On a Theoretical Study of the Properties of Solutions of the Limit Problem for a Magnetically Noninsulated Diode
In this paper, we study the lower and upper bounds for solutions of the limit problem for the plane vacuum diode in the magnetic field in the statement by N. Ben Abdallah, P. Degond, and F. Mehats. This problem was finally set by physicists in the late 1980s and was extensively studied by numerous m...
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irk-123456789-1657512020-02-17T01:25:55Z On a Theoretical Study of the Properties of Solutions of the Limit Problem for a Magnetically Noninsulated Diode Dulov, E.V. Sinitsyn, A.V. Статті In this paper, we study the lower and upper bounds for solutions of the limit problem for the plane vacuum diode in the magnetic field in the statement by N. Ben Abdallah, P. Degond, and F. Mehats. This problem was finally set by physicists in the late 1980s and was extensively studied by numerous mathematicians in the 1990s. Досліджено нижні i вepxнi мєжі для розв'язків граничної задачi плоского вакуумного діода у магнітному полі у постановці Н. Бен Абдалла, П. Дегонда та Р. Мехаца. Ця задача була остаточно поставлена фізиками наприкінці 1980-х років i уважно досліджена багатьма математиками у 1990-х роках. 2005 Article On a Theoretical Study of the Properties of Solutions of the Limit Problem for a Magnetically Noninsulated Diode / E.V. Dulov, A.V. Sinitsyn // Український математичний журнал. — 2005. — Т. 57, № 6. — С. 840–851. — Бібліогр.: 6 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165751 517.9 + 531.19 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Dulov, E.V. Sinitsyn, A.V. On a Theoretical Study of the Properties of Solutions of the Limit Problem for a Magnetically Noninsulated Diode Український математичний журнал |
| description |
In this paper, we study the lower and upper bounds for solutions of the limit problem for the plane vacuum diode in the magnetic field in the statement by N. Ben Abdallah, P. Degond, and F. Mehats. This problem was finally set by physicists in the late 1980s and was extensively studied by numerous mathematicians in the 1990s. |
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| author |
Dulov, E.V. Sinitsyn, A.V. |
| author_facet |
Dulov, E.V. Sinitsyn, A.V. |
| author_sort |
Dulov, E.V. |
| title |
On a Theoretical Study of the Properties of Solutions of the Limit Problem for a Magnetically Noninsulated Diode |
| title_short |
On a Theoretical Study of the Properties of Solutions of the Limit Problem for a Magnetically Noninsulated Diode |
| title_full |
On a Theoretical Study of the Properties of Solutions of the Limit Problem for a Magnetically Noninsulated Diode |
| title_fullStr |
On a Theoretical Study of the Properties of Solutions of the Limit Problem for a Magnetically Noninsulated Diode |
| title_full_unstemmed |
On a Theoretical Study of the Properties of Solutions of the Limit Problem for a Magnetically Noninsulated Diode |
| title_sort |
on a theoretical study of the properties of solutions of the limit problem for a magnetically noninsulated diode |
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Інститут математики НАН України |
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2005 |
| topic_facet |
Статті |
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http://dspace.nbuv.gov.ua/handle/123456789/165751 |
| citation_txt |
On a Theoretical Study of the Properties of Solutions of the Limit Problem for a Magnetically Noninsulated Diode / E.V. Dulov, A.V. Sinitsyn // Український математичний журнал. — 2005. — Т. 57, № 6. — С. 840–851. — Бібліогр.: 6 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT dulovev onatheoreticalstudyofthepropertiesofsolutionsofthelimitproblemforamagneticallynoninsulateddiode AT sinitsynav onatheoreticalstudyofthepropertiesofsolutionsofthelimitproblemforamagneticallynoninsulateddiode |
| first_indexed |
2025-07-14T19:48:44Z |
| last_indexed |
2025-07-14T19:48:44Z |
| _version_ |
1837653067201249280 |
| fulltext |
UDC 517.9 + 531.19
A. V. Sinitsyn (Univ. Nac. de Colombia, Bogotá, Colombia;
Inst. System Dynamics and Control Theory SB RAS, Irkutsk, Russia),
E. V. Dulov (Univ. Nac. de Colombia, Bogotá, Colombia)
ON A THEORETICAL STUDY
FOR THE SOLUTION PROPERTIES OF THE LIMIT PROBLEM
FOR THE MAGNETICALLY NONINSULATED DIODE
∗
ПРО ТЕОРЕТИЧНЕ ДОСЛIДЖЕННЯ
ВЛАСТИВОСТЕЙ РОЗВ’ЯЗКIВ ГРАНИЧНОЇ ЗАДАЧI
ДЛЯ МАГНIТНО НЕIЗОЛЬОВАНОГО ДIОДА
In this paper we study the lower and upper bounds for the solutions of the limit problem of the plane
vacuum diod in the magnetic field in the statement by N. Ben Abdallah, P. Degond and F. M’ehats.
