s)ds

)for all t ∈ I .

If a(t) = α = constant and b(t) = β = con-

stant (> 0 ), then we obtain Gronwall’s original

lemma (1919): if

z(t) ≤ α + β∫ t

0z(s)ds, t ∈ I ,

then

z(t) ≤ α exp(βt), 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) +∫ t

0K(t, s)z(s)ds, t ∈ I ,

then

z(t) ≤ g(t) +∫ t

0R(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) +∫ t

0Ki(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 equa-tions in abstract settings (e.g. in Banach spaces) hasreceived increasing attention. Here are some of the keybooks and papers (see also References / Lecture I):

• A. Friedman & M. Shinbrot, Volterra integral equa-

tions 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 equa-

tions 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. Dif-

ferential 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. Pruss, Evolutionary Integral Equations and Appli-

cations, Birkhauser Verlag, Basel-Boston, 1993.

• M. Vath, Abstract Volterra equations of the second

kind, J. Integral Equations Appl., 10 (1998), 319–362.

• M. Vath, 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 con-

stant λ the linear Volterra integral equation of

the second kind,

u(t) = g(t) + λ∫ t

0K(t, s)u(s)ds, t ∈ [0,T] ,

with continuous g and K, has a unique con-

tinuous solution.

This is in general not true for a linear Fredholm

integral equation of the second kind,

u(t) = g(t) + λ∫ T

0K(t, s)u(s)ds, t ∈ [0,T] .

(2)

→ A (real or complex) value of µ for which

(Fφ)(t) :=∫ T

0K(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 in-

tegral 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

u(t,x) = g(t,x) + λ∫ t

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 equa-

tion.

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 pos-

sesses a unique solution u ∈ C(I×Ω) .

This solution is given by

u(t,x) = g(t,x) +∫ t

0

∫Ω

R(t, s,x, ξ)g(s, ξ)dξ ds ,

where the resolvent kernel R(t, s,x, ξ) is a con-

tinuous 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∫ t

0H(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) +∫ t

0

∂H(t, s)

∂tu(s)ds = f ′(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:

u(t) =f ′(t)

H(t, t)︸ ︷︷ ︸=g(t)

+∫ t

0

−1

H(t, t)

∂H(t, s)

∂t︸ ︷︷ ︸=K(t,s)

u(s)ds .

⇒ Under the above conditions on f and the

kernel H the first-kind VIE has a unique con-

tinuous 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] necessaryfor the existence of a unique solution of∫ t

0

H(t, s)u(s)ds = f(t) ?

(b) Consider the first-kind Volterra integral equation∫ t

0

(t− s)k−1

(k− 1)!︸ ︷︷ ︸=H(t,s)

u(s)ds = f(t), t ∈ [0,T] ,

where k is an integer with k ≥ 1, and f(t) has contin-uous derivatives of at least order k .Does this VIE possess a unique (continuous) solution ?(c) Does the VIE∫ t

0

(2t− 3s)u(s)ds = t2, t ∈ [0,T] ,

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) +∫ t

0(t− s)−αK(t, s)︸ ︷︷ ︸

=:Kα(t,s)

u(s)ds, 0 < α < 1,

where K(t, s) is continuous on D and satis-

fies 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) +∫ t

0log(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 mathe-

matician 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∫ t

0(t− s)−αu(s)ds, t ∈ I . (3)

Definition: (Mittag-Leffler function)

Let β > 0 and z ∈ C. The function

Eβ(z) :=∞∑

j=0

zj

Γ(1 + jβ)

is called the Mittag-Leffler function.

Remark: The Swedish mathematician Gosta Mittag-

Leffler (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 iter-

ation. 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) +∫ t

0Kα(t, s)u(s)ds, t ∈ I,

possesses a unique solution u ∈ C(I) . This so-

lution has the representation

u(t) = g(t) +∫ t

0Rα(t, s)g(s)ds, t ∈ I ,

where the resolvent kernel Rα(t, s) of the ker-

nel 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∈ C′(I) : as t → 0+ the solu-

tion behaves like

u′(t) ∼ Ct−α .

25

Proof of Theorem 1.9:In analogy to the proof of Theorem 1.1 (α = 0 ) we usePicard iteration: setting u0(t) := g(t) we define an in-finite sequence of functions uk(t)k≥1 by

uk(t) := g(t) +

∫ t

0

(t− s)−αK(t, s)uk−1(s)ds, t ∈ I .

