FD-method for solving the nonlinear Klein - Gordon equation
We propose a functional-discrete method for solving the Goursat problem for the nonlinear Klein-Gordon equation. Sufficient conditions for the superexponential convergence of this method are obtained. The obtained theoretical results are illustrated by a numerical example.
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| author | Dragunov, D. V. Makarov, V. L. Sember, D. A. Драгунов, Д. В. Макаров, В. Л. Сембер, Д. А. |
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| description | We propose a functional-discrete method for solving the Goursat problem for the nonlinear Klein-Gordon equation.
Sufficient conditions for the superexponential convergence of this method are obtained. The obtained theoretical results are illustrated by a numerical example. |
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| format | Article |
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UDC 519.633.2
V. L. Makarov, D. V. Dragunov, D. A. Sember (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
FD-METHOD FOR SOLVING THE NONLINEAR KLEIN – GORDON EQUATION
FD-METOД РОЗВ’ЯЗУВАННЯ НЕЛIНIЙНОГО
РIВНЯННЯ КЛЯЙНА – ГОРДОНА
We propose a functional-discrete method for solving the Goursat problem for the nonlinear Klein – Gordon equation.
Sufficient conditions for the superexponential convergence of this method are obtained. The obtained theoretical results are
illustrated by a numerical example.
Запропоновано функцiонально-дискретний метод розв’язування задачi Гурса для нелiнiйного рiвняння Кляйна –
Гордона. Отримано достатнi умови, якi забезпечують суперекспоненцiальну швидкiсть збiжностi методу. Одержанi
теоретичнi результати проiлюстровано на числовому прикладi.
1. Introduction. It is well known that the Klein – Gordon equation (KGE)
∂2v(ξ, t)
∂t2
− ∂2v(ξ, t)
∂ξ2
+ N(v(ξ, t)) = Φ(ξ, t) (1)
has extensive applications in modern physics and engineering. Particulary, it arises when studying the
scalar massive field in the de Sitter and anti-de Sitter spacetime [39, 40], the propagation of intense
ultra-short optical pulses in low density dielectrics [13], the pionic atoms [27] et al. Furthermore, a
partial case of the nonlinear KGE — the Sine – Gordon equation (SGE) — alone has a great number
of applications in physics. One encounters the SGE when studying the propagation of a “slip” in
an infinite chain of elastically bound atoms lying over a fixed lower chain of similar atoms [11],
the magnetic flux propagation in a large Josephson junction, the domain wall dynamics in magnetic
crystals [2] et al. A nonlinear theory for strong interactions has also been developed in which the
SGE appears as a simplified classical model [30, 35]. In geometry, the Goursat and Cauchy problems
for the SGE are related to the existence of special nets on surfaces in E3,which are called Chebyshev
nets [32].
The whole range of the methods for solving the KGE can be conditionally divided into the two
groups: the analytical methods and the discrete methods. The analytical methods allow us to express
the exact solution to the equation through the elementary functions and convergent functional series.
This methods a very useful for studying nonlinear physical phenomena, such as traveling waves and
solitons [8]. Among the analytical methods there are the polynomial approximation method [31, 24],
the extended tanh method (see [25]), the sine-cosine method (see [38]), the variational iteration
method (see [3, 4]), the homotopy methods (see [33, 1, 26] and references therein), the infinite series
methods and many other methods and techniques (see [36, 34, 10] and references therein). However,
when the qualitative analysis of a solution is not the main target, then discrete methods can be useful
as well. The methods of this group approximate the exact solution on a finite set of distinct points
(i.e., on the mesh). The discrete methods for solving KGE take their origins mainly from the finite-
difference methods (see [5, 29, 9] and references therein) and Runge – Kutta methods (see [28, 7] and
references therein).
Apparently, the border line between the two groups of methods introduced above is very fuzzy
and some methods can be considered as belonging to both groups at once. In the present paper
c© V. L. MAKAROV, D. V. DRAGUNOV, D. A. SEMBER, 2012
1394 ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10
FD-METHOD FOR SOLVING THE NONLINEAR KLEIN – GORDON EQUATION 1395
we offer a numerical-analytical method which possesses the mein properties of both analytical and
discrete methods simultaneously. This method (hereinaftr referenced to as the FD-method) is based
on the FD-approach described in [12, 21] and takes its origins from the functional-discrete method
for solving Sturm – Liouville problems (see [22, 23]).
The paper is organized as follows. The Goursat problem for nonlinear KGE is introduced in
Section 2. Section 3 is devoted to the description of the FD-method’s algorithm for the given Goursat
problem. In Section 4 an important auxiliary statement about approximating properties of a hyperbolic
differential equation with piece-wise constant argument is proved (see Theorem 1). Theorem 1 plays
a key role in the proof of Theorem 2 containing sufficient convergence conditions for the proposed
FD-method (Section 5). A numerical example and conclusions are presented in Sections 6 and 7
respectively.
2. Problem statement. Let us consider KGE (1) in a slightly modified form
∂2u(x, y)
∂x∂y
+ N(u(x, y)) = f(x, y), (2)
which is more suitable for application of the FD-method. Equation (2) can be obtained from (1) via
the transform of variables
t = x− y, ξ = x+ y, (3)
assuming that
u(x, y) = v(x− y, x+ y), f(x, y) = Φ(x− y, x+ y).
