Integral Representation of Hyperparabolic Equation
In this work, for functions that satisfy a Cauchy problem for hyperparabolic equations, we write integral equations and solve them using the method of successive approximations.
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Інститут математики НАН України
2001
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| Цитувати: | Integral Representation of Hyperparabolic Equation / I.V. Samoilenko // Нелінійні коливання. — 2001. — Т. 4, № 3. — С. 368-375 . — Бібліогр.: 5 назв. — англ. |
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irk-123456789-1746232021-01-27T01:26:09Z Integral Representation of Hyperparabolic Equation Samoilenko, I.V. In this work, for functions that satisfy a Cauchy problem for hyperparabolic equations, we write integral equations and solve them using the method of successive approximations. 2001 Article Integral Representation of Hyperparabolic Equation / I.V. Samoilenko // Нелінійні коливання. — 2001. — Т. 4, № 3. — С. 368-375 . — Бібліогр.: 5 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/174623 en Нелінійні коливання Інститут математики НАН України |
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In this work, for functions that satisfy a Cauchy problem for hyperparabolic equations, we write
integral equations and solve them using the method of successive approximations. |
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Article |
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Samoilenko, I.V. |
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Samoilenko, I.V. Integral Representation of Hyperparabolic Equation Нелінійні коливання |
| author_facet |
Samoilenko, I.V. |
| author_sort |
Samoilenko, I.V. |
| title |
Integral Representation of Hyperparabolic Equation |
| title_short |
Integral Representation of Hyperparabolic Equation |
| title_full |
Integral Representation of Hyperparabolic Equation |
| title_fullStr |
Integral Representation of Hyperparabolic Equation |
| title_full_unstemmed |
Integral Representation of Hyperparabolic Equation |
| title_sort |
integral representation of hyperparabolic equation |
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Інститут математики НАН України |
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2001 |
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http://dspace.nbuv.gov.ua/handle/123456789/174623 |
| citation_txt |
Integral Representation of Hyperparabolic Equation / I.V. Samoilenko // Нелінійні коливання. — 2001. — Т. 4, № 3. — С. 368-375 . — Бібліогр.: 5 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT samoilenkoiv integralrepresentationofhyperparabolicequation |
| first_indexed |
2025-07-15T11:39:05Z |
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2025-07-15T11:39:05Z |
| _version_ |
1837712849608114176 |
| fulltext |
Nonlinear Oscillations, Vol. 4, No. 3, 2001
INTEGRAL REPRESENTATION
OF HYPERPARABOLIC EQUATION
I. V. Samoilenko
Institute of Mathematics, NAS of Ukraine
Tereshchenkivs’ka St., 3, Kyiv, 01601, Ukraine
In this work, for functions that satisfy a Cauchy problem for hyperparabolic equations, we write
integral equations and solve them using the method of successive approximations.
AMS Subject Classification: 60J30, 35Q40, 45L10
Introduction
The Riemman method is a well-known method for solving a Cauchy problem for the telegraph
equation [1]. But this method can’t be used for an analogue of the telegraph equation in Rn
(hyperparabolic equations), and also nonlinear hyperparabolic equations that are defined later.
1. Cauchy Problem for Telegraph Equation
The telegraph process and the corresponding random evolution is a model for the motion
of a physical particle on a line, when the particle changes its direction of motion to the opposite
in random moments of time. Besides, the telegraph process defines a “rectangular wave” as an
oscillating process in problems of radioengineering [2].
It’s well-known [3] that a special form of a functional of the Markov random evolution
satisfies the Cauchy problem
∂2
∂t2
u(x, t) = −2λ ∂
∂t
u(x, t) + V 2 ∂
2
∂x2
u(x, t),
(1)
u(x, 0) = f(x),
∂
∂t
u(x, t)
∣∣∣
t=0
= V
d
dx
f(x).
But, for the functional u(x, t), it is possible to write an integral equation from the first jump
of the process,
u(x, t) = e−λtf(x+ V t) + λe−λt
t∫
0
f(x− V (t+ 2s))ds
+ λ2
t∫
0
t−s∫
0
e−λ(l+s)u(x+ V (s− l), t− (l + s))dlds. (2)
368 c© I. V. Samoilenko, 2001
INTEGRAL REPRESENTATION OF HYPERPARABOLIC EQUATION 369
The problem (1) can be obtained from (2). To do this, let us change the variables in (2),
u(x, t) = e−λtf(x+ V t) + λe−λt
t∫
0
f(x− V (t+ 2s))ds
− λ2e−λt
t∫
0
∫ t−f
0
eλfu(x+ V (2s+ f − t), f)dsdf .
