Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 4 Issue 2(2012), Pages 174-185.
THE ONE-DIMENSIONAL HEAT EQUATION AS A
FIRST-ORDER SYSTEM :
FORMAL SOLUTIONS BY MEANS OF THE LAPLACE
TRANSFORM
(COMMUNICATED BY MARTIN HERMAN)
HEINZ TOPARKUS
Abstract. In this paper an extended heat equation problem as a linear firstorder system of partial differential equations is considered. The classical problems in a strip S are assigned to our problems. Formal solutions are given by
one-dimensional Laplace transform.
1. The problem
We consider a first-order system of partial differential equations for two real valued functions u(x, t), v(x, t) and given functions fi (x, t) :
ux =
v + f1 (x, t) ,
u, v ∈ C1 (R+ × R+ ) ,
u t − vx =
f2 (x, t) ,
fi ∈ C(R+ × R+ ) , i = 1, 2 .
(1)
In the case f1 (x, t) ̸≡ 0 this system is a more general problem as the conventional
one-dimensional heat equation [1, 12] with the inhomogeneous term f2 (x, t) and
the solution u(x, t). We will see this below, when we look at the formal solutions.
We are looking for solutions of (1) in the strip S = (0, l) × R+ of the first quadrant
of the (x,t)-plane , l ∈ R+ .
Just like in the classical case of the heat equation [6, 8] we consider in S the four
Problems Pi j :
Give a solution [u, v]T of (1) in the domain
0 < x < l, 0 < t < ∞, with :
u satisf ies limt→+0 u(x, t) = f (x), 0 < x < l, f (x) ∈ C(0, l), and
f or a f ixed Pi j , i, j ∈ {1, 2}, the solution satisf ies
2010 Mathematics Subject Classification. 35C05, 35C15, 35E15, 35F40 .
Key words and phrases. first-order parabolic systems, Greens function, real theta functions,
distributions, heat equation, Laplace transform.
c
⃝2012
Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted May 16, 2012. Published June 15, 2012.
174
THE ONE-DIMENSIONAL HEAT EQUATION AS A FIRST-ORDER SYSTEM
P11 :
lim u(x, t) = u0 (t),
x→+0
P22 :
12
P
lim v(x, t) = v0 (t),
x→+0
:
lim u(x, t) = u0 (t),
x→+0
P21 :
lim v(x, t) = v0 (t),
x→+0
175
lim u(x, t) = ul (t),
x→l−0
lim v(x, t) = vl (t),
x→l−0
(2)
lim v(x, t) = vl (t),
x→l−0
lim u(x, t) = ul (t),
x→l−0
where f and the f unctions u0 , ul , v0 , vl ∈ C(R+ ) are given.
2. The problem in the image range, the solution in the image range
We now assume that the Laplace transform exists for all quantities which we are
using. We transform the problems (1), (2) from the t−domain (originals) into the
s−range (images) by Laplace transform:
Let
∫
L[u(x, t); t](s) =
∞
e−st u(x, t)dt = w1 (x, s) , shortly u(x, t) ◦→• w1 (x, s).
0
The transformation of the other quantities from (1), (2) into the s−range is described in the following manner :
v(x, t) ◦→• w2 (x, s) ,
ut (x, t) ◦→• s · w1 (x, s) − f (x) ,
vx (x, t) ◦→• w2,x (x, s) ,
ux (x, t) ◦→• w1,x (x, s) ,
v0 (t) ◦→• w20 (s) ,
vl (t) ◦→• w2l (s) ,
u0 (t) ◦→• w10 (s)
ul (t) ◦→• w1l (s)
,
(3)
,
fi (x, t) ◦→• φi (x, s) , i = 1, 2.
In the s−range we get the following system of ordinary differential equations
[
] [
][ ] [
] [
][ ] [
]
w1,x
0 1 w1
φ1 (x, s)
0 1 w1
r (x, s)
=
+
=
+ 1
.
w2,x
s 0 w2
−f (x) − φ2 (x, s) Df s 0 w2
r2 (x, s)
Considering (2) we are now looking for solutions of the four boundary value
problems Πi j (4), (4a) in the s−range:
(4)
176
H. TOPARKUS
Π11
:
w1 (0, s) = w10 (s),
w1 (l, s) = w1l (s),
Π22
:
w2 (0, s) = w20 (s), w2 (l, s) = w2l (s),
Π12
:
w1 (0, s) = w10 (s), w2 (l, s) = w2l (s),
Π21
:
w2 (0, s) = w20 (s), w1 (l, s) = w1l (s).
(4a)
The functions wi0 (s), wil (s), i = 1, 2, are well-known from (3).
We can write problem Π11 using the ”boundary matrices” B011 , Bl11 and the righthand side β 11 :
[
1 0
0 0
][
] [
][
]
[ 0 ]
w (s)
w1 (0, s)
0 0 w1 (l, s)
11
11
+
= B w(0, s) + Bl w(l, s) = 1l
= β 11.
w2 (0, s)
1 0 w2 (l, s) Df 0
w1 (s) Df
Similarly we rewrite the remaining problems Πi j from (4a) which have the corresponding boundary matrices B0ij , Blij and the right-hand sides β ij .
A particular solution of the initial value problem for a first-order system of linear inhomogeneous ordinary differential equations with constant coefficients can be
found by the method referred to as ”Variation of the Parameters”.
For boundary value problems of the form (4),(4a) (linear inhomogeneous ordinary
first-order systems with constant coefficients and totally separated boundary conditions) the method works too [4]. This is used in the following.
To construct the columns of the fundamental matrix W which contains the principal solutions
of the
√
√ homogeneous system from (4) we make use of the eigenvalues
λ1 = s, λ2 = − s of the coefficient matrix of the homogeneous system from (4)
and of the corresponding eigenvectors
[
[
√ ]T
√ ]T
e1 = 1 , s , e2 = −1 , s :