Here, the resulting iterated kernels Kα,j(t, s) ofKα(t, s) := (t− s)−αK(t, s) are defined by Kα,1(t, s) := Kα(t, s)and

Kα,j(t, s) :=

∫ t

s

Kα(t,v)Kα,j−1(v, s)dv (j ≥ 2; (t, s) ∈ D).

For example,

Kα,2(t, s) =

∫ t

s

(t− v)−α(v − s)−αK(t,v)K(v, s)dv .

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-decreasingon I .(b) The function z ∈ C(I) satisfies the inequality

z(t) ≤ g(t) + λ

∫ t

0

(t− s)−αz(s)ds, t ∈ I ,

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) ≤ g(t) + λ

∫ t

0

(t− s)−αz(s)ds, t ∈ I ,

with λ > 0 and 0 < α < 1 .(b) State and prove the analogue of Lemma 1.10 for theintegral inequality

z(t) ≤ g(t) + λ

∫ t

0

log(t− s)z(s)ds, t ∈ I .

Exercise 1.7:Analyze the regularity of the solution of the VIE

u(t) = tβ + λ

∫ t

0

(t− s)−αu(s)ds, t ∈ I ,

when β > 0, β 6∈ IN and 0 < α < 1 .

27

• Linear Volterra integro-differential equa-

tions (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

u′(t) = a(t)u(t) + b(t) +∫ t

0K(t, s)u(s)ds, t ∈ I,

has a unique solution u ∈ C1(I) satisfying u(0) = u0.

This solution is given by

u(t) = r(t, 0)u0 +∫ t

0r(t, s)b(s)ds, 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: inte-

grate 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,

u′(t) = a(t)u(t) + b(t) +∫ 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 equa-tion possesses a unique solution u ∈ C1(I) sat-isfying u(0) = u0 .(b) If a,b ∈ Cd(I) and K ∈ Cd(D) (for anyd ≥ 1 ), then

u ∈ C(I) ∩Cd+1(0,T],

with

u′′(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

u′(t) = g(t) + λ

∫ t

0

(t− s)−αu(s)ds, 0 < α < 1 ,

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) :=∫ 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 op-

erator.)

This is in general not true if Vα is replaced by

(Ap,αu)(t) :=∫ t

0(tp − sp)−αK(t, s)u(s)ds

with p > 1, α ∈ (0,1) .

Example: The generalized Abel integral oper-

ator (from C[0, 1] → C[0, 1]),

(A2,1/2u)(t) =∫ 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 = λβtβ, 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 (inItalian) [On the invertibility of definite integrals], Atti R.Accad. Sci. Torino, 31 (1896), 311-323.

• V. Volterra, Theory of Functionals and of Integraland 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, VolterraIntegral and Functional Equations, Cambridge UniversityPress, Cambridge, 1990.[Most comprehensive monograph on linear and nonlinearVIEs]

• C. Corduneanu, Integral Equations and Applications,Cambridge University Press, Cambridge, 1991.

• H. Brunner, Collocation Methods for Volterra Inte-gral and Related Functional Differential Equations, Cam-bridge University Press, Cambridge, 2004. (Chapters 2and 6)

(→ See also the handout ”References: Lecture I ” foradditional papers and books on the theory and applica-tions of Volterra integral equations.)

31

HIT Summer Seminar: 5-16 July 2010

Lecture 2 of

Theory and numerical solution ofVolterra functional integral

equations

Hermann Brunner

Department of Mathematics and StatisticsMemorial University of Newfoundland

St. John’s, NL

Canada

Department of MathematicsHong 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) +∫ t

0(t− s)−αk(t, s,u(s))ds (0 ≤ α < 1) .

• Volterra-Hammerstein integral equation:

u(t) = g(t) +∫ t

0(t− s)−αK(t, s)G(s,u(s))ds

(0 ≤ α < 1) ).

• Implicit Volterra integral equation:

F(u(t)) = g(t) +∫ t

0k(t, s,u(s))ds (0 ≤ α < 1) .

Example: G(s,u) = up :

u(t) = g(t) +∫ t

0k(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) +∫ t

0K(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

u′(t) = F(t,u(t)), t ∈ [0,T]; u(0) = u0,

is equivalent to the nonlinear VIE

u(t) = u0 +∫ t

0F(s,u(s))ds, t ∈ [0,T] :

here, K(t, s) = 1 for all (t, s) ∈ D .