Since the FD-method is a numerical-analytical method (not mere analytical) it cannot be applied
to equation (2) without an initial or boundary condition. In the paper we confine ourself to considering
a Goursat problem (see [14]), supplementing equation (2) with the following boundary conditions:
u(x, 0) = ψ(x), u(0, y) = φ(y), ψ(0) = φ(0). (4)
We assume that nonlinear function N(u) can be expressed in the form of
N(u) = N(u)u, N(u) =
∞∑
s=0
νsu
s ∀u ∈ R, νs ∈ R,
and1
ψ(x) ∈ C(1) (D1) ∩ C
(
D̄1
)
, φ(y) ∈ C(1) (D2) ∩ C
(
D̄2
)
, f(x, y) ∈ C(D̄),
D = {(x, y) : 0 < x < X, 0 < y < Y } , D1 = (0;X) , D2 = (0;Y ) .
Given assumptions imply that the solution u(x, y) ∈ C1,1(D) ∩ C(D̄) to the Goursat problem (2),
(4) exists and is unique (see [18]).
3. General description of the FD-method’s algorithm for solving KGE. According to the
FD-method’s algorithm for solving operator equations described in [12], the FD-method for solving
Goursat problem (2), (4) can be constructed in the following way.
1Hereinafter a horizontal bar above a letter indicates the closure of the set denoted by the letter.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10
1396 V. L. MAKAROV, D. V. DRAGUNOV, D. A. SEMBER
We approximate the exact solution u(x, y) to problem (2), (4) by the function
m
u(x, y) defined as
the finite sum
m
u(x, y) =
m∑
k=0
(k)
u (x, y), (5)
where m ∈ N. In the rest part of the paper the function
m
u (x, y) can be also referenced to as the
FD-approximation of rank m.
To define the functions
(k)
u(x, y) we have to introduce a mesh
xi = h1i, yj = h2j, h1 =
X
N1
, h2 =
Y
N2
, (6)
i ∈ 0, N1, j ∈ 0, N2, N1, N2 ≥ 1.
For a while we assume that the positive integers N1 and N2 are chosen arbitrary, however, later it
will be shown that decreasing the value of parameter h =
√
h2
1 + h2
2 (that is, increasing both N1 and
N2) we can increase the accuracy of the FD-method.
As soon as mesh (6) is fixed we can define function
(0)
u (x, y) ∈ C(D̄) as the solution to the
nonlinear Goursat problem with piece-wise constant argument (hereinafter referenced to as the basic
problem)
∂2
(0)
u(x, y)
∂x∂y
+N
( (0)
u (xi−1, yj−1)
) (0)
u (x, y) = f(x, y) ∀(x, y) ∈ P̄i,j , (7)
(0)
u (x, 0) = ψ(x),
(0)
u (0, y) = φ(y) ∀(x, y),∈ D̄, ψ(0) = φ(0), (8)
where
Pi,j = (xi−1, xi)× (yj−1, yj) , i ∈ 1, N1, j ∈ 1, N2. (9)
Once the basic problem (7), (8) is solved, the functions
(k)
u(x, y) ∈ C(D̄), k ∈ 1,m, can be found
as the solutions to the following sequence of linear Goursat problems
∂2
(k)
u(x, y)
∂x∂y
+N(
(0)
u(xi−1, yj−1))
(k)
u(x, y) = −N ′
((0)
u(x, y)
) (0)
u(x, y)
(k)
u(xi−1, yj−1)−
−
k−1∑
s=1
Ak−s
(
N ;
(0)
u(xi−1, yj−1), . . . ,
(k−s)
u (xi−1, yj−1)
) (s)
u(x, y)−
−Ak
(
N ;
(0)
u(xi−1, yj−1), . . . ,
(k−1)
u (xi−1, yj−1), 0
) (0)
u(x, y) +
+
k−1∑
s=0
[
Ak−1−s
(
N ;
(0)
u(xi−1, yj−1), . . . ,
(k−1−s)
u (xi−1, yj−1)
)
−
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10
FD-METHOD FOR SOLVING THE NONLINEAR KLEIN – GORDON EQUATION 1397
−Aj−1−s
(
N ;
(0)
u(x, y) , . . . ,
(k−1−s)
u (x, y)
)] (s)
u(x, y) =
(k)
F(x, y) (10)
∀(x, y) ∈ P̄i,j ∀i ∈ 1, N1 ∀j ∈ 1, N2.
(k)
u (0, y) =
(k)
u (x, 0) = 0 ∀x ∈ [0, X] ∀y ∈ [0, Y ] . (11)
Here An
(
N ; v0, v1, . . . , vn
)
denotes the Adomian polynomial of n-th order for the function N(·)
(see, for example, [37, 16, 20]), which can be calculated by the formulas
An
(
N ; v0, v1, . . . , vn
)
=
1
n!
dn
dτn
N
( ∞∑
s=0
vsτ
s
)∣∣∣∣∣
τ=0
=
=
∑
α1+...+αn=n
α1≥...≥αn+1=0
αi∈N∪{0}
N (α1)(v0)
vα1−α2
1
(α1 − α2)!
. . .
v
αn−αn+1
n
(αn − αn+1)!