Differentiating the last expression with respect to x and t we have
∂2
∂x2
u(x, t) = e−λt
d2
dx2
f(x+ V t) + λe−λt
t∫
0
d2
dx2
f(x− V (t+ 2s))ds
− λ2e−λt
t∫
0
t−f∫
0
eλf
∂2
∂x2
u(x+ V (2s+ f − t), f)dsdf,
∂
∂t
u(x, t) =− λ
[
e−λtf(x+ V t) + λe−λt
t∫
0
f(x− V (t− 2s))ds
− λ2e−λt
t∫
0
t−f∫
0
eλfu(x+ V (2s+ f − t), f)dsdf
]
+
[
V e−λt
d
dx
f(x+ V t)
− λV e−λt
t∫
0
d
dx
f(x− V (t− 2s))ds+ λ2V e−λt
t∫
0
t−f∫
0
eλf
× ∂
∂x
u(x+ V (2s+ f − t), f)dsdf
]
+ λe−λtf(x+ V t)− λ2e−λt
t∫
0
eλf
× u(x+ V t− V f, f)df = −λu(x, t) +
[
V e−λt
d
dx
f(x+ V t)
− λV e−λt
t∫
0
d
dx
f(x− V (t− 2s))ds+ λ2V e−λt
t∫
0
t−f∫
0
eλf
370 I.V. SAMOILENKO
× ∂
∂x
u(x+ V (2s+ f − t), f)dsdf
]
+ λe−λtf(x+ V t)
− λ2e−λt
t∫
0
eλfu(x+ V t− V f, f)df,
∂2
∂t2
u(x, t) =− λ ∂
∂t
u(x, t)− λ
[
V e−λt
d
dx
f(x+ V t)− λV e−λt
×
t∫
0
d
dx
f(x− V (t− 2s))ds+ λ2V e−λt
t∫
0
t−f∫
0
eλf
∂
∂x
u(x+ V (2s
+ f − t), f)dsdf + λe−λtf(x+ V t)− λ2e−λt
t∫
0
eλfu(x+ V t− V f, f)df
]
+ V 2
[
e−λt
d2
dx2
f(x+ V t) + λe−λt
t∫
0
d2
dx2
f(x− V (t− 2s))ds
− λ2e−λt
t∫
0
t−f∫
0
eλf
∂2
∂x2
u(x+ V (2s+ f − t), f)dsdf
]
− λV e−λt d
dx
f(x
+ V t) + λ2V e−λt
t∫
0
eλf
d
dx
u(x+ V t− V f, f)df + λV e−λt
d
dx
f(x
+ V t)− λ2V e−λt
t∫
0
eλf
d
dx
u(x+ V t− V f, f)df + λ2e−λteλtu(x, t)
=− λ ∂
∂t
u(x, t)− λ
[
∂
∂t
u(x, t) + λu(x, t)
]
+ V 2 ∂
2
∂x2
u(x, t) + λ2u(x, t)
=− 2λ
∂
∂t
u(x, t) + V 2 ∂
2
∂x2
u(x, t).
We thus obtained the Cauchy problem (1). The following theorem holds.
INTEGRAL REPRESENTATION OF HYPERPARABOLIC EQUATION 371
Theorem 1. The function u(x, t), satisfying (1) under the condition of integrability of u(x, t)
and f(x), satisfies equation (2). The function u(x, t), satisfying (1) under the condition of di-
fferentiability with respect to x and t, satisfies the Cauchy problem (1).
The Cauchy problem (1) and equation (2) are equivalent in this sense.
Let us consider the question of whether there exist a solution of (2) in the space of functions
φ(x, t) = φ0(x, t) + c, (3)
where c = const, φ0(x) → 0, x, t → ∞ . This Banach space with sup-norm was studied in the
works of V.S.Koroljuk and A.F. Turbin (see, for example, [4]).