Example 2:

The second-order differential equation

u′′(t) = F(t,u(t)), u(0) = u0, u′(0) = v0,

is equivalent to the nonlinear VIE

u(t) = u0 + v0t +∫ 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 or-

dinary differential equations (ODEs) the solu-

tion of a nonlinear VIE may not be unique, or it

exists only on some subinterval [0, t) with t < T .

Recall that when proving the (local) existence

and uniqueness of a solution to the initial-value

problem for an ODE,

u′(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 +∫ t

0f(s,u(s))ds, t ≥ 0,

The general nonlinear VIE

u(t) = g(t) +∫ t

0k(t, s,u(s))ds, t ≥ 0 ,

can be treated in the same way, and thus Theo-

rem 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) +∫ t

0

r∑j=1

Aj(t)bj(s,u(s))ds , (4)

where Aj = Aj(t) and bj = bj(t, z) are contin-uous functions.(A kernel of the form

k(t, s, z) =r∑

j=1

Aj(t)bj(s, z)

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 be-

low), the local existence and uniqueness of its

solution can be established by using ODE the-

ory.

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,

u′(t) = λu(t) + εup(t) (t ≥ 0), u(0) = u0 ,

is equivalent to the semi-linear VIE

u(t) = u0 +∫ 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

limt→T−b

u(t) = +∞ (6)

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 isgiven by

Tb =1

λ(p− 1)ln

1 +λ

εup−10

.

Proof: Since the above ODE is a Bernoulli differentialequation, its exact solution is easily found; it is

u(t) =

(1

u1−p0 e−λ(p−1)t − (ε/λ)[1− e−λ(p−1)t]

)1/(p−1)

.

If there exists a finite t = Tb > 0 so that the denomi-nator becomes zero, then the solution exists only in theinterval [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, z)| ≤ LB|z− z|

for all (t, s, z), (t, s, z) ∈ ΩB .

Then the nonlinear VIE

u(t) = g(t) +∫ t

0k(t, s,u(s))ds, t ≥ 0 ,

possesses a unique solution u ∈ C(I0) where

I0 := [0, δ0], δ0 := minT,B/MB .

(A detailed proof can also be found in the book by Brun-

ner (2004), Section 2.1.5.)

38

Exercise 2.2:(a) Let p > 1 and u0 > 0. Does the solution of thenonlinear VIE

u(t) = u0 +

∫ t

0

(t− s)up(s)ds, t ≥ 0,

blow up in finite time?(b) (hard!) Answer (a) for the VIE with weakly singularkernel,

u(t) = u0 +

∫ t

0

(t− s)−αup(s)ds,

where 0 < α < 1 .

Exercise 2.3: Extend Theorem 2.2 to nonlinear VIEswith weakly singular kernels:

u(t) =

∫ t

0

(t− s)−αk(t, s,u(s))ds, 0 < α < 1 ,

where k(t, s,u) = K(t, s)G(u) .

39

Volterra-Hammerstein integral equations (VHIEs):

u(t) = g(t) +∫ t

0K(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) +

∫ t

0K(t, s)z(s)ds

), t ≥ 0,

(8)

and

u(t) = g(t) +∫ t

0K(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) +∫ t

0K(t, s)G(u(t),u(s))ds, t ∈ I .

(Zhang Ran, Guan Qingguang & Zou Yongkui (2010))

• Auto-convolution VIEs:

u(t) = g(t) +∫ t

0K(t, s)G(u(t− s),u(s))ds .

(von Wolfersdorf & Janno (1995), Berg & von Wolfers-

dorf (2005))

Exercise 2.4: Does the auto-convolution VIE

u(t) = g(t) +

∫ t

0

u(t− s)u(s)ds, t ∈ I,

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) +∫ t

0k(t− s)u(s)ds, t ≥ 0

(where the kernel K(t, s) = k(t− s) is a convo-

lution 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 in-

vestment 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: limt→∞

u(t) = ?