. (12)
4. Approximating properties of the basic problem. It is well known (see, for example, [6]) that
problem (7), (8) possesses a unique solution, which can be represented in the following form:
(0)
u (x, y) = R (x, yj−1, x, y)
(0)
u(x, yj−1)+
+R(xi−1, y, x, y)
(0)
u(xi−1, y)−R(xi−1, yj−1, x, y)
(0)
u(xi−1, yj−1)−
−
x∫
xi−1
[
∂
∂ξ
R(ξ, yj−1, x, y)
]
(0)
u(ξ, yj−1)dξ−
−
y∫
yj−1
[
∂
∂η
R(xi−1, η, x, y)
]
(0)
u(xi−1, η)dη+
+
x∫
xi−1
y∫
yj−1
R(ξ, η, x, y)f(ξ, η)dξdη ∀(x, y) ∈ P̄i,j , (13)
where
R(x, y; ξ, η) = J0
(√
4Ni,j(x− ξ)(y − η)
)
= 0F1 (1;−(x− ξ)(y − η)Ni,j) ,
∂
∂x
R(x, y; ξ, η) = 0F1 (2;−(x− ξ)(y − η)Ni,j)Ni,j(η − y),
∂
∂y
R(x, y; ξ, η) = 0F1 (2;−(x− ξ)(y − η)Ni,j)Ni,j(ξ − x)
∀(x, y), (ξ, η) ∈ P̄i,j , Ni,j =
∣∣∣N((0)
u i,j
)∣∣∣, i ∈ 1, N1, j ∈ 1, N2,
(14)
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10
1398 V. L. MAKAROV, D. V. DRAGUNOV, D. A. SEMBER
and J0, 0F1 denote the Bessel function of the first kind and the confluent hypergeometric function
respectively (see [17]). Using integration by parts we can rewrite formula (13) as follows:
(0)
u(x, y) =
(0)
u(xi−1, y) +
x∫
xi−1
R(ξ, yj−1, x, y)
[
∂
∂ξ
(0)
u(ξ, yj−1)
]
dξ−
−
y∫
yj−1
[
∂
∂η
R(xi−1, η, x, y)
]
(0)
u(xi−1, η)dη+
+
x∫
xi−1
y∫
yj−1
R(ξ, η, x, y)f(ξ, η)dξdη ∀(x, y) ∈ P̄i,j . (15)
Theorem 1. Suppose that u(x, y) and
(0)
u(x, y) are the solutions to problems (2), (4) and (7),
(8) respectively. Then for the sufficiently small values of h1 and h2 there exists an independent on h1
and h2 constant κ, such that∥∥u(x, y)−
(0)
u(x, y)
∥∥
D̄
≤ hκ, h =
√
h2
1 + h2
2, (16)
where
∥∥u(x, y)−
(0)
u(x, y)
∥∥
D̄
= max(x,y)∈D |u(x, y)−
(0)
u(x, y)|.
Proof. Let us consider the auxiliary function
z(x, y) = u(x, y)−
(0)
u(x, y).
It is easy to see that this function is continuous on D̄ and satisfies the equation
∂2z(x, y)
∂x∂y
+N(
(0)
u(xi−1, yj−1))z(x, y)+
+
[
N(u(x, y))−N(
(0)
u(xi−1, yj−1))
]
u(x, y) = 0 ∀(x, y) ∈ P̄i,j , (17)
together with the boundary conditions
z(x, 0) = 0, x ∈ D̄1, z(0, y) = 0, y ∈ D̄2.
To prove the theorem it is enough to find a positive real constant κ, independent on h, such that
‖z(x, y)‖P̄i,j
≤ hκ ∀i ∈ 1, N1 ∀j ∈ 1, N2, (18)
where ‖z(x, y)‖P̄i,j
= max(x,y)∈P̄i,j
|z(x, y)| .
For further convenience we have to introduce the notation
(0)
u i,j=
(0)
u(xi−1, yj−1), ui,s = u(xi−1,yj−1), N ′i,j =
∣∣∣N ′((0)
u i,j)
∣∣∣, (19)
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10
FD-METHOD FOR SOLVING THE NONLINEAR KLEIN – GORDON EQUATION 1399
Li,j = max
(x,y)∈P̄i,j
θ∈[0,1]
∣∣∣N′((0)
u i,j −θ
((0)
u i,j −u(x, y)
))∣∣∣, (20)
zi,j = z(xi−1, yj−1), ‖z‖i,j = ‖z(x, y)‖P̄i,j
, (21)
Ri,j = 0F1(1;Ni,jh1h2), R′i,j = 0F1(2;Ni,jh1h2)Ni,j , (22)
‖u‖ = ‖u(x, y)‖D̄ ,
∥∥ψ′∥∥ =
∥∥ψ′(x)
∥∥
[0,X]
,
∥∥φ′∥∥ =
∥∥φ′(y)
∥∥
[0,Y ]
. (23)
It is worth to emphasize, that the principal role in the proof is assigned to the constants Nα and
Lα defined in the following way
Nα = max
u∈(ρ1,ρ2)
|N (u)| , Lα = max
u∈(ρ1,ρ2)
∣∣N′ (u)
∣∣ , (24)
ρ1 = min
(x,y)∈D̄
u(x, y)− α, ρ2 = max
(x,y)∈D̄
u(x, y) + α.