This problem was finally set by a physists in late 1980-s and was attentively studied by a number of the
mathematitians in 1990-s.
Дослiджено нижнi i верхнi межi для розв’язкiв граничної задачi плоского вакуумного дiода у
магнiтному полi у постановцi Н. Бен Абдалла, П. Дегонда та Р. Мехаца. Ця задача була оста-
точно поставлена фiзиками наприкiнцi 1980-х рокiв i уважно дослiджена багатьма математиками у
1990-х роках.
1. Introduction. This paper is aimed at studying the stationary self-consistent problem
of magnetic insulation under space-charge limitation via the asymptotics of the Vlasov –
Maxwell system. This approach has been introduced by Langmuir and Compton [1]
and recently developed by Degond and Raviart [2], N. Ben Abdallah, P. Degond and
F. M’ehats [3] to analyze the space charge limited operation of a vacuum diode. In
a dimensionless form of the Vlasov – Poisson system, the ratio of the typical particle
velocity at the cathode to that reached at the anode appears as a small parameter [2].
The associated perturbation analysis provides a mathematical framework to the results
of Langmuir and Compton [1], stating that the current flowing through the diode cannot
exceed a certain value called the Child – Langmuir current. This paper is concerned with
an extension of this approach, based on the Child – Langmuir asymptotics to magnetized
flows [3]. In particular, the value of the space charge limited current is determined
when the magnetic field is small (noninsulated diode). Since the arising model could
not be solved analytically, it is very important to discover its properties in noninsulated
and nearly-insulated cases first.
For better understanding of the discussed mathematical problem and the obtained
results with a rising physical effects in vacuum diode, first we need to introduce the
description, how it really works.
The excellent description of this processes found in [4] is brought here.
1.1. Description of vacuum diode. The vacuum diode consists of a hot cathode
surrounded by a metal anode inside an evacuated enclosure. At suffciently high
temperatures electrons are emitted from the cathode and are attracted to the positive
anode. Electrons moving from the cathode to the anode constitute a current; they do
∗ This work is supported by INTAS No. 2000-15 and grant of the National University of Colombia.
c© A. V. SINITSYN, E. V. DULOV, 2005
840 ISSN 1027-3190. Укр. мат. журн., 2005, т. 57, № 6
ON A THEORETICAL STUDY FOR THE SOLUTION PROPERTIES OF THE LIMIT . . . 841
so when the anode is positive with respect to the cathode. When the anode is negative
with respect to the cathode, electrons are repelled by the anode and the reverse current
is almost zero (due to the tail of the Maxwellian distribution of the electrons it is
greater than zero). The space between the anode and the cathode is evacuated, so that
electrons may move between the electrodes unimpeded by collisions with gas molecules.
If Vf = 0 and no emission takes place, the diode may be regarded as a parallel-plate
capacitor whose potential difference is Vp. In this case, the potential distribution in the
cathode-plate space is represented by a straight line which joins the points corresponding
to cathode potential Vk = 0 and the plate potential Vp. When the filament voltage rises,
the electrons leaving the cathode gang up in the interelectrode space as a cloud called a
space charge. This charge alters the potential distribution. Since the electrons making
up the space charge are negative, the potential in the cathode-plate space goes up, though
all points remain at positive potential. The vector of the electric field is directed from
the plate to the cathode, so all the electrons escaping from the cathode make for the
plate. In this case, the plate current equals the emission current. One could say the all
electrons are being sucked away from the cathode by the anode. This region is known
as the emission-limited region. As the filament voltage is increased, emission increases,
and so does the space charge. Electrons having low initial velocities are driven back
to the cathode by the negative space charge due to the electrons. The density of the
electron cloud near the cathode increases to the point where it forms a negative potential
region whose minimum, Vmin, is usually within a few hundredth or tenths of a millimetre
of the cathode surface. Thus, there is a high retarding electric field near the cathode
(0 < x < xmin); the vector is directed away from the cathode to the plate. To overcome
this field, an initial velocity v0 of the electrons leaving the cathode should exceed a
certain value determined by Vmin, v0 >
√
2
e
m
Vmin.
