CUBO A Mathematical Journal
Vol.11, No¯ 03, (65–77). August 2009

A Localized Heat Source Undergoing Periodic Motion:

Analysis of Blow-Up and a Numerical Solution

C.M. Kirk

Department of Mathematics, California Polytechnic State University,

San Luis Obispo, Ca 93407

email: ckirk@calpoly.edu

ABSTRACT

A localized heat source moves with simple periodic motion along a one-dimensional

reactive-diffusive medium. Blow-up will occur regardless of the amplitude or frequency

of motion. Numerical results suggest that blow-up is delayed by increasing the am-

plitude or by increasing the frequency of motion. A brief survey is presented of the

literature concerning numerical studies of nonlinear Volterra integral equations with

weakly singular kernels that exhibit blow-up solutions.

RESUMEN

Una fuente de calor localizada se mueve con un movimiento periódico simple a lo

largo de un medio reactivo-difuso unidimensional. “Blow-up” ocurrirá considerando la

amplitud o frecuencia del movimiento. Resultados numéricos sugieren que el “Blow-up”

es retardado por aumento de amplitud o frecuencia de movimiento. Un breve informe es

presentado de la literatura al respecto de estudios numéricos de ecuaciones integrales de

Volterra no lineales con nucleo débilmente singular que exhiben soluciones “Blow-up”.



66 Colleen M. Kirk CUBO
11, 3 (2009)

Key words and phrases: Moving heat source, blow-up, integral equations, numerical simulation.

Math. Subj. Class.: 45D05, 45M99, 35K60, 65-04, 65R20.

Introduction

A number of studies have addressed blow-up in a reactive-diffusive medium due to a localized heat

source. Generally this problem is modeled by a parabolic partial differential equation (PDE) with

a nonlinear source term. The highly localized nature of the heat source can be represented by a

Dirac delta function. See [30] and [31] for surveys of this literature. The general model often takes

the form:

x
q
Tt (x, t) − Txx(x, t) = δ(x − x0)F [T (x, t)] for 0 < t and x ∈ D (1)

T (x, 0) = T0(x) f or x ∈ D

with appropriate boundary conditions on D. Usually T0 is non-negative and continuous. In most

studies, q = 0. [10] and [12] address degenerate problems with q 6= 0. Variations of (1) include
systems of equations ([20], [26], [23]), higher dimensions ([11], [19]), and problems that include

non-local features ([27], [12]). The model can also include motion of sources ([25], [18], [19], [20])

and sources of varying size and shape [19]. The delta function δ(x − x0) reflects the intense
localization. Some studies have allowed for less intense, perhaps more realistic, localization [19].

One question of interest is whether or not the solution undergoes blow-up in finite time.

In this article we will examine a localized heat source that moves with simple periodic mo-

tion (x0(t) = A cos(ωt)) along a one-dimensional reactive-diffusive medium. Blow-up will occur

regardless of the amplitude and frequency of motion. This particular motion is suggested by [18]

which addresses various types of motion in one-dimension. This problem is initially modeled as a

nonlinear parabolic PDE. The analysis is carried out by converting to the corresponding Volterra

integral equation (VIE). We develop a numerical method to model the solution. The results of the

numerical study suggest that blow-up is delayed either by increasing the amplitude or by increasing

the frequency of motion. Intuitively this makes sense since increasing the amplitude allows the

heat source to oscillate over a wider spatial domain, allowing the heat more time to dissipate as the

source moves into cooler surroundings. Increasing the frequency causes the source to move more

quickly, generally allowing the heat less opportunity to accumulate. Furthermore the numerical

results agree with the analytical results that can be obtained for this specific kind of motion.

Numerical modeling of nonlinear VIEs with blow-up solutions can be a difficult problem. Little

research has been done in this area. Specifically, rigorous numerical analysis of such schemes is

essentially non-existent. A brief survey of the literature regarding numerical studies of nonlinear

VIEs with weakly singular kernels that exhibit blow-up solutions is presented in the last section of

this paper.