Here α denotes an arbitrary positive real number fixed throughout the proof. It is easy to see that
according to the definition of Nα (24) we have the inequality ‖N(u((x, y))‖D̄ ≤ Nα.
To prove Theorem 1 we need the following auxiliary statement.
Lemma 1. Suppose that u(x, y) and
(0)
u(x, y) are the solutions to problems (2), (4) and (7), (8)
respectively. Then the following inequalities hold true
‖z‖i,j ≤ ‖z‖i−1,j
(
1 + h1h2R
′
i,j
)
+
+h1Ri,j
{
h2
j−1∑
s=1
[
(2Ni,s + Li,s) ‖z‖i,s + h (Ni,s + Li,s)A
]}
+
+h1h2Ri,j (Li,j +Ni,j) ‖z‖i,j−1 + h1h2hRi,j (Ni,j + Li,j)A, (25)
‖z‖i,j ≤ ‖z‖i,j−1
(
1 + h1h2R
′
i,j
)
+
+h2Ri,j
{
h1
i−1∑
s=1
[
(2Ns,j + Ls,j) ‖z‖s,j + h (Ns,j + Ls,j)A
]}
+
+h1h2Ri,j (Li,j +Ni,j) ‖z‖i−1,j + h1h2hRi,j (Ni,j + Li,j)A, (26)
for all (x, y) ∈ P̄i,j , i ∈ 1, N1, j ∈ 1, N2, where
A =
{[∥∥ψ′∥∥+ Y (‖f‖+Nα ‖u‖)
]2
+
[∥∥φ′∥∥+X (‖f‖+Nα ‖u‖)
]2
}1/2
,
‖z‖0,j = ‖z‖i,0 = 0 ∀i ∈ 1, N1 ∀j ∈ 1, N2.
(27)
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10
1400 V. L. MAKAROV, D. V. DRAGUNOV, D. A. SEMBER
Proof. Unless otherwise stated we assume that (x, y) ∈ Pi,j for some fixed positive integers
i ∈ 1, N1 and j ∈ 1, N2.
We begin with the proof of inequality (25). As it was mentioned above, function z(x, y) can be
represented by virtue of the Reimann function in the following form:
z(x, y) = z(xi−1, y)+
+
x∫
xi−1
R(ξ, yj−1, x, y)
[
∂
∂ξ
z(ξ, yj−1)
]
dξ−
−
y∫
yj−1
[
∂
∂η
R(xi−1, η, x, y)
]
z(xi−1, η)dη+
+
x∫
xi−1
y∫
yj−1
R(ξ, η, x, y)
[
N(u(ξ, η))−N(
(0)
u(xi−1, yj−1))
]
u(ξ, η)dξdη. (28)
Equality (28) yields the estimate (see notation (19) – (22))
‖z‖i,j ≤ ‖z‖i−1,j + h1Ri,j
∥∥∥∥∂z(x, yj−1)
∂x
∥∥∥∥
i,j−1
+ h1h2R
′
i,j ‖z‖i−1,j +
+h1h2Ri,j
∥∥∥N(u(ξ, η))− N(
(0)
u i,j)
∥∥∥
i,j
+
+h1h2Ri,j
∥∥∥N((0)
u i,j
) (0)
u i,j −N
((0)
u i,j
)
u(ξ, η)
∥∥∥
i,j
≤
≤ ‖z‖i−1,j
(
1 + h1h2R
′
i,j
)
+ h1Ri,j
∥∥∥∥∂z(x, yj−1)
∂x
∥∥∥∥
i,j−1
+
+h1h2Ri,j (Li,j +Ni,j)
∥∥u(x, y)−
(0)
u i,j
∥∥
i,j
≤
≤ ‖z‖i−1,j
(
1 + h1h2R
′
i,j
)
+ h1Ri,j
∥∥∥∥∂z(x, yj−1)
∂x
∥∥∥∥
i,j−1
+
+h1h2Ri,j (Li,j +Ni,j)
[
‖u(x, y)− ui,j‖i,j + |zi,j |
]
. (29)
Taking into account the obvious inequality2 |zi,j | ≤ ‖z‖i,j−1 , we can summarize inequalities (29) in
the following way
‖z‖i,j ≤ ‖z‖i−1,j
(
1 + h1h2R
′
i,j
)
+ h1Ri,j
∥∥∥∥∂z(x, yj−1)
∂x
∥∥∥∥
i,j−1
+
2The inequality |zi,j | ≤ ‖z‖i−1,j is valid as well and it will be used in the proof of inequality (26).