If the electron is below this value, the electron will not be able to overcome the
potential barrier. It will slow down to a stop, and the field will push it back to the
cathode. Accordingly, the retarding field region (from 0 to xmin) contains not only
electrons traveling away from the cathode, but also those falling back towards the
cathode. At a constant filament voltage, a dynamic equilibrium sets in, so that the
number of electrons reaching the plate and the number falling back to the cathode is
equal to the number of electrons emitted by the cathode. Therefore, plate current is
smaller than emission current, or the cathode produces more electrons than the anode
can.
1.2. Description of the mathematical model. We consider a plane diode consisting of
two perfectly conducting electrodes, a cathode (X = 0) and an anode (X = L) supposed
to be infinite planes, parallel to (Y ;Z). The electrons, with charge −e and mass m, are
emitted at the cathode and submitted to an applied electromagnetic field Eext = EextX;
Bext = BextZ such that Eext ≤ 0 and Bext ≥ 0. Such an electromagnetic field does
not act on the PZ component of the particle momentum. Hence, we shall consider a
situation where this component vanishes, leading to a confinement of electrons to the
plane Z = 0. The relationship between momentum and velocity is then given by the
relativistic relations
ISSN 1027-3190. Укр. мат. журн., 2005, т. 57, № 6
842 A. V. SINITSYN, E. V. DULOV
V(P) =
P
γm
, γ =
√
1 +
|P|2
m2c2
,
V = (VX , VY ), P = (PX , PY ), |P|2 = P 2
X + P 2
Y ,
(1)
which can also be written
V(P) = 5PE(P), (2)
where E is the relativistic kinetic energy
E(P) = mc2(γ − 1), (3)
and c is the speed of light. We shall moreover assume that the electron distribution
function F does not depend on Y and that the flow is stationary and collisionless. The
injection profile G(PX , PY ) at the cathode is assumed to be given whereas no electron
is injected at the anode. The system is then described by the so called 1, 5 dimensional
Vlasov – Maxwell model
VX
∂F
∂X
+ e
(
dΦ
dX
− VY
∂F
∂PX
)
+ eVX
dA
dX
∂F
∂PY
= 0, (4)
d2Φ
dX2
=
e
ε0
N(X), X ∈ (0, L), (5)
d2A
dX2
= −µ0JY (X), X ∈ (0, L), (6)
subject to the following boundary conditions :
F (0, PX , PY ) = G(PX , PY ), PX > 0, (7)
F (L,PX , PY ) = 0, PX < 0, (8)
Φ(0) = 0, Φ(L) = ΦL = −LEext, (9)
A(0) = 0, A(L) = AL = LBext. (10)
In this system, the macroscopic quantities, namely the particle density N, X and Y are
the components of the current density JX , JY . In the above equations, ε0 and µ0 are
respectively the vacuum permittivity and permeability.
The boundary conditions are justified by the fact that the electric field E = −d
Φ
dX
and the magnetic field B = − dA
dX
are exactly equal to the external fields when self-
consistent effects are ignored (N = JY = 0).
The 1, 5 dimensional model (4) – (10) ignores the self-consistent magnetic field due
to JX , which would introduce two-dimensional effects, and is only an approximation of
the complete stationary Vlasov – Maxwell system. In this paper we especially interested
in the case, when the applied magnetic field is not strong enough to insulate the diode,
JX does not vanish and our model can be viewed as an approximation of the Maxwell
equations.
ISSN 1027-3190. Укр. мат. журн., 2005, т. 57, № 6
ON A THEORETICAL STUDY FOR THE SOLUTION PROPERTIES OF THE LIMIT . . . 843
In order to get a better insight in the behaviour of the diode, we write the model in
dimensionless variables in the spirit of [2, 5]. We first introduce the following units
respectively for position, velocity, momentum, electrostatic potential, vector potential,
particle density, current and distribution function:
X̄ = L, V̄ = c, P̄ = mc, E = mc2,
Φ̄ =
mc2
e
, Ā =
mc
e
, N̄ =
ε0Φ̄
xX̄2
, J̄ = −ecN̄ , F̄ =
N̄
P̄ 2
,
and the corresponding dimensionless variables
x =
X
X̄
, p =
P
P̄
= (px, py),
v = (vx, vy) =
V
V̄
=
p√
1 + p2
, ε =
E
Ē
=
√
1 + p2 − 1,
ϕ =
Φ
Φ̄
, a =
A
Ā
, n =
N
N̄
, j =
J
J̄
, f =
F
F̄
.