CUBO
11, 3 (2009)

A Localized Heat Source Undergoing Periodic ... 67

Theory

Consider the following nonlinear PDE:

Tt (x, t) − Txx(x, t) = δ(x − x0)F [T (x, t)] for 0 < t and x ∈ (−∞, ∞)

T (x, 0) = T0(x) for x ∈ (−∞, ∞)

T (x, t) → 0 as |x| → ∞

This problem models the heating of a reactive-diffusive material by a highly-localized source. The

delta function reflects the strong localization of the heat source. T0 is non-negative, continuous,

and approaches 0 as |x| → ∞.

Here we present the conversion from the PDE to the corresponding integral equation. First

apply the appropriate one-dimensional free space Green’s function.

T (x, t) =

t
∫

0

∞
∫

−∞

G(x, t|ξ, s)δ(ξ − x0)F [T (x0, s)]dξds

+

∞
∫

−∞

G(x, t|ξ, 0)T0(ξ)dξ, −∞ < x < ∞, t ≥ 0

Let x = x0 and utilize the delta function property:

T (x0, t) =

t
∫

0

G(x0, t|x0, s)F [T (x0, s)]ds

+

∞
∫

−∞

G(x0, t|ξ, 0)T0(ξ)dξ, −∞ < x < ∞, t ≥ 0

Define u(t) ≡ T (x0, t) and h(t) ≡
∞
∫

−∞

G(x0, t|ξ, 0)T0(ξ)dξ and

k(t, s) ≡ G(x0, t|x0, s) = H(t−s)
2

√
π(t−s)

exp

[

−(x0(t)−x0(s))
2

4(t−s)

]

so that:

u(t) = h(t) +

t
∫

0

k(t, s)F (u(s))ds (2)

Of interest is whether or not (2) has a blow-up solution. A solution u(t) is a blow-up solution

if u(t) → ∞ as t → ̂t < ∞. The following theorem is used to prove the existence of a unique,
continuous solution to (2) up to some lower bound time t∗. Various versions of this theorem appear

in [32], [24], [18] and [19].

Theorem 1. (Existence Theorem): Equation (2) has a unique, continuous solution for 0 ≤ t < t∗,
where t∗ < ∞ can be determined as the smallest root of I(t∗) = Λ ≡ max

0≤M<∞

[

M
F (M+h)

]

and t∗ = ∞
if I(t) < Λ for all t ≥ 0.



68 Colleen M. Kirk CUBO
11, 3 (2009)

The proof of this theorem involves the creation of a mapping from the space of continuous

functions u(t) that satisfy:

0 ≤ u(t) ≤ M < ∞, 0 ≤ t ≤ t1

where M is the smallest root of M
F (M+h̄)

=
1

F ′(M)
.The mapping is defined as the integral operator

K where:

K[u(t)] = h(t) +

t
∫

0

k(t, s)F [u(s)]ds

It can be shown that K is a contraction mapping (with the sup norm) from the space into itself if

certain conditions are satisfied. Then by the Contraction Mapping Theorem, a unique continuous

solution exists. Those required conditions lead to the lower bound on the blow-up time.

The following theorem provides an upper bound on the blow-up time.

Theorem 2. (Non-existence Theorem): Let k(t, s) be a non-increasing function in t. Whenever

there exists a t∗∗ < ∞ such that I(t∗∗) = κ ≡
∫

∞

h
dz

F (z)
, it follows that (2) can not have a continuous

solution for t ≥ t∗∗.

A contradiction argument is used to prove non-existence of the solution. This method exploits

the non-increasing nature of the typical kernel. The theorem must be modified if the kernel does

not have this property. Various versions of this theorem appear in [32], [24], [18] and [19]. The

bounds on the blow-up time can then be summarized:

If u(t) → ∞ as t → ̂t < ∞, then t∗ < ̂t < t∗∗.

Now consider a moving source: x0 = x0(t) in one-dimension along an infinite rod. Assume

x0(t) is continuously differentiable. This is a reasonable assumption since x0(t) is a position

function. Then:

u(t) = h(t) +

t
∫

0

1

2

√
π(t−s)

exp

(

− (x0(t)−x0(s))
2

4(t−s)

)

F [u(s)]ds

Specifically we consider the special case of simple periodic motion as suggested by [18]:

x0(t) = A cos(ωt), A > 0, ω > 0.