Let us write (2) in the following form: u(x, t) = Au(x, t), where Au(x, t) is equal the right-
hand side of (2). A acts in the space (3), when f(x) = f0(x) + c. Indeed,
A(φ0 + c) =e−λt(f0(x) + c) + λe−λt
t∫
0
[f0(x)(x− V (t− 2s)) + c]ds
+ λ2
t∫
0
t−s∫
0
e−λ(l+s)[φ0(x+ V (s− l), t− (s+ l)) + c]dlds =
{
f0(x)e
−λt
+ λe−λt
∫ t
0
f0(x)(x− V (t− 2s))ds+ λ2
t∫
0
t−s∫
0
e−λ(l+s)φ0(x+ V (s− l), t
− (s+ l))dlds
}
+
{
e−λtc+ λe−λttc+ λ2
(
− 1
λ
e−λtct− 1
λ2
(e−λt − 1)c
)}
=
{
f0(x)e
−λt + λe−λt
t∫
0
f0(x)(x− V (t− 2s))ds+ λ2
t∫
0
t−s∫
0
e−λ(l+s)
× φ0(x+ V (s− l), t− (s+ l))dlds
}
+ c,
where the expression in braces converges to 0 as x, t → ∞. Indeed,
e−λt
t∫
0
f(x)dx < e−λt
t∫
0
sup
x
|f(x)|dx = sup
x
|f(x)|te−λt → 0,
x, t → ∞, t > 0,
372 I.V. SAMOILENKO
t∫
0
t−s∫
0
e−λ(l+s)φ0(x+ V (s− l), t− (s+ l))dlds
< sup
x,t
|φ0(x+ V (s− l), t− (s+ l))|
t∫
0
t−s∫
0
e−λ(l+s)dlds
= sup
x,t
∫ t
0
(
− 1
λ
)[
e−λt − e−λs
]
ds
= sup
x,t
[
− 1
λ
te−λt − 1
λ2
(
e−λt − 1
)]
→ 0, x, t → ∞.
Let us show that A is a contraction. We have
ρ(Aφ1, Aφ2) = sup
x,t
|λ2
t∫
0
t−s∫
0
e−λ(l+s)φ1(x+ V (s− l), t− (s+ l))dlds
− λ2
t∫
0
t−s∫
0
e−λ(l+s)φ2(x+ V (s− l), t− (s+ l))dlds ≤ λ2
t∫
0
t−s∫
0
e−λ(l+s)
× |φ1(x+ V (s− l), t− (s+ l))− φ2(x+ V (s− l), t− (s+ l))|dlds
= ρ(φ1, φ2)[−λte−λt − e−λt + 1],
where [−λte−λt − e−λt + 1] < 1, so A is indeed a contraction. By the fixed point theorem, a
solution of (2) exists and it is unique.
Theorem 2. For f(x) from the space (3), equation (2) has a unique solution lying in (3).
The solution is lim
n→∞
un(x, t) = lim
n→∞
Aun−1(x, t),where u0(x, t) is an arbitrary function from
(3). But convergence also takes place in other spaces.
Example 1. f(x) = x,
u0(x, t) = 0, u1(x, t) = e−λt(x+ xt+ V t),
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
un(x, t) = e−λt
(
x
2n−1∑
0
tk
k!
λk +
V
λ
2n−1∑
0
(λt)k
k!
)
,
INTEGRAL REPRESENTATION OF HYPERPARABOLIC EQUATION 373
where k is even in the second sum,
lim
n→∞
un(x, t) = x+
V
2λ
(1− e−2λt),
which coincides with the first moment of the Markov random evolution found in [5].
It should be noted that this method can be used for hyperparabolic equations in Rn.
2. Nonlinear Hyperparabolic Equation
A fading telegraph process and a fading Markov random evolution defines a motion of a
particle on a line in the field of gravity, if the particle is attracted to some point on the line. From
the point of view of radioengineering a fading telegraph process defines a fading “rectangular
wave”.
When we consider a generalization of a telegraph process, a fading telegraph process, another
integral equation appears,
u(V, x, t) = e−λtf(x+ V t) + λe−λt
t∫
0
f
(
x+ V s− V
c
(t− s)
)
+ λ2
t∫
0
t−s∫
0
e−λ(l+s)u
(
V
c2
, x+ V s− V
c
l, t− (s+ l)
)
dlds. (4)
Let us note that for c = 1, equation (4) coincides with (2), where u(V, x, t) = u(x, t).