(Feller (1941), ... , Miller (1975), ... ;

Diekmann, Gyllenberg & Thieme (1991) [Abstract

framework])

42

• Population growth models (I):(Brauer (1975), Brauer & Castillo-Chavez (2001) )

u(t) = g(t) +∫ t

0P(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 popu-lation size, and with a probability of death thatdepends only on age.Here, G(u) is the number of members added to the pop-

ulation (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 timerate, the resulting mathematical model is theVIE

u(t) = g(t) +∫ t

0P(t− s)G)u(s))ds−Φ(t), t ≥ 0 ,

where Φ(t) represents the number of membersof 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(∞) := limt→∞

Φ(t) exists.)

43

• Population growth models (II):

(Cooke & Yorke (1973), Cooke (1976), Smith (1977),

Torrejon (1990), ... )

→ Mathematical models of single-species pop-

ulation growth with immigration and given

age distribution:

u(t) =∫ t

t−τP(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 Torrejon (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)−

∫ t

0a(t− s)u(s)ds

)

×(g(t) +

∫ t

0b(t− s)u(s)ds

), t ≥ 0.

Here, u(t) is the rate at which individuals that

are susceptible to the disease have become in-

fected up to time t, and c > 0 is a given con-

stant.

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 popu-

lation living in a habitat Ω (a closed subset of IRn)

is given by

u(t,x) =∫ t

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 sus-

ceptibles 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

u(t) = γ∫ t

0

(1 + s)q[u(s) + 1]p√π(t− s)

ds

where γ,p,q are positive parameters, arises as

a mathematical model in steel production: for-

mation of shear bands in steel, when subjected

to very high strain rates ⇒ huge rise in tem-

perature u(t) .

→ Behaviour of solution of model VIE:

(I) Finite-time blow-up:

limt→T−b

u(t) = ∞ for some Tb < ∞ ?

(II) Quenching (= rapid cooling) of solutions

(Roberts (2007)):

limt→T−q

u(t) < ∞ and limt→T−q

u′(t) = ±∞

for some Tq < ∞ ?

47

• Optimal control problems involving VIEs

(Gripenberg (1983), ... )

Find w(t) on IR+ so that the functional

J[w] :=∫ ∞0

w(s)ds

is minimized under the condition that the solu-

tion u(t) of the VIE

u(t) = w(t) +∫ t

0k(t− s)G(u(s))ds, t ∈ IR+,

(10)

satisfies

limt→∞

u(t) ≥ infβ ∈ IR+ : G(β)∫ ∞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)).

→ Problem: Minimize total inputs J[w] =

∫ ∞

0w(s) ds

(returns on investments suffice for consumption and re-

investments).

(Theory/applications of optimal control problems involv-

ing VIEs: see, e.g., Corduneanu (1991))

48

VIE models: other areas of applications

• Inverse problems in viscoelasticity:Identification (recovery) of the memory kernelk(t− s) in the hyperbolic PDE

r(x)utt(t,x) = div (β(x)∇u(t,x))

−∫ t

0k(t− s)div (β(x)∇u(s,x))ds + g(t,x)

(with appropriate initial and boundary condi-tions, plus some additional condition).⇒ The problem can be reduced to a Volterraintegral equation (of the first kind) for the un-known memory kernel k(t− s) .(Janno & von Wolfersdorf (1997, 2001), ... )

• Inverse problems related to wave propa-gation:(Geophysics, accoustics, electrodynamics, ... )→ Most inverse problems for such hyperbolicPDEs 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 homo-

geneous diffusion equation (on bounded Ω ⊂ IR2

with smooth boundary Γ ), Dirichlet boundary

data g and vanishing initial data is a Volterra-

Fredholm integral equation of the first kind (Ham-

ina & Saranen (1994)):∫ t

0

∫ 1

0E(x(θ)− x(ϕ), t− τ)u(ϕ, τ)dϕdτ = 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) :=

(4πt)−1 exp(−|x|2/(4t)), t > 00, t ≤ 0.

→ Convergence analysis of collocation method

in S(−1)m−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) =∫ 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 collo-

cation 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) (Belair

(1990)):

u(t) =∫ 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 Belair 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 populationsgoverned 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 forthe geographical spread of infection, J. Math. Biology,6 (1978), 109-130.

• G. Gripenberg, S.-O. Londen & O. Staffans, VolterraIntegral and Functional Equations, Cambridge UniversityPress, 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 diffusivemedium, J. Integral Equations Appl., 5 (1993), 531-546.