The next step is to express that particle emission at the cathode occurs in the Child –
Langmuir regime: in such a situation, the thermal velocity VG is much smaller than
the typical drift velocity supposed to be of the order of the speed of light c. Letting
ε = VG/c, we shall assume that
f(0, px, py) = gε(px, py) =
1
ε3
g
(px
ε
,
py
ε
)
, px > 0,
where g is a given profile. The scaling factor ε3 insures that the incoming current
remains finite independently of ε, whereas the dependence on
p
ε
expresses the fact that
electrons are emitted at the cathode with a very small velocity. We refer to [2, 5] for a
detailed discussion of the scaling. The dimensionless system reads
vx
∂fε
∂x
+
(
dϕε
dx
− vy
daε
dx
)
∂fε
∂px
+ vx
daε
dx
∂fε
∂py
= 0, (11)
(x, px, py) ∈ (0, 1)× R2,
d2ϕε
dx2
= nε(x), x ∈ (0, 1), (12)
d2aε
dx2
= jε
y(x), x ∈ (0, 1). (13)
Here nε(x) is a particle density, jε
y(x) is a current density in Y direction. The initial
and boundary conditions are also transformed
fε(0, px, py) = gε(px, py) =
1
ε3
g
(px
ε
,
py
ε
)
, px > 0, (14)
fε(1, px, py) = 0, px < 0, (15)
ϕε(0) = 0, ϕε(1) = ϕL, (16)
ISSN 1027-3190. Укр. мат. журн., 2005, т. 57, № 6
844 A. V. SINITSYN, E. V. DULOV
aε(0) = 0, aε(1) = ϕL. (17)
Omitting the complete derivation of the limit system, when ε → 0, we need to
introduce some notions and notations used ahead.
Definition 1. We define θ(x) = (1 + ϕ(x))2 − 1− a2(x) as an effective potential.
It is readily seen that electrons do not enter the diode unless the effective potential
θ is nonnegative in the vicinity of the cathode. Therefore, we always have θ′(0) ≥ 0.
The limiting case θ′(0) = 0 is the space charge limited or the Child – Langmuir regime.
In view (16), (17) (it still hold in the limit ε → 0), this condition is equivalent to the
standard Child – Langmuir condition
dϕ
dx
(0) = 0.
Let θL be the value of θ at the anode θL = (1 + ϕL)2− 1−a2
L. If θL < 0, electrons
cannot reach the anode x = 1, they are reflected by the magnetic forces back to the
cathode and the diode is said to be magnetically insulated. This enables us to define the
Hull cut-off magnetic field, which is the relativistic version of the critical field introduced
in [6] in the nonrelativistic case:
aH
L =
√
ϕ2
L + 2ϕL.
The diode is magnetically insulated if aL > aH
L , and is not insulated if aL < aH
L In
dimensional variables, the Hull cut-off magnetic field is given by
BH =
1
Lc
√
Φ2
L +
2mc2
e
ΦL.
Thus our primary goal is a stugy of noninsulated, or nearly insulated diodes, which
means Bext < BH . The complete derivation of the model is given in [3], while we
need only its formal expressions
d2ϕ
dx2
(x) = jx
1 + ϕ(x)√
(1 + ϕ(x))2 − 1− a2(x)
, (18)
d2a
dx2
(x) = jx
a(x)√
(1 + ϕ(x))2 − 1− a2(x)
, (19)
with a corresponding Cauchy and boundary conditions
ϕ(0) = 0, ϕ(1) = ϕL, (20)
dϕ
dx
(0) = 0, (21)
a(0) = 0, a(1) = aL. (22)
Let us recall that the unknowns are the electrostatic potential ϕ, the magnetic potential
a and the current jx (which does not depend on x).
It is to be noticed that the whole construction of this model depends heavily on the
assumption that the effective potential is positive. Actually, θ could vanish at some
points in the diode, leading to closed trajectories and trapped particles.
ISSN 1027-3190. Укр. мат. журн., 2005, т. 57, № 6
ON A THEORETICAL STUDY FOR THE SOLUTION PROPERTIES OF THE LIMIT . . . 845
2. Solution trajectory, upper and lower solutions.
Finally, the limit model of magnetically noninsulation diode is described by the
system of two second order ordinary differential equations (18), (19) with conditions
(20) – (22).
2.1. Existence of semitrivial solutions for the problem. Let us introduce the
definition of cone in a Banach space X.
Definition 2. Let X be a Banach space. A nonempty convex closed set P ⊂ X is
called a cone, if it satisfies the conditions:
(i) x ∈ P, λ ≥ 0 implies λx ∈ P ;
(ii) x ∈ P, −x ∈ P implies x = O, where O denotes zero element of X.