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10
FD-METHOD FOR SOLVING THE NONLINEAR KLEIN – GORDON EQUATION 1401
+h1h2Ri,j (Li,j +Ni,j) ‖z‖i,j−1 +
+h1h2Ri,j (Li,j +Ni,j) ‖u(x, y)− ui,j‖i,j . (30)
Let us estimate the expressions
∥∥∥∥∂z(x, yj−1)
∂x
∥∥∥∥
i,j−1
and ‖u(x, y)− u (xi−1, yj−1)‖i,j arising in the
right-hand side of inequality (30). We start with the latter one. The mean value theorem provides us
with the inequality
|u(ξ, η)− u(xi−1, yj−1)| ≤ |u(x, y)− u(xi−1, y)|+
+ |u(xi−1, y)− u(xi−1, yj−1)| ≤
≤ |x− xi−1|
∥∥∥∥∂u(x, y)
∂x
∥∥∥∥
D̄
+ |y − yj−1|
∥∥∥∥∂u(x, y)
∂y
∥∥∥∥
D̄
. (31)
Furthermore, equation (2) yields us the equalities
∂u(x, y)
∂x
= ψ′(x) +
y∫
0
[f(x, η)−N(u(x, η))u(x, η)] dη, (32)
∂u(x, y)
∂y
= φ′(y) +
x∫
0
[f(ξ, y)−N(u(ξ, y))u(ξ, y)] dξ. (33)
Combining (31) with (32) and (33) we get the estimate
|u(x, y)− u(xi−1, yj−1)| ≤
≤ h1
(∥∥ψ′(x)
∥∥
[0,X]
+ Y
[
‖f(x, y)‖D̄ + ‖N(u(x, y))‖D̄ ‖u(x, y)‖D̄
])
+
+h2
(∥∥φ′(y)
∥∥
[0,Y ]
+X
[
‖f(x, y)‖D̄ + ‖N(u(x, y))‖D̄ ‖u(x, y)‖D̄
])
≤
≤ A
√
h2
1 + h2
2 = Ah. (34)
Now let us pass to the estimation of
∣∣∣∣∂z(x, yj−1)
∂x
∣∣∣∣ . Equation (17) implies the equality
∂z(x, yj−1)
∂x
=
= −
j−1∑
s=1
ys∫
ys−1
(
N
((0)
u i,s
)
z(x, η) +
[
N(u(x, η))−N
((0)
u i,s
)]
u(x, η)
)
dη. (35)
Using notation (20) from (35) it is easy to get the estimate (∀x ∈ [xi−1, xi])
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10
1402 V. L. MAKAROV, D. V. DRAGUNOV, D. A. SEMBER
∣∣∣∣∂z(x, yj−1)
∂x
∣∣∣∣ ≤ j−1∑
s=1
Ni,s
ys∫
ys−1
|z(x, η)| dη+
+
j−1∑
s=1
ys∫
ys−1
[
(Li,s +Ni,s) {|u(x, η)− ui,s|+ |zi,s|}
]
dη. (36)
Combining inequalities (34) and (36) we obtain∣∣∣∣∂z(x, yj−1)
∂x
∣∣∣∣ ≤ h2
j−1∑
s=1
(2Ni,s + Li,s) ‖z‖i,s + h2hA
j−1∑
s=1
(Ni,s + Li,s) . (37)
Finally, using estimates (34) and (37) we get from (30) the target inequality (25).
The proof of inequality (26) is mostly similar to the proof of inequality (25). However, to obtain
inequality (26) we have to use the formula
z(x, y) = z(x, yj−1)+
+
x∫
xi−1
[
∂
∂ξ
R(ξ, yj−1, x, y)
]
z(ξ, yj−1)dξ−
−
y∫
yj−1
R(xi−1, η, x, y)
[
∂
∂η
z(xi−1, η)
]
dη+
+
x∫
xi−1
y∫
yj−1
R(ξ, η, x, y)
[
N(u(ξ, η))−N(
(0)
u (xi−1, yj−1))
]
u(ξ, η)dξdη (38)
instead of formula (28). Formula (38) leads us to the inequality
‖z‖i,j ≤ ‖z‖i,j−1
(
1 + h1h2R
′
i,j
)
+ h2Ri,j
∥∥∥∥∂z(xi−1, y)
∂y
∥∥∥∥
i−1,j
+
+h1h2Ri,j (Li,j +Ni,j) ‖z‖i−1,j +
+h1h2Ri,j (Li,j +Ni,j) ‖u(x, y)− ui,j‖i,j . (39)
Instead of inequality (37) we have to use the following one (∀y ∈ [yj−1, yj ]) :∣∣∣∣∂z(xi−1, y)
∂y
∣∣∣∣ ≤ h1
i−1∑
s=1
(2Ns,j + Ls,j) ‖z‖s,j + h1hA
i−1∑
s=1
(Ns,j + Ls,j) , (40)
which can be obtained in a similar way. Finally, estimates (34), (39) and (40) lead us to inequa-
lity (26), which was to be proved.
Lemma 1 is proved.