e
√
sx
W(x, s) = 
√ √sx
se
√
−e−
sx
√ −√sx
se


(5)
·
As an exemplary case, in the sequel we give focus on the problem Π11 from (4),
[
]T
(4a), where r(x, s) = r1 (x, s) , r2 (x, s) :
Using (5) and the boundary matrices B011 , Bl11 defined above, we define
[
M11 ≡
B011 W(0, s)
+
Bl11 W(l, s)
1
=
0
]
[
]
0
0 0
W(0, s) +
W(l, s) .
0
1 0
(6)
Thus, we obtain the solution of the inhomogeneous problem Π11 from (4), (4a) :
THE ONE-DIMENSIONAL HEAT EQUATION AS A FIRST-ORDER SYSTEM
w(x, s) =
−1
W(x, s) M11
∫
β
11
l
G 11 (x, ξ, s) r(ξ, s) dξ .
+
177
(7)
0
Since the function G 11 describes the influence of the left-hand and the right-hand
boundary of S on the solution of the problem Π11 at the position x, x ∈ (0, l), it is
named influence function or Greens function.
With (7) we have
G 11 (x, ξ, s) =
[
]


1 0

−1

W(x, s) M11
W(0, s) W −1 (ξ, s) ,



0 0



[
]



0
0

−1

−W(x, s) M11
W(l, s) W −1 (ξ, s) ,

1 0
ξ≤x,
(8)
x<ξ.
Let us abbreviate
√
√
S(x) = sinh( sx), C(x) = cosh( sx).
Df
Df
Then, Green’s function can be formulated as:


1
√ {−C(x−ξ − l)+C(x+ξ − l)}

s
1 
11


=
G ξ≤x

2 · S(l)  √
− s{C(x−ξ −l)+C(x + ξ − l)} −S(x−ξ − l)+S(x+ξ −l) ,

−S(x−ξ − l)−S(x+ξ − l)