Here ≤ is the order in X induced by P, i.e., x ≤ y if and only if y− x is an element
of P.
We will also assume that the cone P is normal in X, i.e., order intervals are norm
bounded.
In X ≡ {(u, v) : u, v ∈ C1(Ω̄), u = v = 0} we introduce the norm |U |X =
= |u|C1 + |v|C1 , and the norm |U |X = |u|∞ + |v|∞ in C, where U = (u, v). Here a
cone P is given by P = {(u, v) ∈ X : u ≥ 0, v ≥ 0 for all x ∈ Ω}. So, if u 6= 0,
v 6= 0 belong to P, then −u,−v does not belong. We will work with classical spaces
on the intervals Ī = [a, b], Î =]a, b], I = (a, b) :
C(Ī) with norm ‖u‖∞ = max{|u(x)| : x ∈ Ī};
C1(Ī) = ‖u‖∞ + ‖u′‖∞;
Cloc(I), which contains all functions that are locally absolutely continuous in I. We
introduce a space Cloc(I) because the limit problem is singular for ϕ = 0. The order
≤ in cone P is understood in the weak sense, i.e., y is increasing if a ≤ b implies
y(a) ≤ y(b) and y is decreasing if a ≤ b implies y(a) ≥ y(b).
Theorem 1 (comparison principle in cone). Let y ∈ C(Ī)
⋂
Cloc(I). The function
f is defined on I ×R. Let f(x, y) is increasing in y function, then
v′′ − f(x, v) ≥ w′′ − f(x,w) in mean on I, (23)
v(a) ≤ w(a), v(b) ≤ w(b) implies v ≤ w on Ī .
For the convenience of defining an ordering relation in cone P, we make a transfor-
mation for the problem (18) – (22). Let F (ϕ, a) and G(ϕ, a) be defined by (18) – (22).
Then through the transformation ϕ = −u the limit problem is reduced to the form
−d2u
dx2
= jx
1− u√
(1− u)2 − 1− a2
4
= F̃ (jx, u, a), u(0) = 0, u(1) = ϕL,
d2a
dx2
= jx
a√
(1− u)2 − 1− a2
4
= G̃(jx, u, a), a(0) = 0, a(1) = aL.
(24)
We note that all solutions of the initial problem, as well the problem (24), are symmetric
with respect to the transformation of sign for the magnetic potential a : (ϕ, a) = (ϕ,−a)
or the same (u, a) = (u,−a). Thus we must search only positive solutions ϕ > 0,
a > 0 in cone P or only negative ones: ϕ < 0, a < 0. Thanks to the symmetry of
problem it is equivalently and does not yields the extension of the types of sign-defined
solutions of the problem (18) – (22) (respect. (24)). Once more, we note that introduction
of negative electrostatic potential in problem (24) is connected with more convenient
ISSN 1027-3190. Укр. мат. журн., 2005, т. 57, № 6
846 A. V. SINITSYN, E. V. DULOV
relation between order in cone and positiveness of Green function for operator −u′′ that
we use below.
Definition 3. A pair [(ϕ0, a0), (ϕ0, a0)] is called
a) sub-super solution of the problem (18) – (22) relative to P, if the following
conditions are satisfied:
(ϕ0, a0) ∈ Cloc(I)
⋂
C(Ī)× Cloc(I)
⋂
C(Ī),
(ϕ0, a0) ∈ Cloc(I)
⋂
C(Ī)× Cloc(I)
⋂
C(Ī),
(25)
ϕ
′′
0 − jx
1 + ϕ0√
(1 + ϕ0)2 − 1− a2
4
= F (ϕ0, a) ≤ 0 in I,
(ϕ0)′′ − jx
1 + ϕ0√
(1 + ϕ0)2 − 1− a2
4
= F (ϕ0, a) ≥ 0 in I ∀a ∈ [a0, a
0];
a
′′
0 − jx
a0√
(1 + ϕ)2 − 1− a2
0
4
= G(ϕ, a0) ≤ 0 in I,
(a0)′′ − jx
a0√
(1 + ϕ)2 − 1− (a0)2
4
= G(ϕ, a0) ≥ 0 in I ∀ϕ ∈ [ϕ0, ϕ
0];
ϕ0 ≤ ϕ0, a0 ≤ a0 in I
and on the boundary
ϕ0(0) ≤ 0 ≤ ϕ0(0), ϕ0(1) ≤ ϕL ≤ ϕ0(1),
a0(0) ≤ 0 ≤ a0(0), a0(1) ≤ aL ≤ a0(1);
b) sub-sub solution of the problem (18) – (22) relative to P, if a condition (25) is
satisfied and
ϕ
′′
0 − F (jx, ϕ0, a0) ≤ 0 in I,
a
′′
0 −G(jx, ϕ0, a0) ≤ 0 in I
(26)
and on the boundary
ϕ0(0) ≤ 0, ϕ0(1) ≤ ϕL, a0(0) ≤ 0, a0(1) ≤ aL. (27)
Remark 1. In Definition 3 the expressions with square roots we take by modulus of
effective potential θ(·).