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FD-METHOD FOR SOLVING THE NONLINEAR KLEIN – GORDON EQUATION 1403
To use the results of Lemma 1 we have to make an assumption about the correlation between h1
and h2. It is precisely this assumption that determines which of the estimates, (25) or (26), will be
used in the further reasoning. Without loss of generality we assume that3
h1 ≤ h2 ⇔ N2 ≤
Y N1
X
. (41)
Taking into account estimate (25) together with the definitions of constants Nα, and Lα (24), we
can conclude that inequalities
ρ1 ≤
(0)
u k,l≤ ρ2 ∀k ∈ 0, N1 − 1 ∀l ∈ 0, j, j < N2, (42)
imply the estimates
Nk,l ≤ Nα, Lk,l ≤ Lα,
‖z‖k,l ≤
(
1 + h1h2R
′
α
)
‖z‖k−1,l +
+h1Rα(2Nα + Lα)h2
l−1∑
s=1
‖z‖i,s + h1h2Rα(Lα +Nα) ‖z‖k,l−1 +
+h1hRαA(Nα + Lα)(Y + h2) ∀l ∈ 1, j + 1, k ∈ 1, N1, (43)
where
Rα = 0F1
(
1;Nαh
2
2
)
, R′α = 0F1
(
2;Nαh
2
2
)
Nα. (44)
However, generally speaking, conditions (42) could not be satisfied for all l ∈ 1, N2 unless some
restriction on the value of h2 is imposed. To find out this restriction we have to study some properties
of the sequence of real numbers µi,j ∀i ∈ 0, N1, ∀j ∈ 0, N2 defined by formulas
µi,j = aµi−1,j + bµi,j−1 + c, µ0,j = µi,0 = 0 (45)
with
a = 1 + h1a1(h2) = 1 + h1h2R
′
α,
b = h1b1(h2) = h1
[
RαY (2Nα + Lα) + h2Rα (Nα + Lα)
]
, (46)
c = h1hc1(h2) = h1h
[
RαA (Nα + Lα) (Y + h2)
]
.
3Under this assumption inequality (26) is not useful for us and we need to use inequality (25) instead. However, if we
were assumed that h1 > h2 we would be forced to use inequality (26) to prove the theorem.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10
1404 V. L. MAKAROV, D. V. DRAGUNOV, D. A. SEMBER
Lemma 2. Suppose that real numbers µi,j ∀i ∈ 0, N1 ∀j ∈ 0, N2 are defined by formulas (45)
with
a = 1 + h1a1, b = h1b1, c = h1hc1
and assumption (41) holds. Then
µi,j ≤ hXc1 exp
(
(X + Y )b1 +Xa1
)
∀i ∈ 1, N1, j ∈ 1, N2. (47)
Proof. Using the method of mathematical induction, it is not hard to prove the explicit formula
for calculation of µi,j
µi,j =
0 if i = 0, j ∈ 1, N2 or j = 0, i ∈ 1, N1,
c
j−1∑
k=0
i−1∑
p=0
(k + p)!
k!p!
apbk ∀i ∈ 1, N1, j ∈ 1, N2.
(48)
Using formula (48) and assumption (41) we get
µi,j ≤ µN1,N2 = c
N1−1∑
p=0
ap
N2−1∑
k=0
(k + p)!
k!p!
bk =
= c
N1−1∑
p=0
ap
N2−1∑
k=0
1
k!
(p+ 1)(p+ 2) . . . (p+ k)hk1
(
b
h1
)k
≤
≤ c
N1−1∑
p=0
ap
N2−1∑
k=0
1
k!
(N1 +N2)khk1
(
b
h1
)k
≤
≤ c
N1−1∑
p=0
ap
N2−1∑
k=0
1
k!
(
N1 +
Y N1
X
)k ( X
N1
)k ( b
h1
)k
=
= c
N1−1∑
p=0
ap
N2−1∑
k=0
((X + Y ) b1)k
k!
≤ hXc1a
N1−1 exp
(
(X + Y )b1
)
=
= hXc1
(
1 +
X
N1
a1
)N1−1
exp
(
(X + Y )b1
)
≤ hXc1 exp
(
(X + Y )b1 +Xa1
)
.
Lemma 2 is proved.
To obtain the required restriction on the maximum value of h2 mentioned above, we have to
consider the auxiliary function
E(h1, h2) =
√
h2
1 + h2
2c1(h2)X exp
(
(X + Y )b1(h2) +Xa1(h2)
)
.
Taking into account assumption (41) we arrive at the inequality
E(h1, h2) ≤ E(h2, h2) = E(h2).
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10
FD-METHOD FOR SOLVING THE NONLINEAR KLEIN – GORDON EQUATION 1405
Function E(h2) is strictly increasing function on [0,+∞], limh2→+∞ E(h2) = +∞, E(0) = 0. This
fact implies the existence and uniqueness of the constant Hα > 0, such that
E(Hα) = α and E(h2) < α ∀h2 ∈ [0, Hα) .
Henceforward we assume that
h2 ≤ Hα. (49)
It follows from Lemma 2 that inequality (49) provides the estimates
µi,j ≤ hK ≤ α, K = c1(Hα)X exp ((X + Y )b1(Hα) +Xa1(Hα)) , (50)
i ∈ 1, N1, j ∈ 1, N2.
Now we are in position to prove inequality (18) with κ = K (see notation (50)) via the method
of mathematical induction. We will use induction with respect to j.
The base case, j = 1. Taking into account that u(x, 0) =
(0)
u(x, 0) ∀x ∈ [0, X] we arrive at the
conclusion that inequalities (42) are valid for j = 0. This fact provides inequalities (43) for j = 1,
which, using notations (44) and (46), can be represented in the form of
‖z(x, y)‖i,1 ≤ ‖z‖i−1,1 a+ c, i ∈ 1, N1. (51)
Taking into account formula (45) it is easy to see that
‖z(x, y)‖i,1 ≤ µi,1, i ∈ 1, N1, (52)
and, consequently, from (50) it follows that
‖z(x, y)‖i,1 ≤ hK, i ∈ 1, N1. (53)
Inequalities (53) prove inequality (18) with κ = K for j = 1 and for all i ∈ 1, N1.