1
√ {−C(x−ξ + l) + C(x+ξ − l)}

s
1 
11


G x<ξ
=

2 · S(l) √
− s{C(x−ξ +l) +C(x + ξ − l)}
−S(x−ξ +l) + S(x+ξ −l) ·
−S(x−ξ + l)−S(x+ξ − l)
So we have in accordance with (7) the complete solution of Π11 :
178
H. TOPARKUS
w111 (x, s) =
∫
1
−
2
1
2
−
−
0
∫
∫
1
2
l
0
1
2
0
−
1
2
1
2
∫
0
∫
√
√
∫
sinh s(x + ξ − l)
1 l cosh s(x + ξ − l)
√
√
√
r1 (ξ, s)dξ +
r2 (ξ, s)dξ ,
2 0
sinh sl
s sinh sl
√
√
∫ √
s cosh s(x−ξ −l)
1 l s cosh s(x −ξ +l)
√
√
r1 (ξ, s)dξ −
r1 (ξ, s)dξ
2 x
sinh sl
sinh sl
x
0
l
(9)
√
√
√
√
s cosh s(l − x) 0
s cosh sx l
√
√
w1 (s) +
w1 (s)
=−
sinh sl
sinh sl
∫ x√
−
√
√
∫
sinh s(x − ξ − l)
1 l sinh s(x − ξ + l)
√
√
r1 (ξ, s)dξ −
r1 (ξ, s)dξ
2 x
sinh sl
sinh sl
√
√
∫
cosh s(x − ξ − l)
1 l cosh s(x − ξ + l)
√
√
√
√
r2 (ξ, s)dξ −
r2 (ξ, s)dξ
2 x
s sinh sl
s sinh sl
x
0
w211 (x, s)
−
x
√
√
sinh s(l − x) 0
sinh sx l
√
√ w (s)
w1 (s) +
sinh sl
sinh sl 1
√
√
∫
sinh s(x − ξ − l)
1 l sinh s(x − ξ + l)
√
√
r2 (ξ, s)dξ −
r2 (ξ, s)dξ
2 x
sinh sl
sinh sl
(9a)
√
√
√
∫
s cosh s(x + ξ − l)
1 l sinh s(x + ξ − l)
√
√
r1 (ξ, s)dξ +
r2 (ξ, s)dξ .
2 0
sinh sl
sinh sl
Inserting the expressions (9), (9a) into the system (4) and looking at the boundary
T
conditions of Π11 , we see that [w111 , w211 ] is indeed the solution of the inhomogeneous problem Π11 .
In a similar manner we discuss the remaining problems Π22 , Π12 , Π21 from (4),(4a);
we introduce the corresponding quantities
(B022 , Bl22 , β 22 , M22 ),
(B012 , Bl12 , β 12 , M12 ),
(B021 , Bl21 , β 21 , M21 )
and obtain in analogy with (6), (7), (8), (9), (9a) the solutions
[
w122 (x, s)
w222 (x, s)
]
,
[ 12
]
w1 (x, s)
w212 (x, s) ,
[
]
w121 (x, s)
w221 (x, s) .
(10)
Here, we don’t formulate the complete solutions (10) of the problems Π22 , Π12 , Π21
in analogy with (9), (9a). We give only the solutions of the corresponding homogeneous problems (r1 (x, s) = r2 (x, s) ≡ 0 in (4)). So we can demonstrate at least,
which combinations of hyperbolic functions play an important part in the individual
THE ONE-DIMENSIONAL HEAT EQUATION AS A FIRST-ORDER SYSTEM
179
problems Πi j , because only those quotients of hyperbolic functions, which appear
in the solution of the homogeneous problem also play a role in the solution of the
inhomogeneous problem. From these quotients we need the originals of the Laplace