Let F (z) = ez and h(t) = 0 so that

u(t) =

t
∫

0

1

2
√

π(t − s)
exp

(

− (A cos(ωt) − A cos(ωs))
2

4(t − s)

)

e
u(s)

ds (3)

For this motion, blow-up will always occur. To see this, apply a theorem from [18]:



CUBO
11, 3 (2009)

A Localized Heat Source Undergoing Periodic ... 69

Theorem 3. Let x0(t) be bounded and Lipschitz continuous, with constants x
∗∗

> 0, v
∗∗

> 0 such

that |x0(t) − x0(t′)| ≤ v∗∗|t − t′|, 0 ≤ t < ∞, 0 ≤ t′ < ∞. Then a continuous solution of (3) can
not exist for t ≥ t∗∗ = πκ2 exp(x∗∗v∗∗).

Note, of course, that in our example

|x0(t)| ≤ A, |x0(t) − x0 (t′) | ≤ Aω |t − t′| , 0 ≤ t < ∞, 0 ≤ t′ < ∞

Hence Theorem 3 guarantees that blow-up will necessarily occur. The upper bound is obtained

directly from Theorem 3: t∗∗ = πκ2eA
2ω

. Recall:

κ ≡
∞
∫

h

dz
F (z)

, which for this example becomes: κ = 1. So then t∗∗ = πeA
2ω. Now consider

the lower bound on the blow-up time. A comparison kernel can be introduced in order to modify

Theorem 1 for problems such as this one for which I(t) is difficult to obtain. (Note that a compar-

ison kernel was also needed to prove Theorem 3 since the kernel in this problem is not necessarily

non-increasing.) Apply the following theorem from [18]:

Theorem 4. Let k(t, s) ≤ k(t, s), 0 ≤ s < t < ∞, where k(t, s) is continuous for 0 ≤ s < t and
integrable as s → t. Then Theorem 1 holds with I(t) replaced by I(t) where

I(t) ≡
∫ t

0
k(t, s)ds, 0 ≤ t < ∞

The appropriate comparison kernel in this case is

k(t, s) =
1

2

√
π(t−s)

Since h = 0 and F (z) = ez, we have Λ = 1
e

. The lower bound on the blow-up time is obtained

from I(t∗) = 1
e

. So t∗ = π
e2

. Together the bounds on the blow-up time are:

π
e2

< ̂t < πe
A2ω

Numerical Method

The numerical computation of the solution to problem (3) is based on a standard product quadra-

ture approach. Discretize the interval of integration (0, t) into subintervals: (ti, ti+1), i = 1...n.

ui = u(ti) is the approximation of u(t) at t = ti. The nonlinear portion of the integrand is repre-

sented by the canonical Lagrange functions. We have exp(u(s)) ≈ s−ti+1
ti−ti+1

e
u(ti) +

s−ti
ti+1−ti

e
u(ti+1) for

s ∈ (ti, ti+1). The typical integration rule for un+1 can then be written:



70 Colleen M. Kirk CUBO
11, 3 (2009)

un+1 =
1

2
√

π

n
∑

i=1

ti+1
∫

ti

1√
tn+1 − s

exp

{

−A2[cos(ωtn+1) − cos(ωs)]2
4(tn+1 − s)

}

exp(u(s))ds

=
1

2
√

π

n
∑

i=1

e
u(ti)

ti − ti+1

ti+1
∫

ti

s − ti+1√
tn+1 − s

exp

{

−A2[cos(ωtn+1) − cos(ωs)]2
4(tn+1 − s)

}

ds

+
1

2
√

π

n
∑

i=1

e
u(ti+1)

ti+1 − ti

ti+1
∫

ti

s − ti√
tn+1 − s

exp

{

−A2[cos(ωtn+1) − cos(ωs)]2
4(tn+1 − s)