By analogy with (26), (27), we may introduce the definition of super-super solution
in cone.
Definition 4. The functions Φ(x, xai , jx), Φ1(x, xϕj , jx) we shall call a semitrivial
solutions of the problem (18) – (22), if Φ(x, xai
, jx) is a solution of the scalar boundary-
value problem
ϕ′′ = F (jx, ϕ, xai) = jx
1 + ϕ√
(1 + ϕ)2 − 1− (xai)2
,
ϕ(0) = 0, ϕ(1) = ϕL,
(28)
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ON A THEORETICAL STUDY FOR THE SOLUTION PROPERTIES OF THE LIMIT . . . 847
and Φ1(x, xϕj
, jx) is a solution of the scalar boundary-value problem
a′′ = G(jx, xϕj
, a) = jx
a√
(1 + xϕj )2 − 1− a2
,
a(0) = 0, a(1) = aL.
(29)
Here xai
, i = 1, 2, 3, and xϕj
, j = 1, 2, are respectively, the indicators of semitrivial
solutions Φ(x, xai
, jx), Φ1(x, xϕj
, jx) defined by the following way:
xa1 = 0, if a(x) = 0;
xa2 = a0, if a = a0quadbe upper solution of the problem (29);
xa3 = a0, if a = a0quadbe lower solution of the problem (29);
xϕ1 = ϕ0, if ϕ = ϕ0quadbe upper solution of the problem (28);
xϕ2 = ϕ0, if ϕ = ϕ0quadbe upper solution of the problem (28).
From Definition 4, we obtain the following types of scalar boundary-value problems
for semitrivial (in sense of Definition 4) solutions are
(A1) ϕ′′ = F (ϕ, 0) = jx
1 + ϕ√
(1 + ϕ)2 − 1
, ϕ(0) = 0, ϕ(1) = ϕL,
(A2) ϕ′′ = F (ϕ, a0) = jx
1 + ϕ√
(1 + ϕ)2 − 1− (a0)2
, ϕ(0) = 0, ϕ(1) = ϕL,
(A3) ϕ′′ = F (ϕ, a0) = jx
1 + ϕ√
(1 + ϕ)2 − 1− (a0)2
, ϕ(0) = 0, ϕ(1) = ϕL,
(A4) a′′ = G(ϕ0, a) = jx
a√
(1 + ϕ0)2 − 1− a2
, a(0) = 0, a(1) = aL,
(A5) a′′ = G(ϕ0, a) = jx
a√
(1 + ϕ0)2 − 1− a2
, a(0) = 0, a(1) = aL.
We shall find the solutions of problems (A1) − (A3) for ϕ0 < ϕ0, where ϕ0(xa1),
ϕ0(xa2) are respectively, lower and upper solutions of problem (A1). The solution (ϕ, a)
of the initial problem should be belong to the interval
ϕ ∈ Φ(ϕ, 0)
⋂
Φ(ϕ, a0)
⋂
Φ(ϕ, a0),
a ∈ Φ1(ϕ0, a)
⋂
Φ1(ϕ0, a).
Moreover, the ordering of lower and upper solutions of problems (A1)−(A3) is satisfied
ϕ0(xa1) < ϕ0(xa2) < ϕ0(xa3) < ϕ0(xa2) < ϕ0(xa1).
We shall seek the solution of problems (A4), (A5) for a0 < a0. In this case the following
ordering of lower and upper solutions of problems (A4), (A5) :
a0(xϕ1) < a0(xϕ2) < a0(xϕ2) < a0(xϕ1)
is satisfied.
We go over to the direct study of the problem (28) which includes the cases (A1)−
(A3). Let us consider the boundary-value problem (28) with
ISSN 1027-3190. Укр. мат. журн., 2005, т. 57, № 6
848 A. V. SINITSYN, E. V. DULOV
(B1) F (x, ϕ) : (0, 1]× (0,∞) → (0,∞).