transform to get the formal solution of the problems Πi j :
√
√
1
1
1
√ {− √ cosh s(l − x)·w20 (s) + √ cosh sx·w2l (s)} ,
sinh sl
s
s
(11)
√
√
1
22
0
l
√ {sinh s(l − x) · w2 (s) + sinh sx · w2 (s)} ,
w2,hom (x, s) =
sinh sl
22
w1,hom
(x, s) =
√
√
1
1
√ {cosh s(l − x) · w10 (s) + √ sinh sx · w2l (s)} ,
cosh sl
s
√
√
√
1
12
√ {− s sinh s(l − x) · w10 (s) + cosh sx · w2l (s)} ,
w2,hom
(x, s) =
cosh sl
(11a)
√
√
1
1
21
√ {− √ sinh s(l − x) · w20 (s) + cosh sx · w1l (s)} ,
w1,hom
(x, s) =
cosh sl
s
√
√
√
1
21
√ {cosh s(l − x) · w20 (s) + s sinh sx · w1l (s)} .
w2,hom
(x, s) =
cosh sl
12
w1,hom
(x, s) =
3. Formal solutions of the problems Pij : The solutions of the
problems Πij will be transformed back
We know that under the Laplace transform the originals of particular quotients
of hyperbolic functions are mainly Jacobian theta functions. We make use of theta
functions as real valued functions of real arguments and we write them as follows [7] :
∞
∞
∑
(x+n− 1 )2
2
1 2
1 ∑
2
n −
t
Θ1 (x, t) = √
=2
(−1) e
(−1)n e−π t(n+ 2 ) · sin[(2n+1)πx],
πt n=−∞
n=0
∞
∞
∑
(x+n)2
2
1 2
1 ∑
n −
t
√
Θ2 (x, t) =
(−1) e
= 2
e−π t(n+ 2 ) · cos[(2n + 1)πx] ,
πt n=−∞
n=0
(12)
∞
∑
1
e−
Θ3 (x, t) = √
πt n=−∞
(x+n)2
t
=
∞
∑
εn e−π
2
tn2
· cos[2πnx] ,
n=0
∞
∞
∑
2
2
1 ∑ − (x+n+ 12 )2
t
Θ4 (x, t) = √
e
=
(−1)n εn e−π tn · cos[2πnx] ,
πt n=−∞
n=0
ε0 = 1, εn = 2, n > 0 .
180
H. TOPARKUS
Such theta functions are entire, transcendent functions of their arguments in the
upper half-plane H = {(x, t)| − ∞ < x < ∞, t > 0} . See the charts of the theta
2
functions in [10] and put ibidem q = e−π t .
For t → +0 we have a characteristic behavior on the boundary of H. We observe,
see [3], that the terms, appearing in (12) in Θ3 (x, t) ,
(x+n)2
1
kt (x, n) = √ e− t
,
Df
πt
n ∈ Z,
establish w.r.t. the variable t a ” sequence of type δ ”, i.e. they converge for t → +0
to the δ-distribution δ(x + n) in the sense of convergence, which is valid in D′ (R)
( D′ space of distributions) . The sum δ(x+n) converges on its part in D′ [11], thus
lim Θ3 (x, t) = δΘ3 (x) =
t→+0
∞
∑
δ(x + n)
n=−∞
is a distribution (Dirac comb), ∑
which assigns each φ ∈ D (D space of test functions)
∞
the (per definition) finite sum k=−∞ φ(k) .
Similar statements hold for the remaining Θ-functions in (12).
x t
The theta functions Θi ( 2l