}

ds (4)

for the approximation of the solution at time tn+1. To address the singular behavior when i = n,

apply the Mean Value Theorem to the term cos(ωt) − cos(ωs) to obtain:

exp

{

−A2[cos(ωt)−cos(ωs)]2

4(t−s)

}

= exp

{

−A2ω2[sin(ω˜t)(t−s)]2

4(t−s)

}

= exp

{

−A2ω2[sin(ω˜t)]2

4
(t − s)

}

where s < ˜t < t. Then choose a small value 0 < ∆Mn << (tn, tn+1). Consider just the integral

from the last term in equation (4) to illustrate this step for i = n :

tn+1
∫

tn

s−tn
√

tn+1−s
exp

{

−A2[cos(ωtn+1)−cos(ωs)]
2

4(tn+1−s)

}

ds

=

tn+1−∆Mn
∫

tn

s−tn
√

tn+1−s
exp

{

−A2[cos(ωtn+1)−cos(ωs)]
2

4(tn+1−s)

}

ds

+

tn+1
∫

tn+1−∆Mn

s−tn
√

tn+1−s
exp

{

−A2[cos(ωtn+1)−cos(ωs)]
2

4(tn+1−s)

}

ds

Evaluate analytically the integral for (tn+1 − ∆Mn, tn+1) :
tn+1
∫

tn+1−∆Mn

s−tn
√

tn+1−s
exp

{

−A2[cos(ωtn+1)−cos(ωs)]
2

4(tn+1−s)

}

ds

=

tn+1
∫

tn+1−∆Mn

s−tn
√

tn+1−s
exp

{

−A2ω2[sin(ω˜t)]2

4
(tn+1 − s)

}

ds

=
1

R5/2
[
√

∆MnR
3/2

exp(−R∆Mn) + R
√

π erf(
√

R
√

∆Mn)
{

R(tn+1 − tn) − 12
}

]

where R ≡ A
2ω2 sin2 ˜t

4
and tn+1 − ∆Mn < ˜t < tn+1. Then the governing equation becomes:



CUBO
11, 3 (2009)

A Localized Heat Source Undergoing Periodic ... 71

un+1 =
1

2
√

π

n
∑

i=1

[
eu(ti )

ti−ti+1

ti+1
∫

ti

s−ti+1
√

tn+1−s
exp

{

−A2[cos(ωtn+1)−cos(ωs)]
2

4(tn+1−s)

}

ds]

+
1

2
√

π

n
∑

i=1

[
e

u(ti+1)

ti+1−ti

ti+1
∫

ti

s−ti
√

tn+1−s
exp

{

−A2[cos(ωtn+1)−cos(ωs)]
2

4(tn+1−s)

}

ds]

=
1

2
√

π

n−1
∑

i=1

[
eu(ti )

ti−ti+1

ti+1
∫

ti

s−ti+1
√

tn+1−s
exp

{

−A2[cos(ωtn+1)−cos(ωs)]
2

4(tn+1−s)

}

ds]

+
1

2
√

π
eu(tn )

tn−tn+1

tn+1
∫

tn

s−tn+1
√

tn+1−s
exp

{

−A2[cos(ωtn+1)−cos(ωs)]
2

4(tn+1−s)

}

ds

+
1

2
√

π

n−1
∑

i=1

[
e

u(ti+1)

ti+1−ti

ti+1
∫

ti

s−ti
√

tn+1−s
exp

{

−A2[cos(ωtn+1)−cos(ωs)]
2

4(tn+1−s)

}

ds]

+
1

2
√

π
e

u(tn+1)

tn+1−ti

tn+1
∫

tn

s−tn
√

tn+1−s
exp

{

−A2[cos(ωtn+1)−cos(ωs)]
2

4(tn+1−s)

}

ds

un+1 =
1

2
√

π

n−1
∑

i=1

[
eu(ti )

ti−ti+1

ti+1
∫

ti

s−ti+1
√

tn+1−s
exp

{

−A2[cos(ωtn+1)−cos(ωs)]
2

4(tn+1−s)

}

ds]