Induction step. Assume that inequality (18) and auxiliary inequality
‖z(x, y)‖i,j ≤ µi,j (54)
are proved for all j ∈ 1, n, i ∈ 1, N1, 1 < n < N2. This assumption implies the validity of
inequalities (42) for j = n and, consequently, we obtain inequalities (43) for j = n. Combining the
auxiliary inequality (54) together with obvious inequalities
µk,l−1 ≤ µk,l, k ∈ 1, N1, l ∈ 1, N2,
and inequalities (43) with j = n, we arrive at the following estimate for zi,n+1:
‖z‖i,n+1 ≤ aµi−1,n+1 + c+
+h1Rα(2Nα + Lα)h2
n∑
s=1
µi,s + h1h2Rα(Lα +Nα)µi,n ≤
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10
1406 V. L. MAKAROV, D. V. DRAGUNOV, D. A. SEMBER
≤ aµi−1,n+1 + bµi,n + c = µi,n+1. (55)
Estimate (55) is valid for all i ∈ 1, N1 and proves inequality (54) for j = n+ 1. Furthermore, taking
into account estimates (50) and estimate (55) we immediately obtain the validity of inequality (18)
with κ = K, for j = n+ 1 and for all i ∈ 1, N1, which was to be demonstrated:
‖z(x, y)‖i,n+1 ≤ µi,n+1 ≤ hK ≤ α, i ∈ 1, N1.
By the principle of mathematical induction it follows that inequality (18) is valid for all i ∈ 1, N1,
j ∈ 1, N2. Thereby, the theorem is proved.
5. The FD-method for nonlinear Goursat problem: the convergence result. In this section we
intend to study the question of sufficient conditions providing convergence of the FD-method (5),
(6), (7), (8), (10), (11) to the exact solution of the Goursat problem (2), (4). In other words, given
that the parameters h1 and h2 are sufficiently small 4 we will prove that
lim
m→∞
m∑
k=0
∥∥(k)
u(x, y)
∥∥
1,D̄
<∞, (56)
and
u(x, y) =
∞∑
k=0
(k)
u(x, y), (57)
where
∥∥f(x, y)
∥∥
1,D̄
= max
∥∥f(x, y)
∥∥
D̄
, max
i∈1,N1,j∈1,N2
[∥∥∥∥ ∂∂xf(x, y)
∥∥∥∥2
Pi,j
+
∥∥∥∥ ∂∂yf(x, y)
∥∥∥∥2
Pi,j
] 1
2
,
for all f(x, y), such that f(x, y) ∈ C(D) and f(x, y) ∈ C1,1(Pi,j), i ∈ 1, N1, j ∈ 1, N2. To achieve
that we will use the method of generating functions and the main part of this section is devoted to
the derivation of an appropriate equation for a generating function.
We begin with the estimation of
∥∥(k)
u(x, y)
∥∥
1,D̄
.
The piece-wise constant function
(s)
u⊥(x, y) , (x, y) ∈ D̄ defined as
(s)
u⊥(x, y) =
(s)
u (xi−1, yj−1) , if (x, y) ∈ [xi−1, xi)× [yj−1, yj), i ∈ 1, N1, j ∈ 1, N2,
allows us to represent equations (10) in the form which is valid for all (x, y) ∈ D̄, i.e.,
∂2
(k)
u (x, y)
∂x∂y
+N
( (0)
u⊥(x, y)
) (k)
u (x, y) =
= −N ′
( (0)
u⊥(x, y)
) (0)
u (x, y)
(k)
u⊥ (x, y)−
−
k−1∑
s=1
Ak−s
(
N ;
(0)
u⊥(x, y), . . . ,
(k−s)
u⊥ (x, y)
) (s)
u (x, y)+
4We assume that h1 ≤ h2 ≤ 1 and inequality (49) holds true.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10
FD-METHOD FOR SOLVING THE NONLINEAR KLEIN – GORDON EQUATION 1407
+
k−1∑
s=0
[
Ak−1−s
(
N,
(0)
u⊥(x, y) , . . . ,
(k−1−s)
u⊥ (x, y)
)
−
−Ak−1−s
(
N,
(0)
u (x, y) , . . . ,
(k−1−s)
u (x, y)
)] (s)
u (x, y)−
−Ak
(
N,
(0)
u⊥(x, y) , . . . ,
(k−1)
u⊥ (x, y) , 0
) (0)
u (x, y) =
(k)
F (x, y). (58)
As it was mentioned above (see previous section or [19]), the solution to the k-th equation of
system (10) (k > 1) on P̄i,j can be represented in the following form:
(k)