, l2 ) , i = 1, 2, 3, 4, (x, t) ∈ H, satisfy the differential
equation
x t
x t
, 2 ) = Θxx ( , 2 ) .
2l l
2l l
This can be demonstrated with the help of (12) .
Now we have to transform back quotients of hyperbolic functions which are relevant
for our problem (4),(4a). For this purpose we give two tables [5, 7, 9]. We suppose
|ν| < l , ν ∈ R.
The tables T ab.1, T ab.2 present the orginals of hyperbolic functions in which we
are interested: the headers for a general argument ν, the subsequent five lines for
the arguments named concretly in the first column.
Θt (
ν
√
sinh sν
1 ∂
ν t
√ •→◦
Θ4 ( , 2 )
l ∂ν
2l l
sinh sl
√
1
cosh sν
ν t
√ •→◦ Θ4 ( , 2 )
√
l
2l l
s sinh sl
x
1
x t
Θ4, x ( , 2 )
l
2l l
1
x t
Θ4 ( , 2 )
l
2l l
l−x
x t
1
− Θ3, x ( , 2 )
l
2l l
1
x t
Θ3 ( , 2 )
l
2l l
x−ξ−l
1
x−ξ t
Θ3, x (
, )
l
2l l2
1
x−ξ t
Θ3 (
, )
l
2l l2
x−ξ+l
1
x−ξ t
Θ3, x (
, )
l
2l l2
1
x−ξ t
Θ3 (
, )
l
2l l2
x+ξ−l
x+ξ t
1
Θ3, x (
, )
l
2l l2
1
x+ξ t
Θ3 (
, )
l
2l l2
THE ONE-DIMENSIONAL HEAT EQUATION AS A FIRST-ORDER SYSTEM
181
T ab.1
ν
√
cosh sν
1 ∂
ν t
√ •→◦
Θ1 ( , 2 )
l ∂ν
2l l
cosh sl
√
sinh sν
1
ν t
√
√ •→◦ Θ1 ( , 2 )
l
2l l
s cosh sl
x
1
x t
Θ1, x ( , 2 )
l
2l l
1
x t
Θ1 ( , 2 )
l
2l l
l−x
1
x t
− Θ2, x ( , 2 )
l
2l l
x t
1
Θ2 ( , 2 )
l
2l l
x−ξ−l
1
x−ξ t
− Θ2, x (
, )
l
2l l2
1
x−ξ t
− Θ2 (
, )
l
2l l2
x−ξ+l
x−ξ t
1
Θ2, x (
, )
l
2l l2
x−ξ t
1
Θ2 (
, )
l
2l l2
x+ξ−l
1
x+ξ t
− Θ2, x (
, )
l
2l l2
1
x+ξ t
− Θ2 (
, )
l
2l l2
T ab.2
The terms of the solutions in (9), (9a), (11), (11a) contain products of functions
depended on the variable s. Therefore we have convolutions in the t-domain.
Furthermore we have not considered hitherto in T ab.1, T ab.2 terms of such kind:
√
√
sinh sν
cosh sν
√ · h(s) ,
√ · h(s).
s· √
s· √
s sinh sl
s cosh sl
We treat such terms by the product rule pertaining to the s-range [7]
{∫ t
}
d
s · g(s) = s · g1 (s) · g2 (s) •→◦
f1 (t − τ )f2 (τ )dτ ,
dt
0
∫ t
=
f1,t (t − τ )f2 (τ )dτ + f1 (0)f2 (t),
(13)
0
gi (s) •→◦ fi (t) , i = 1, 2.
With (13) and T ab.1 we have
√
∫
1 t
ν t−τ
1
ν
cosh sν
√ ·h(s) •→◦
Θ4, t ( , 2 )H(τ )dτ + Θ4 ( , 0) H(t) ,
s· √
l 0
2l l
l
2l
s sinh sl
h(s) •→◦ H(t) ,
and especially in the t−domain for
(14)
182
H. TOPARKUS
ν = x , h(s) = w1l (s)
ν = l − x , h(s) = w10 (s)
:
1
l
1
l
:
∫
t
Θ4, t (
∫
0
t
Θ3, t (
0
x t−τ