+
1

2
√

π
eu(tn )

tn−tn+1

tn+1−∆Mn
∫

tn

s−tn+1
√

tn+1−s
exp

{

−A2[cos(ωtn+1)−cos(ωs)]
2

4(tn+1−s)

}

ds

+
1

2
√

π
eu(tn )

tn−tn+1
1

R5/2
[
√

∆MnR
3/2

exp(−R∆Mn) + − R2
√

π erf(
√

R
√

∆Mn)]

+
1

2
√

π

n−1
∑

i=1

[
e

u(ti+1)

ti+1−ti

ti+1
∫

ti

s−ti
√

tn+1−s
exp

{

−A2[cos(ωtn+1)−cos(ωs)]
2

4(tn+1−s)

}

ds]

+
1

2
√

π
eu(tn+1)

tn+1−tn

tn+1−∆Mn
∫

tn

s−tn
√

tn+1−s
exp

{

−A2[cos(ωtn+1)−cos(ωs)]
2

4(tn+1−s)

}

ds

+
1

2
√

π
eu(tn+1)

tn+1−tn
1

R5/2
[
√

∆MnR
3/2

exp(−R∆Mn) + R
√

π erf(
√

R
√

∆Mn)
{

R(tn+1 − tn) − 12
}

]

where R ≡ A
2ω2 sin2 ˜t

4
and tn+1 − ∆Mn < ˜t < tn+1. Newton’s method is then used to solve the

implicit nonlinear equations that arise.

The results of the numerical calculations follow here. For these calculations, n = 100 and

∆Mn =
1

100
(

T
n
) = T

10,000
where T is the stopping time. The numerical code is run in MatLab. The

intervals (ti, ti+1) are chosen to be uniform. First we compare various amplitude values. In Plot

1, we keep frequency fixed at ω = 1 and plot results for A = 1, A = 10 and A = 20. Note that the

blow-up time seems to increase from about ̂t ≈ 1.2 to ̂t ≈ 9.5 to ̂t ≈ 16 as the amplitude increases.
Intuitively this makes sense since increasing the amplitude allows the heat source to oscillate over

a wider spatial domain, allowing the heat more time and space to dissipate as the source moves

into cooler surroundings. Note the oscillatory behavior for A = 10 and A = 20. This behavior is

expected since the oscillating source moves more quickly near the center of its path of motion and

moves more slowly near the extremes of its path. Hence one would expect the medium (where the

source is located) to heat up less readily when the source is near the center of the path and to heat



72 Colleen M. Kirk CUBO
11, 3 (2009)

up more readily when the source is near the extremes of the path.

Figure 1:

0 5 10 15
0

1

2

3

4

5

t

u
(t

)

 

 

u(t):  A =   1,  w = 1
u(t):  A = 10,  w = 1
u(t):  A = 20,  w = 1

We do not see this oscillatory behavior when A = 1. This is because blow-up occurs very quickly,

before the source changes direction even once. See Plot 2 for more detailed pre-blow-up behavior

for A = 1.

Figure 2:

0 0.2 0.4 0.6 0.8 1 1.2
0

0.5

1

1.5

2

2.5

3

t

u
(t

)

 

 

 

u(t):  A =  1,  w = 1

Now vary the frequency ω while keeping the amplitude A = 10 fixed. In Plot 3, we show

results for ω = 0.1, ω = 0.5 and ω = 1. As frequency increases, the blow-up time seems to increase

from about ̂t ≈ 1.1 to ̂t ≈ 6.2 to ̂t ≈ 9.5. This behavior is consistent with intuition. Increasing the
frequency causes the source to move more quickly, generally allowing the heat less opportunity to

accumulate.



CUBO
11, 3 (2009)

A Localized Heat Source Undergoing Periodic ... 73

Figure 3:

0 2 4 6 8 10
0

1

2

3

4

5

t

u
(t

)

 

 

u(t):  A = 10,  w = 0.1
u(t):  A = 10,  w = 0.5
u(t):  A = 10,  w = 1.0

If the frequency is small enough, the source may move so slowly that the heat accumulates and

blow-up occurs before the source changes direction even once. Indeed this seems to be the case

with ω = 0.1, where blow-up occurs well before t = π
0.1

and no oscillatory behavior is seen. See

Plot 4. When ω = 1, the source completes an oscillation before blow-up occurs.