• F. Brauer & C. Castillo-Chavez, Mathematical Mod-els in Population Biology and Epidemiology, Springer-Verlag, 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 ofVolterra functional integral

equations

Hermann Brunner

Department of Mathematics and StatisticsMemorial University of Newfoundland

St. John’s, NL

Canada

Department of MathematicsHong 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) +∫ t

0K(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 := maxhn : 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) +∫ t

0K(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 inte-

gral by some (high-order) quadrature formula:∫ tn

0K(tn, s)u(s)ds ≈ h

n∑`=0

wn,`K(tn, t`)u(t`) .

If we denote by un an approximation to the

(unknown) value u(tn), then we obtain a sys-

tem of linear algebraic equations for un :

[1− hwn,nK(tn, tn)]un = g(tn)− hn−1∑`=0

wn,`K(tn, t`)u`

(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

S(−1)m−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 .

S(−1)m−1(Ih) is called the space of piecewise poly-

nomials of degree less than or equal to m− 1.

⇒ dimS(−1)m−1(Ih) = Nm . (12)

Example 1: m = 1

⇒ S(−1)0 (Ih) : piecewise constant functions.

(→ such a function contains N unknown coefficients)

Example 2: m = 2

⇒ S(−1)1 (Ih) : piecewise linear functions.

(→ 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 coef-

ficients, 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 .

→ Find uh ∈ S(−1)m−1(Ih) so that it satisfies the

given VIE at the points Xh :

uh(t) = g(t) +∫ t

0K(t, s)uh(s)ds, t ∈ Xh.

This function uh(t) is called the collocation

solution for the VIE

u(t) = g(t) +∫ t

0K(t, s)u(s)ds, t ∈ [0,T].

After we have computed the collocation solu-

tion uh(t) we can define the iterated colloca-

tion solution uith(t) :

uith(t) := g(t) +

∫ t

0K(t, s)uh(s)ds, t ∈ [0,T].

This may be viewed as a post-processing of the colloca-

tion solution uh(t): the accuracy (order of convergence)

of uith(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 con-

stant γ < ∞ (independent of N ) so that

max(n) hn

min(n) hn≤ γ for all N ≥ 1 .

(⇒ Nh ≤ γT )

• Graded mesh Ih :

tn =(

n

N

)rT (n = 0,1, . . . , 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] themesh 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 := maxhn : 0 ≤ n ≤ N − 1 .For given integers r ≥ 1 and 0 ≤ d < r we de-fine the general space of piecewise polynomialsof degree r by

S(d)r (Ih) := v ∈ Cd(I) : v|en ∈ Pr (0 ≤ n ≤ N−1) .

(13)Thus, S(d)

r (Ih) ⊂ Cd(I) with

dimS(d)r (Ih) = N(r− d) + (d + 1) .

As we shall see (e.g. in collocation methods forODEs and Volterra integro-differential equa-tions (VIDEs)), an important special case of(13) corresponds to r = m + d : the dimensionof this linear space is given by

dimS(d)m+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 diame-

ter.)

→ Collocation space for first-order ODEs:

Definition: For given integer m ≥ 1 ,

S(0)m (Ih) := v ∈ C(I) : v|en ∈ Pm (0 ≤ n ≤ N−1)

denotes the space of globally continuous piece-

wise polynomials (with respect to the given

mesh Ih) of degree m.

⇒ dim S(0)m (Ih) = Nm + 1 .

→ The collocation solution uh ∈ S(0)m (Ih) for

u′(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 initial-

value problem for

u′(t) = f(t,u(t)) (t ∈ I), with u(0) = u0,

by an element uh in the collocation space S(0)m (Ih) ,

satisfying the initial-condition uh(0) = u0 .

Since dimS(0)m (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 num-

bers (the collocation parameters) in (0, 1] .

⇒ |Xh| = Nm .

→ Questions:• Computational form of the collocation equation

u′h(t) = f(t,uh(t)), t ∈ Xh ?

• Optimal global order of convergence (on I):

‖u− uh‖∞ ≤ 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 u′h(t) = f(t,uh(t)), t ∈ Xh :

Let

Lj(v) :=m∏

k 6=j

v − ck

cj − ck, v ∈ [0, 1] (j = 1, . . . , m),

denote the Lagrange canonical polynomials with

respect to the collocation parameters ci .