1
x
,
)ul (τ )dτ + Θ4 ( , 0) ul (t),
2l l2
l
2l
x t−τ
1
x
,
)u0 (τ )dτ + Θ3 ( , 0) u0 (t).
2l l2
l
2l
In the same way we get the originals of the remaining ν-values (see T ab.1) with
h(s) = φ1 (ξ, s) in (14) .
In exactly the same manner we obtain with (13) and T ab.2
√
∫
sinh sν
1 t
ν t−τ
1
ν
√ ·h(s) •→◦
s· √
Θ1, t ( , 2 )H(τ )dτ + Θ1 ( , 0) H(t) ,
l 0
2l l
l
2l
s cosh sl
(15)
h(s) •→◦ H(t) ,
and especially in the t−domain for
ν = x , h(s) = w1l (s)
ν = l − x , h(s) = w10 (s)
:
:
1
l
1
l
∫
t
Θ1, t (
∫
0
t
Θ2, t (
0
1
x
x t−τ
,
)ul (τ )dτ + Θ1 ( , 0) ul (t),
2l l2
l
2l
1
x
x t−τ
, 2 )u0 (τ )dτ + Θ2 ( , 0) u0 (t).
2l l
l
2l
In the same way we get the originals of the remaining ν-values (see T ab.2) with
h(s) = φ1 (ξ, s) in (15) .
Now that the transformation into the t − domain has been accomplished for the
solutions (11), (11a) of the homogeneous problems and for the solutions of the corresponding inhomogeneous problems, we translate the solutions (9),(9a) as well as
the not explicitely listed solutions (10) term by term into the t−domain and obtain
formally explicit solutions for the problems Pi j given in (1), (2).
Setting
Θ⊕
i (x, ξ, t, τ ) = Θi (
x−ξ t−τ
x+ξ t−τ
, 2 ) + Θi (
, 2 ),
2l
l
2l
l
Θ⊖
i (x, ξ, t, τ ) = Θi (
x−ξ t−τ
x+ξ t−τ
, 2 ) − Θi (
, 2 ),
2l
l
2l
l
i ∈ {2, 3},
THE ONE-DIMENSIONAL HEAT EQUATION AS A FIRST-ORDER SYSTEM
183
we can write the formal solutions for our problems in the strip S as:
u11 (x, t) = −
−
1
2l
∫ l∫
0
1
2l
+
t
∫
1
l
t
Θ3, x (
0
Θ⊕
3, x (x, ξ, t, τ )f1 (ξ, τ )dτ dξ +
0
∫
l
∫
x t−τ
1
,
)u0 (τ )dτ +
2l l2
l
1
2l
t
Θ4, x (
0
∫ l∫
0
t
x t−τ
,
)ul (τ )dτ
2l l2
Θ⊖
3 (x, ξ, t, τ )f2 (ξ, τ )dτ dξ
0
Θ⊖
3 (x, ξ, t, 0)f (ξ)dξ ,
0
(P11 )
v 11 (x, t) = −
+
∫
0
1
2l
t
Θ3, t (
0
∫ l∫
1
2l
−
1
l
t
Θ⊕
3, t (x, ξ, t, τ )f1 (ξ, τ )dτ dξ −
0
∫ l∫
0
t
∫
x t−τ
1
,
)u0 (τ )dτ +
2l l2
l
t
Θ4, t (
0
∫
1
2l
0
Θ⊕
3 (x, ξ, τ, τ )f1 (ξ, t)dξ
0
1
2l
Θ⊖
3, x (x, ξ, t, τ )f2 (ξ, τ )dτ dξ +
l
x t−τ
,
)ul (τ )dτ
2l l2
∫
l
Θ⊖
3, x (x, ξ, t, 0)f (ξ)dξ
0
1
x
1
x
− Θ3 ( , 0)u0 (t) + Θ4 ( , 0)ul (t) ,
l
2l
l
2l
u22 (x, t) = −
−
1
2l
∫ l∫
0
1
+
2l
t
1
2l
∫
1
+
2l
Θ3 (
0
1
x t−τ
,
)v0 (τ )dτ +
2l l2
l
Θ⊖
3, x (x, ξ, t, τ )f1 (ξ, τ )dτ dξ +
l
1
2l
∫
t
Θ4 (
0
∫ l∫
0
t
x t−τ
,
)vl (τ )dτ
2l l2
Θ⊕
3 (x, ξ, t, τ )f2 (ξ, τ )dτ dξ
0
Θ⊕
3 (x, ξ, t, 0)f (ξ)dξ ,