Figure 4:

0 0.2 0.4 0.6 0.8 1 1.2
0

0.5

1

1.5

2

2.5

3

t

u
(t

)

 

 

u(t):  A = 10,  w = 0.1

Furthermore these numerical results agree with the analytical results that can be obtained for

this specific kind of motion. For example, consider the case ω = 1, A = 1, with ̂t ≈ 1.2 in Plot
2. The numeric blow-up time lies within the analytical bounds: 0.425 < ̂t < 8.539. These bounds,

however, are fairly crude. Also note that the numerical solution remains bounded: u(t) < M = 1

for t < 0.425, as required by another analytical result we obtained in the Theory section of this

paper.



74 Colleen M. Kirk CUBO
11, 3 (2009)

Numerical Approaches to Nonlinear VIEs with Blow-Up Solutions

We have not yet carried out convergence or error analysis of our numerical scheme. In fact, the

numerical modeling of nonlinear Volterra integral equations with blow-up solutions can be a difficult

problem. Little research has been done in this area. Specifically, rigorous numerical analysis of

such schemes is essentially non-existent. Here we present a brief review of the literature regarding

numerical studies of nonlinear VIEs of the second kind with weakly singular kernels that exhibit

blow-up solutions.

There is a solid body of research on the numerical analysis of nonlinear VIEs of the second

kind with weakly singular kernels that do not exhibit blow-up solutions. See [4], [5], [6], [7], [1],

[16]. These papers provide very good summaries of the existing literature. They also describe in

detail some of the important techniques used to address these problems. One successful approach

involves collocation solutions, with approximations in the space of piecewise polynomials. Appro-

priate quadrature formulas are usually used to approximate the integrals involved. Generally large

nonlinear systems must then be solved. Special issues arise with the introduction of weakly singular

kernels. For example, the order of convergence is reduced near the singularity when polynomial

splines and uniform meshes are used. These issues can often be successfully addressed using graded

meshes which are finer near the region of the singularity; by carrying out a variable transformation;

or by using other basis functions which more closely reflect the nature of the solution near the sin-

gularity. See [5], [9], [8], [28], [17]. The convergence, superconvergence, and stability properties of

these collocation solutions are then studied. In [5], H. Brunner includes extensive details of these

methods. He also lists many references for these methods, as well as for other methods used to

address these problems.

Research on numerical methods for nonlinear VIEs of the second kind with weakly singular

kernels that do exhibit blow-up in finite time is still at the early stages. The development of

computational approaches and rigorous analysis of these approaches is limited. See [2] and [3]

and [5]. Despite this, researchers have attempted to compute solutions numerically to various

interesting applied problems. Without rigorous analysis of errors and convergence, however, these

authors often rely upon available analytical information to help verify their numerical results.

It can be useful to employ knowledge of the asymptotic behavior of the solution near the time

of blow-up. If the solution is known to exhibit certain asymptotic behavior near blow-up, then the

numerical solution can be tracked until it reflects this behavior, suggesting the onset of blow-up.

For example, in [21], D. G. Lasseigne and W. E. Olmstead analyze the ignition of a combustible

solid due to excessive reactant consumption. An integral equation is obtained that models the

perturbation of temperature in the reaction zone. Before carrying out the numerical analysis, an

asymptotic analysis is performed on the governing integral equation. Then the behavior of the

evolving numerical solution is compared to the expected asymptotic solution form. Eventually,

the numerical solution matches the known behavior pattern. This provides extra assurance that

thermal runaway is indeed occurring. L. R. Ritter, W. E. Olmstead and V. A. Volpert follow a



CUBO
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A Localized Heat Source Undergoing Periodic ... 75

similar approach in [29] to model numerically the initiation of a polymerization wave. Their goal

is to understand how various parameters in a frontal polymerization process determine whether

the onset of a wave front will be initiated or inhibited.