Setting Yn,j := u′h(tn + cjhn) and

u′h(tn + vhn) =m∑

j=1

Lj(v)Yn,j, v ∈ (0, 1],

we obtain the local representation of the col-

location solution uh ∈ S(0)m (Ih) on the subinter-

val [tn, tn+1]:

uh(tn + vhn) = uh(tn) + hn

m∑j=1

βj(v)Yn,j, v ∈ [0, 1],

with βj(v) :=∫ v

0Lj(s)ds .

→ Computation of Yn,j (0 ≤ n ≤ N − 1):

Yn,i = f

tn + cihn,yn + hn

m∑j=1

ai,jYn,j

(i = 1, . . . , m),

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∑j=1

βj(v)Yn,j, v ∈ [0, 1]

(local representation of the collocation solu-

tion uh ∈ S(0)m (Ih) on the subinterval [tn, tn+1]),

and

Yn,i = f

tn + cihn,yn + hn

m∑j=1

ai,jYn,j

(i = 1, . . . , m)

(collocation equations for t = tn + cihn )

represents an m-stage continuous implicit

Runge-Kutta method for solving the ODE

initial-value problem

u′(t) = f(t,u(t)), t ∈ [0,T]; u(0) = u0.

For arbitrary ci (and u ∈ Cd(I) with d ≥ m + 1):

‖u(j) − u(j)h ‖∞ ≤ Chm (j = 0,1).

→ Question:

Is a global order p > m possible for special

choice(s) of the collocation parameters ci ?

64

Convergence results: uh ∈ S(0)m (Ih)

Define

Jν :=∫ 1

0sν

m∏i=1

(s− ci)ds (0 ≤ ν ≤ m− 1).

• If u ∈ Cm+2(I) and J0 = 0 :

‖u− uh‖∞ ≤ 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 ⇒ ci are the Gauss (-Legendre)

points:

max|u(t)− uh(t)| : t ∈ Ih ≤ Ch2m,

but:

max|u′(t)− u′h(t)| : t ∈ Ih ≤ Chm.

→ Why O(h2m)-convergence on Ih for uh

but not for u′h ?

→ Other choices of collocation parameters cj?

65

Illustration: uh ∈ S(0)m (Ih) for

u′(t) = a(t)u(t) + b(t), t ∈ I, u(0) = u0 .

→ The collocation equation can be written as

u′h(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

e′h(t) = a(t)eh(t) + δh(t), t ∈ I, eh(0) = 0 .

Thus:

eh(t) =∫ t

0r(t, s)δh(s)ds, t ∈ I ,

where r(t, s) := exp

(∫ t

sa(z)dz

).

(a) t = tn + vhn (v ∈ [0,1]) :

eh(t) =∫ tn

0r(t, s)δh(s)ds +

∫ t

tnr(t, s)δh(s)ds

=n−1∑`=0

h`

∫ 1

0r(t, t` + sh`)δh(t` + sh`)ds︸ ︷︷ ︸

+hn

∫ v

0r(t, tn + shn)δh(tn + shn)ds︸ ︷︷ ︸ .

66

(b) t = tn (1 ≤ n ≤ N) :

eh(tn) =n−1∑`=0

h`

∫ 1

0r(tn, t` + sh`)δh(t` + sh`)ds︸ ︷︷ ︸.

→ Connection with optimal m-point interpo-

latory 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∫ 1

0

fn(t` + sh`)ds =m∑

j=1

wjfn(t` + cjh`) + En,` .

Here, En,` denotes the quadrature error, and we have

fn(t` + cjh`) = 0 since δh(t` + cjh`) = 0 .

Thus:

eh(tn) =n−1∑`=0

h`En,` (n = 1, . . . , N).

67

Special sets cj of collocation parameters:

The optimal collocation parameters correspond

to special abscissas in m-point interpolatory quadra-

ture formulas of the form

Q(f) :=∫ 1

0f(s)ds =

m∑j=1

wm,jf(cj)︸ ︷︷ ︸=:Qm(f)

+Em(f) ,

with 0 ≤ c1 < · · · < cm ≤ 1 and quadrature weights

wm,j =∫ 1

0Lj(s)ds (j = 1, . . . , m).

Definition: The quadrature formula Qm(f) has

degree of precision ≥ q if

Em(f) = 0 for all f ∈ Pq .

Define

Jν :=∫ 1

0sν

m∏i=1

(s− ci)ds (ν = 0, . . . , m− 1).

Lemma 3.1: (Optimal degree of precision)

(a) The degree of precision of an m-point inter-

polatory 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 :=∫ 1

0sν

m∏i=1

(s− ci)ds = 0 .