0
∫ l∫
0
t
0
(P22 )
v 22 (x, t) = −
−
∫
1
l
1
l
t
∫
t
Θ3, x (
0
x t−τ
1
,
)v0 (τ )dτ +
2l l2
l
Θ⊖
3, t (x, ξ, t, τ )f1 (ξ, τ )dτ dξ −
0
∫ l∫
0
0
t
Θ⊕
3, x (x, ξ, t, τ )f2 (ξ, τ )dτ dξ
1
2l
∫
t
Θ4, x (
0
∫
l
x t−τ
,
)vl (τ )dτ
2l l2
Θ⊖
3 (x, ξ, τ, τ )f1 (ξ, t)dξ
0
1
+
2l
∫
0
l
Θ⊕
3, x (x, ξ, t, 0)f (ξ)dξ ,
184
H. TOPARKUS
u12 (x, t) = −
−
1
2l
∫ l∫
0
1
+
2l
t
∫
Θ2, x (
0
l
∫
x t−τ
1
,
)u0 (τ )dτ +
2l l2
l
Θ⊕
2, x (x, ξ, t, τ )f1 (ξ, τ )dτ dξ +
1
2l
t
Θ1 (
0
∫ l∫
0
t
x t−τ
,
)vl (τ )dτ
2l l2
Θ⊖
2 (x, ξ, t, τ )f2 (ξ, τ )dτ dξ
0
Θ⊖
2 (x, ξ, t, 0)f (ξ)dξ ,
0
1
l
∫
0
1
2l
t
Θ2, t (
0
∫ l∫
1
2l
+
t
0
v 12 (x, t) = −
−
∫
1
l
t
0
∫ l∫
0
t
∫
x t−τ
1
,
)u0 (τ )dτ +
2l l2
l
Θ⊕
2, t (x, ξ, t, τ )f1 (ξ, τ )dτ dξ −
0
t
Θ1, x (
0
1
2l
Θ⊖
2, x (x, ξ, t, τ )f2 (ξ, τ )dτ dξ +
(P12 )
∫
l
x t−τ
,
)vl (τ )dτ
2l l2
Θ⊕
2 (x, ξ, τ, τ )f1 (ξ, t)dξ
0
1
2l
∫
l
Θ⊖
2, x (x, ξ, t, 0)f (ξ)dξ
0
1
x
− Θ2 ( , 0)u0 (t) ,
l
2l
u21 (x, t) = −
−
1
2l
∫ l∫
0
1
+
2l
t
∫
1
2l
+
t
Θ2 (
0
x t−τ
1
,
)v0 (τ )dτ +
2l l2
l
Θ⊖
2, x (x, ξ, t, τ )f1 (ξ, τ )dτ dξ +
0
l
1
2l
1
2l
∫
t
Θ1, x (
0
∫ l∫
0
t
x t−τ
,
)ul (τ )dτ
2l l2
Θ⊕
2 (x, ξ, t, τ )f2 (ξ, τ )dτ dξ
0
Θ⊕
2 (x, ξ, t, 0)f (ξ)dξ ,
0
v 21 (x, t) = −
−
∫
1
l
1
l
∫
Θ2, t (
0
∫ l∫
0
t
x t−τ
1
,
)u0 (τ )dτ +
2l l2
l
Θ⊕
2, t (x, ξ, t, τ )f1 (ξ, τ )dτ dξ −
0
∫ l∫
0
t
t
0
∫
t
Θ1, x (
0
1
2l
Θ⊖
2, x (x, ξ, t, τ )f2 (ξ, τ )dτ dξ +
(P21 )
∫
1
2l
l
x t−τ
,
)vl (τ )dτ
2l l2
Θ⊕
2 (x, ξ, τ, τ )f1 (ξ, t)dξ
0
∫
l
Θ⊖
2, x (x, ξ, t, 0)f (ξ)dξ
0
1
x
− Θ2 ( , 0)u0 (t) .
l
2l
Each permitted input f , u0 , v0 , ul , vl , on the three sides of the strip S according
to (1),(2) leads to a formal solution (u(x, t), v(x, t)), 0 < t ≤ te , x ∈ (0, l), i.e.
THE ONE-DIMENSIONAL HEAT EQUATION AS A FIRST-ORDER SYSTEM
185
each permitted initial state has a solution for the interior of the strip and for the
closure of the strip at t = te , x ∈ (0, l).
The problem (1),(2) therefore describes evolutionary problems. The point is that
the problems Pi j are initial value problems in the strip S.
11
22
12
It needs to be explained in full detail that the formal solutions (P ), (P ), (P ),
21
(P ) solve the problem (1), (2), i.e. that they satisfy the assumption of a theorem
concerning existence and unity. We recall that [1, 12] contain proofs of theorems