By making the changes of variables, similar to the one in Section 1, and differentiating we
get a Cauchy problem corresponding to (4),
∂3
∂t3
u(V, x, t) + 3λ
∂2
∂t2
u(V, x, t) + 3λ2
∂
∂t
u(V, x, t)− λ2 ∂
∂t
u
(
V
c2
, x, t
)
− V 2
c2
∂3
∂x2∂t
u(V, x, t)− V 2λ
c2
∂2
∂x2
u(V, x, t)
− c− 1
c
λ2V u
(
V 2
c2
, x, t
)
+ λ3u(V, x, t)− λ3u
(
V
c2
, x, t
)
= 0, (5)
u(V, x, t) = f(x),
∂
∂t
u(x, t)
∣∣∣
t=0
= V
d
dx
f(x),
∂2
∂t2
u(x, t)
∣∣∣
t=0
= −c+ 1
c
λV
d
dx
f(x) + V 2 d
2
dx2
f(x).
374 I.V. SAMOILENKO
For c = 1, we have
∂3
∂t3
u(x, t) + 3λ
∂2
∂t2
u(x, t) + 2λ2
∂
∂t
u(x, t)
− V 2 ∂3
∂x2∂t
u(x, t)− V 2λ
∂2
∂x2
u(x, t) = 0 (6)
or {
∂3
∂t3
u(x, t) + 2λ
∂2
∂t2
u(x, t)− V 2 ∂3
∂x2∂t
u(x, t)
}
+
{
λ
∂2
∂t2
u(x, t) + 2λ2
∂
∂t
u(x, t)− V 2λ
∂2
∂x2
u(x, t)
}
= 0,
∂
∂t
{
∂2
∂t2
u(x, t) + 2λ
∂
∂t
u(x, t)− V 2 ∂
2
∂x2
u(x, t)
}
+ λ
{
∂2
∂t2
u(x, t) + 2λ
∂
∂t
u(x, t)− V 2 ∂
2
∂x2
u(x, t)
}
= 0,
{
∂
∂t
+ λ
}{
∂2
∂t2
+ 2λ
∂
∂t
− V 2 ∂
2
∂x2
}
u(x, t) = 0.
This is a factorized equation, one component of which coincides with (1). Correspondingly,
if u(x, t) satisfies (5) for c = 1, then it satisfies (1).
The proof of existence of a solution of (4) is similar to that in Section 1 for the space of
functions φ(V, x, t) = φ0(V, x, t) + c. The following theorem holds.
Theorem 3. The Cauchy problem for nonlinear hyperparabolic equation (5) is equivalent to
integral equation (4) that has a solution in the space of functions φ(V, x, t) = φ0(V, x, t) + c,
where c = const, φ0(V, x, t) → 0, V, x, t → ∞.
As in Section 1, the method of successive approximations converges for the functions
f(x) = xk.
Example 2. f(x) = x,
u0(V, x, t) = 0, u1(V, x, t) = e−λt
(
x+ xt+ V t− V t2
2c
+
V t2
2
)
,
INTEGRAL REPRESENTATION OF HYPERPARABOLIC EQUATION 375
u2(V, x, t) = u1(V, x, t) + λ2e−λt
(
xt2
2
+
xt3
6
+
λ2t3V
6
c2 − c+ 1
c2
+
λ3t4V
24
c3 − c2 + c− 1
c3
)
,
lim
n→∞
un(V, x, t) = x+
V c
λ(c+ 1)
(
1− e−λt
1+c
c
)
.
For c = 1, we have x+
V
2λ
(
1− e−2λt
)
(see Example 1).
REFERENCES
1. Glinner E.B. and Koshliakov N.S. Partial Differential Equations of Mathematical Physics [in Russian], Vysshaya
Shkola, Moscow (1970).
2. Levin B.R. Theoretical Bases of Statistical Radio Engineering [in Russian], Soviet Radio, Moscow (1966).
3. Kac M. Probability and Related Questions in Physics [in Russian], Mir, Moscow (1965).
4. Koroljuk V.S. and Turbin A.F. Mathematical Foundation of State Lumping of Large Systems, Kluver Academic
Press, Amsterdam (1990).
5. Turbin A.F. and Samoilenko I.V. “Probabilistic method of solution of telegraph equation with real-analytical
initial conditions,” Ukr. Mat. Zh., 52, No. 8, 1127 – 1134 (2000).
Received 05.12.2000
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