In [10] and [12], C. Y. Chan and H. Y. Tian employ computational methods to estimate

blow-up time. In [10] they examine a degenerate parabolic problem with a nonlinear source in a

one-dimensional material of finite length. They obtain analytical expressions for the bounds on

the blow-up time as functions of the length of the domain. Then they use an iterative approach to

approximate the solution over the discretized spatial domain at the estimated blow-up time. This

estimated blow-up time is then refined, shifted upward or downward, depending on whether the

growth of the iterated solution lies within or exceeds certain tolerances. They provide numerical

results indicating the blow-up time is a decreasing function of the length of the domain, as expected.

These same authors apply a similar technique in [12] to a problem with a nonlinear source of local

and nonlocal features. Again they use an iterative process to refine the analytical bounds.

Note that in none of these studies is a rigorous numerical analysis carried out. The authors

instead acquire confidence in their numerical solutions partly by making judicious use of known

analytical results. Clearly some understanding of the analytical solution is important in choosing

the best numerical approach and in acquiring confidence in the numerical results.

H. Brunner proposes in [5] another possible approach to these problems using collocation

methods. He suggests the possibility of obtaining two different collocation solutions vh and uh

generated from different sets of collocation points such that the corresponding iterated numerical

solutions satisfy: vith ≤ u(t) ≤ uith for a chosen mesh Ih.

Furthermore, if the VIE of interest stems from an originating PDE, one might instead examine

the original PDE for blow-up behavior. The research into these corresponding PDEs is more

advanced than that of analogous VIEs. See [2] and [3] and [5] for surveys of these studies.

Received: July 28, 2008. Revised: September 5, 2008.

References

[1] Baker C., A perspective on the numerical treatment of Volterra equations, Numerical anal-

ysis 2000, Vol. VI, Ordinary differential equations and integral equations. J. Comput. Appl.

Math., 125(2000), 217–249.

[2] Bandle, C. and Brunner, H., Numerical analysis of semilinear parabolic problems with

blow-up solutions, Rev. Real Acad. Cienc. Exact. Fís. Natur. Madrid, 88(1994), 203–222.

[3] Bandle, C. and Brunner, H., Blow-up in diffusion equations: A survey, J. Comp. Appl.

Math., 97(1998), 3–22.



76 Colleen M. Kirk CUBO
11, 3 (2009)

[4] Blom, J.G. and Brunner, H., The numerical solution of nonlinear Volterra integral equa-

tions of the second kind by collocation and iterated collocation methods, SIAM J. Sci. Statist.

Comput., 8(1987), 806–830.

[5] Brunner, H., Collocation methods for Volterra integral and related functional differential

equations, Cambridge University Press, 2004.

[6] Brunner, H., The numerical analysis of functional integral and integro-differential equations

of Volterra type, Acta Numer., 13(2004), 55–145.

[7] Brunner, H., The discretization of Volterra functional integral equations with proportional

delays, Differences and differential equations, 3–27, Fields Inst.Commun., 42, Amer. Math.

Soc., Providence, RI, 2004.

[8] Brunner, H., Pedas, A. and Vainikko, G., The piecewise polynomial collocation method

for nonlinear weakly singular Volterra equations, Math. Comp. 68(1999), 1079–1095.

[9] Brunner, H. and van der Houwen, P.J., The numerical solution of Volterra equations,

CWI Monographs 3, North Holland, 1986.

[10] Chan, C.Y. and Tian, H.Y., Single-point blow-up for a degenerate parabolic problem due

to a concentrated nonlinear source, Quart. Appl. Math. 61(2003), 363–385.

[11] Chan, C.Y. and Tian, H.Y., Multi-dimensional explosion due to a concentrated nonlinear

source, J. Math. Anal. Appl., 295(2004), 174–190.

[12] Chan, C.Y. and Tian, H.Y., Single-point blow-up for a degenerate parabolic problem with

a nonlinear source of local and nonlocal features, Appl. Math. Comput., 145(2003), 371–390.

[13] Chan, C.Y. and Carrillo Escobar, J.C., Blow-up of the solution for a singular semilinear

parabolic problem due to a concentrated nonlinear source, Proceedings of Dynamic Systems

and Applications, 5, 2008.