concerning existence and unity of the solution of the homogeneous and the inhomogeneous heat equation of second order in the strip S. In order to prove such a
theorem for the problem (1), (2), more work is needed.
If we reduce (1), (2) to the classical heat equation of second order and we adopt the
assumptions of the theorems in [1, 12] and we look at the remaining parts of the
11
22
solution u11 (x, t) of (P ), u22 (x, t) of (P ), we obtain solutions which are identical
to those in [1, 12].
References
[1] J.R. Cannon, The One-Dimensional Heat Equation, Addison-Wesley Publishing Company,
Menlo Park, 1984, pp. 62-63, pp. 327-340, (Th.6.3.1, Th.6.4.1, Th.19.3.4, Th. 19.3.5) .
[2] G. Doetsch, Einführung in Theorie und Anwendung der Laplace-Transformation, Birkhäuser
Verlag, Basel und Stuttgart, 2. Auflage 1970, pp. 297-308 .
[3] I.M. Gelfand, G.E. Schilov, Verallgemeinerte Funktionen (Distributionen), Band 1, Deutscher
Verlag der Wissenschaften, Berlin, 1960, pp. 45-47.
[4] M. Hermann, Numerik gewöhnlicher Differentialgleichungen, Anfangs- und Randwertprobleme, Oldenbourg Verlag, München, Wien, 2004, pp. 195-209 .
[5] Jahnke-Emde-Lösch, Tafeln höherer Funktionen, 6. Auflage, Revised by F. Lösch, Teubner,
Stuttgart, 1960, pp. 82-85 .
[6] J. Kevorkian, Partial Differential Equations : Analytical Solution Techniques, Springer, New
York, Berlin, Heidelberg, 2000, p. 39 .
[7] F. Oberhettinger, L. Badii, Tables of Laplace Transforms, Springer, Berlin-Heidelberg-New
York, 1973, p. 421,422,295,208,209 .
[8] A.D Polyanin, V.F. Zaitsev, Handbuch der linearen Differentialgleichungen, Spektrum
Akademischer Verlag, Heidelberg, Berlin, Oxford, 1996, pp. 345-358 .
[9] A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, Integrals and Series, Vol. 5, Inverse Laplace
Transforms, Gordon and Breach Science Publishers, New York, Reading, 1992, pp. 69-71 .
[10] W.J. Thompson, Atlas for Computing Mathematical Functions, Wiley and Sons INC. , New
York, 1997, pp. 525-528 .
[11] W. Walter,Einführung in die Theorie der Distributionen, 3. Auflage, BI Wissenschaftsverlag,
Mannheim, 1994, p. 47,57 .
[12] D.V. Widder, The Heat Equation, Academic Press, NewYork-San Francisco, 1975, pp. 86 106 (especially Theorems 9, 10) .
Heinz Toparkus
Maedertal 6, D-07745 Jena, Germany
E-mail address: linsys@gmx.de