Example 1: Gauss (-Legendre) points (κ = m)

→ The ci are the zeros of Pm(2s− 1) (shifted Leg-

endre polynomial of degree m).

m = 1 : c1 = 1/2

m = 2 : c1 = (3−√

3)/6, c2 = (3 +√

3)/6 .

m = 3 : c1 = (5−√

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)P′m−1(2s− 1) .

m = 3 : c1 = 0, c2 = 1/2, c3 = 1 .

69

Superconvergence on I and Ih

Let uh ∈ S(0)m (Ih) be the collocation solution

for the initial-value problem

u′(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 :=∫ 1

0

m∏i=1

(s− ci)ds = 0,

then

‖u− uh‖∞ ≤ Chm+1 .

(b) Let 1 ≤ κ ≤ m . If u ∈ Cd(I) (d ≥ m+κ+1)

and

Jν :=∫ 1

0sν

m∏i=1

(s− ci)ds = 0, ν = 0, . . . , κ− 1,

then

max1≤n≤N

|u(tn)− uh(tn)| ≤ Chm+κ .

70

Corollary 3.3:

(a) If the ci are the Gauss (-Legendre) points

in (0, 1) (note that cm < 1 ), then

max1≤n≤N

|u(tn)− uh(tn)| ≤ Ch2m .

(b) If the ci are the Radau I points in [0, 1)

or the Radau II points in (0, 1], then

max1≤n≤N

|u(tn)− uh(tn)| ≤ Ch2m−1 .

The resulting numerical ODE schemes are, re-

spectively, the continuous m-stage implicit Runge-

Kutta-Gauss method and the continuous m-

stage implicit Runge-Kutta-Radau I/II methods.

Remarks:

• The m-stage continuous implicit Runge-Kutta

methods of orders 2m and 2m− 1 are collo-

cation methods in S(0)m (Ih). (Guillou & Soule

(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 Runge-

Kutta 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

a collocation method in S(0)m (Ih) if, and only

if,

m∑j=1

aijcν−1j =

cνi

ν, ν = 1, . . . , m (i = 1, . . . , m).

(The above condition is known as (order) condition C(m)

in the Runge-Kutta theory; see Hairer, Nørsett & Wan-

ner, Solving Ordinary Differential Equations I, Springer-

Verlag, 1993, p. 212.)

72

ODEs: Collocation in smoother piecewise poly-

nomial spaces ?

• uh ∈ S(m−1)m (Ih) (d = m− 1):

→ uh is divergent (as h → 0) when m ≥ 4 !

(Loscalzo & Talbot (1967))

• uh ∈ S(2)4 (Ih) , 0 < c1 < c2 = 1:

uh is divergent if

1− c1

c1> 1 (or: c1 < 1/2).

• uh ∈ S(2)m (Ih) (m ≥ 4) :

uh is divergent if the ci are the Radau II

points.

(Complete convergence / divergence analysis for ODEs:

Multhei (1979); see also Brunner (BIT, 2004))

Observation:

The natural (and optimal) piecewise polyno-

mial spaces for (first-order) ODEs are the spaces

S(0)m (Ih) with m ≥ 1.

For VIEs (Lecture 4) and VFIEs (Lecture 6: Volterra

functional integral equations), the natural colloca-

tion spaces are S(−1)m−1(Ih) .

73

Basic references:

• P.H.M. Wolkenfelt, The construction of reduciblequadrature rules for Volterra integral and integro-differentialequations, IMA J. Numer. Anal., 2 (1982), 131-152.

• S.P. Nørsett & G. Wanner, Perturbed collocationand Runge-Kutta methods, Numer. Math., 38 (1981),193-208.

• H. Brunner & P.J. van der Houwen, The NumericalSolution of Volterra Equations, CWI Monographs, Vol.3, North-Holland, Amsterdam, 1986.

• J.C. Butcher, The Numerical Analysis of OrdinaryDifferential Equations: Runge-Kutta and General LinearMethods, Wiley, Chichester, 1987.

• E. Hairer, S.P. Nørsett & G. Wanner, SolvingDifferential Equations I: Nonstiff Problems (2nd ed.),Springer-Verlag, Berlin, 1993.

• H. Brunner, On the divergence of collocation insmooth piecewise polynomial spaces for Volterra integralequations, 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