In condition (B1) for F (x, ϕ) we dropped index ai, considering a general case of
nonlinear dependence F of x.
We shall assume that F is a Caratheodory function, i.e.,
(B2) F (·, s) measurable for all s ∈ R,
(B3) F (x, ·) is continuous a.e. for x ∈ ]0, 1],
and the following conditions hold:
(B4)
1∫
0
s(1− s)Fds < ∞,
(B5)
∂F
∂ϕ
> 0, i.e., F is increasing in ϕ.
There are γ(x) ∈ L1(]0, 1]) and α ∈ R, 0 < α < 1 such that
(B6) |F (x, s)| ≤ γ(x)(1 + |s|−α) ∀(x, s) ∈ ]0, 1]×R.
We are intersted in a positive classical solution of equation (28), i.e., ϕ > 0 in P
for x ∈ ]0, 1] and ϕ ∈ C([0, 1])
⋂
C2(]0, 1]). The problem (28) is singular, therefore,
condition (B1) is not fulfilled on the interval ϕ ∈ (0,∞) and in this connection, the
well-known theorems on existence of lower and upper solution in cone P does not work.
It follows from Theorem 1, since F in (28) is increasing in ϕ, then ϕ < w for x ∈ ]0, 1],
where ϕ and w satisfy the differential inequality (23).
Theorem 2. Assume conditions (B2) − (B6). Then there exists a positive solution
ϕ ∈ C([0, 1])
⋂
C2(]0, 1]) of the boundary-value problem (28).
Application of monotone iteration techniques to the equation (28) gives an existence
of maximal solution ϕ̄(x, jx) such that
ϕ(x, xj) ≤ ϕ̄(x, xj) < w(x) for x ∈ ]0, 1].
Proposition 1. Let 0 < c ≤ jx ≤ jmax
x . Then equation (A1)
ϕ
′′
= F (jx, ϕ, 0) = jx
1 + ϕ√
ϕ(2 + ϕ)
,
ϕ(0) = 0, ϕ(1) = ϕL
has a lower positive solution
u0 = δ2x4/3, (30)
if
4δ3 ≥ 9jmax
x (1 + δ2)/
√
2 + δ2 (31)
and an upper positive solution
u0 = α + βx, α, β > 0, (32)
with
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ON A THEORETICAL STUDY FOR THE SOLUTION PROPERTIES OF THE LIMIT . . . 849
x10
ϕ, a
ϕ0
ϕ0
a0
a0
Fig. 1. Location of lower (ϕ0, a0) and upper (ϕ0, a0) solutions.
ϕL ≥ δ2, (33)
where δ is defined from (31).
Remark 2. Square root is taking as
√
|ϕ(2 + ϕ)| in the case of negative solutions.
Here u0 = −εx is an upper solution, and u0 = −2 + ε is a lower solution (0 < ε <
< 1). Hence equation (A1) has the negative solution only for 0 < ϕL < −2 because
F (x,−2) = −∞.
It follows from (31), (33) that a value of current is limited by the value of electrostatic
potential on the anode ϕL
jx ≤ jmax
x ≤ F(ϕL). (34)
Analysis of lower and upper solutions (30), (32) exhibits that for δ2 = ϕL > 2 and
α = β ≤ 1 interval in x between lower and upper solutions is decreased, and for the
large values of the potential ϕL diode makes on regime ϕLx4/3.
Proposition 2. Let 0 < c ≤ jx ≤ jmax
x . Then equation (A4)
a
′′
= G(jx, ϕ0, a) = jx
a√
(1 + ϕ0)2 − 1− a2
, a(0) = 0, a(1) = aL,
with a lower solution a0 = 0 and an upper solution a0 = u0 > 0, has an unique solution
a(x, jx, c), which is positive, moreover
0 ≤ aL ≤
√
ϕ0(2 + ϕ).
Remark 3. The problem (A5) is considered by analogy with problem (A4), change
of an upper solution a0 = u0 to a lower a0 = u0 one and 0 ≤ aL ≤
√
ϕ0L(2 + ϕ0L).