[14] Chan, C.Y. and Boonklurb, R., A blow-up criterion for a degenerate parabolic problem

due to a concentrated nonlinear source, Quart. Appl. Math., 65(2007), 781–787.

[15] Diogo, T., Franco, N.B. and Lima, P., High order product integration methods for

a Volterra integral equation with logarithmic singular kernel, Commun. Pure Appl. Anal.,

3(2004), 217–235.

[16] Diogo, T. and Lima, P., Collocation Solutions of a Weakly Singular Volterra Integral

Equation, TEMA Tend. Mat. Apl. Comput., 8(2007), 229–238.

[17] Diogo, T., Mckee, S., and Tang, T., Collocation methods for second-kind Volterra inte-

gral equations with weakly singular kernels, Proc. Roy. soc. Edinburgh, 124A(1994), 199–201.

[18] Kirk, C.M. and Olmstead, W.E., Blow-up in a reactive-diffusive medium with a moving

heat source, Z. Angew. Math. Phys., 53(2002), 147–159.



CUBO
11, 3 (2009)

A Localized Heat Source Undergoing Periodic ... 77

[19] Kirk, C.M. and Olmstead, W.E., Blow-up solutions of the two-dimensional heat equation

due to a localized moving source, Anal. Appl., 3(2005), 1–16.

[20] Kirk, C.M. and Olmstead, W.E., The influence of two moving heat sources on blow-up

in a reactive-diffusive medium, Z. Angew. Math. Phys., 51(2000), 1–16.

[21] Glenn Lasseigne, D. and Olmstead, W.E, Ignition of a Combustible Solid with Reactant

Consumption, SIAM J. Appl. Math., 47(1987), 332–342.

[22] Levine, H.A., The role of critical exponents in the blow-up theorems, SIAM Rev., 32(1990),

262–288.

[23] Mydlarczyk, W., Okrasi’nski, W. and Roberts, C.A., Blow-up solutions to a system

of nonlinear Volterra equations, J. Math. Anal. Appl., 301(2005), 208–218.

[24] Olmstead, W.E. and Roberts, C.A., Explosion in a diffusive strip due to a concentrated

nonlinear source, Methods Appl. Anal., 1(1994), 435–445.

[25] Olmstead, W.E., Critical speed for the avoidance of blow-up in a reactive-diffusive medium,

Z. angew. Math. Phys., 48(1997), 701–710.

[26] Olmstead, W.E., Roberts, C.A. and Deng, K., Coupled Volterra equations with blow-up

solutions, J. Integral Equations Appl., 7(1995), 499–516.

[27] Olmstead, W.E. and Roberts, C.A., Explosion in a diffusive strip due to a source with

local and nonlocal features, Methods Appl. Analysis, 3(1996), 345–357.

[28] Pedas, A. and Vainikko, G., Smoothing transformation and piecewise polynomial colloca-

tion for weakly singlular Volterra integral equations, Computing, 73(2004), 271–293.

[29] Ritter, L.R., Olmstead, W.E. and Volpert, V.A., Initiation of free radical polymer-

ization waves, SIAM J. Appl. Math., 63(2003), 1831–1848.

[30] Roberts, C.A., Recent results on blow-up and quenching for nonlinear Volterra equations,

J. Comput. Appl. Math., 205(2007), 736–743.

[31] Roberts, C.A., Analysis of explosion for nonlinear Volterra integral equations, J. Comp.

Appl. Math., 97(1998), 153–166.

[32] Roberts, C.A., Lasseigne, D.G. and Olmstead, W.E, Volterra equations which model

explosion in a diffusive medium, J. Integral. Eqn. Appl., 5(1993), 531–546.

[33] Souplet, P., Blow-up in nonlocal reaction-diffusion equations, SIAM J. Math. Anal.,

29(1998), 1301–1334.

[34] Stuart, A.M. and Floater, M,S., On the computation of blow-up, Europ. J. Appl. Math.,

1(1990), 47–71.


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