Following to the Definition 3 and Propositions 1, 2, solutions of the problems (28),
(29) we can write in the form (Fig. 1)
lower-lower (ϕ0, a0)):
ϕ0 = u0 = δ2x4/3, a0 = 0, ϕL ≥ δ2;
upper-lower (ϕ0, a0) :
ϕ0 = u0 = α + βx, a0 = 0, δ2 ≤ ϕL ≤ C, C = max{α, β};
lower-upper (ϕ0, a
0):
ϕ0 = u0 = δ2x4/3, a0 = u0, ϕL ≥ δ2, aL ≤
√
(u0(2 + u0);
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850 A. V. SINITSYN, E. V. DULOV
0
0,2
0,4
0,6
0,8
1
0,2 0,4 0,6 0,8 1 0
0,2
0,4
0,6
0,8
1
0,2 0,4 0,6 0,8 1x
a (x)ϕ(x)
x
Fig. 2. Numerical solution for ϕL = aL = 1; estimated jx = 0,534075023488271,
da
dx
(0) = 0,879738089874635. Function ϕ(x) : upper solution y = x and lower solution y =
7
10
x
4
3 .
Function a(x) : upper solution y = x and lower solution y = x
4
3 .
0
2
4
6
8
0,2 0,4 0,6 0,8 1 0
0,5
1
1,5
2
2,5
3
0,2 0,4 0,6 0,8 1x
a (x)ϕ(x)
x
Fig. 3. Numerical solution for ϕL = 8, aL = 3; estimated jx = 8,93859989164142,
da
dx
(0) = 1,72776197665836. Function ϕ(x) : upper solution y = 8x and lower solution y = 5x
4
3 .
Function a(x) : upper solution y = 3x and lower solution y =
5
2
x
4
3 .
upper-upper (ϕ0, a0) :
ϕ0 = u0 = α + βx, a0 = u0, ϕL ≤ C, aL ≤ a0 ≤ u0.
Thus we have the following main result:
Theorem 3. Assume conditions (B2), (B3), (B6) and inequalities (31), (32) and
aL ≤
jx
2
≤ jmax
x
2
≤ F(ϕL)
2
fulfilled. Then the problem (18) – (22) possesses a positive solution in cone P such that
ϕ
′′
0 ≥ jxF (ϕ0, z2), z2 ∈ [0, ϕ0],
(ϕ0)′′ ≤ jxF (ϕ0, z2), z2 ∈ [0, ϕ0],
a
′′
0 ≥ G(jx, z1, a0), z1 ∈ [ϕ0, ϕ
0],
(a0)′′ ≤ G(jx, z1, a
0), z1 ∈ [ϕ0, ϕ
0],
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ON A THEORETICAL STUDY FOR THE SOLUTION PROPERTIES OF THE LIMIT . . . 851
0
0,05
0,1
0,15
0,2
0,25
0,3
0,2 0,4 0,6 0,8 1 0
0,2
0,4
0,6
0,8
0,2 0,4 0,6 0,8 1x
a (x)ϕ(x)
x
Fig. 4. Numerical solution for ϕL = 0,3, aL = 0,8; estimated jx = 0,0761231763035411,
da
dx
(0) = 0,759092882499624. Function ϕ(x) : upper solution y = 0,3x and lower solution
y = 0,18x
4
3 . Function a(x) : upper solution y = 0,8x and lower solution y = 0,8x
4
3 .
where ϕ0 = δ2x4/3 is a lower solution of problem (A1), ϕ0 = α + βx, α, β > 0, is an
upper solution of problem (A1) with condition ϕL ≥ δ2; a0 = 0 is a lower solution of
problem (A4) with condition 0 ≤ aL ≤
√
ϕ0(2 + ϕ0).
2.2. Numerical trajectories and the upper and lower solutions. Leaving apart the
discussion of numerical solution methods for next the sections, here we provide some
numerical solution trajectory examples both for ϕ(x) and a(x) evaluated for different
boundary conditions, see Fig. 2 – 4. They clearly show that that the theoretical results
fully coincide with the obtained numerical estimations.
Acknowlegment. We kindly thank professor Pierre Degond for his remarks and
support while professor Alexandr Sinitsyn was staying in Universit’e Paul Sabatier,
Toulouse, France where this research paper was finished.
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// Asymptotic Anal. – 1991. – 4. – P. 187 – 214.
3. Ben Abdallah N., Degond P., M’ehats F. Mathematical models of magnetic insulation // Ibid. – 1999.
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4. Kelly O. Shot noise in a diode // El. manuscr. – 1999. – http:\\www.maths.tcd.ie\olly\shot.pdf. –
P. 11.
5. Degond P., Raviart P.-A. On a penalization of the Child – Langmuir emission condition for the
one-dimensional Vlasov – Poisson equation // Asymptotic Anal. – 1992. – 6. – P. 1 – 27.
6. Hull A. W. The effect of a uniform magnetic field on the motion of electrons between coaxial
cylinders // Phys. Rev. – 1921. – 18. – P. 31 – 57.
Received 26.10.2004
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