u (x, y) =
(k)
u (xi−1, y) +
x∫
xi−1
R(ξ, yj−1;x, y)
[
∂
∂ξ
(k)
u (ξ, yj−1)
]
dξ−
−
y∫
yj−1
[
∂
∂η
R(xi−1, η;x, y)
]
(k)
u (xi−1, η)dη+
−N ′
((0)
u i,j
) (k)
u i,j
x∫
xi−1
y∫
yj−1
R(ξ, η;x, y)
(0)
u(ξ, η)dξdη+
+
x∫
xi−1
y∫
yj−1
R(ξ, η;x, y)
(k)
F(ξ, η)dξdη. (59)
As it follows from Theorem 1, the function
(0)
u(x, y) =
(0)
u(h, x, y) tends uniformly to u(x, y) on D̄
as h→ 0. Hence, taking into account the existence and uniqueness of the continuous on D̄ solution
u(x, y) to the Goursat problem (2), (4), we can conclude that there exists an independent on h1 and
h2 constant Mu, such that ∥∥(0)
u(x, y)
∥∥
D̄
≤Mu. (60)
The last fact provides the existence of the independent on h1 and h2 constants MN ,M
′
N ,MR,M
′
R >
> 0, such that ∥∥∥N((0)
u(x, y)
)∥∥∥
D̄
≤MN ,
∥∥∥N ′((0)
u(x, y)
)∥∥∥
D̄
≤M ′N , (61)
∥∥∥0F1
(
1,
∣∣N((0)
u(x, y)
)∣∣)∥∥∥
D̄
≤MR,
∥∥∥0F1
(
2,
∣∣N((0)
u(x, y)
)∣∣)∣∣N((0)
u(x, y)
)∣∣∥∥∥
D̄
≤M ′R.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10
1408 V. L. MAKAROV, D. V. DRAGUNOV, D. A. SEMBER
Using inequalities (60), (61) together with notations (19), (20), (21) from equation (58) we can derive
the estimate (∀x ∈ [xi−1, xi])
∣∣∣∂ (k)
u(x, yj−1)
∂x
∣∣∣ ≤ j−1∑
s=1
ys∫
ys−1
Ni,s
∣∣(k)
u(x, y)
∣∣dy+
+
j−1∑
s=1
ys∫
ys−1
N ′i,s
∣∣(k)
u i,s
∣∣∣∣(0)
u(x, y)
∣∣dy +
yj−1∫
0
∣∣(k)
F(x, y)
∣∣dy ≤
≤ h2
j−1∑
s=1
Ni,s
∥∥(k)
u
∥∥
i,s
+ h2
j−1∑
s=1
N ′i,s
∥∥(k)
u
∥∥
i,s
∥∥(0)
u
∥∥
i,s
+ Y
∥∥(k)
F
∥∥ ≤
≤
(
MN +M ′NMu
)
h2
j−1∑
s=1
∥∥(k)
u
∥∥
i,s
+ Y
∥∥(k)
F
∥∥, (62)
where ∥∥(k)
u
∥∥
i,s
=
∥∥(k)
u (x, y)
∥∥
P̄i,s
,
∥∥(k)
F
∥∥ =
∥∥(k)
F (x, y)
∥∥
D̄
.
Combining representation (59) with estimate (62) we obtain the inequality∥∥(k)
u
∥∥
i,j
≤
(
1 + h1h2M
′
R
) ∥∥(k)
u
∥∥
i−1,j
+ h1h2M
′
NMRMu
∥∥(k)
u
∥∥
i,j−1
+
+h1MR
{(
MN +M ′NMu
)
h2
j−1∑
s=1
∥∥(k)
u
∥∥
i,s
+ Y
∥∥(k)
F
∥∥}+ h1h2MR
∥∥(k)
F
∥∥. (63)
Denoting expression
∥∥(k)
u
∥∥
i,j
∥∥(k)
F
∥∥−1
by
(k)
U i,j , we can rewrite inequality (63) in the form of
(k)
U i,j≤
(
1 + h1h2M
′
R
) (k)
U i−1,j +h1h2M
′
NMRMu
(k)
U i,j−1 +
+h1MR
{
h2
(
MN +M ′NMu
) j−1∑
s=1
(k)
U i,s +Y
}
+ h1h2MR. (64)
Using the method of mathematical induction it is easy to prove (see the proof of Theorem 1) that
Ui,j ≤ µi,j , i ∈ 1, N1, j ∈ 1, N2, (65)
where the real numbers µi,j are defined by formulas (45) with
b = h1b1(h2) = h1MR
(
h2M
′
NMu + Y
(
MN +M ′NMu
))
, (66)
a = 1 + h1h2M
′
R, c = h1MR(h2 + Y ).
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FD-METHOD FOR SOLVING THE NONLINEAR KLEIN – GORDON EQUATION 1409
Inequalities (65) together with Lemma 2 yields us the estimates
Ui,j ≤ µi,j ≤MRX(h2 + Y ) exp
(
(X + Y )b1(h2) +Xh2M
′
R
)
=
= E(h2) ≤ E(1) = σ1, i ∈ 1, N1, j ∈ 1, N2. (67)
Returning to the estimation of
(k)
u(x, y) we get
∥∥(k)
u
∥∥ df
=
∥∥(k)
u(x, y)
∥∥
D̄
= max
i∈1,N1
j∈1,N2
∥∥(k)
u
∥∥
i,j
≤ σ1
∥∥(k)
F
∥∥. (68)
Using estimate (68) and equation (58) it is not hard to obtain the inequalities
∥∥∥∂ (k)
u (x, y)
∂x
∥∥∥
D̄
≤ Y σ2
∥∥(k)
F
∥∥, ∥∥∥∂ (k)
u (x, y)
∂y
∥∥∥
D̄
≤ Xσ2
∥∥(k)
F
∥∥, (69)
where
σ2 = σ1(MN +M ′NMu) + 1.
Combining inequalities (68) and (69) we get the following estimate:
∥∥(k)
u
∥∥
1
=
∥∥(k)
u (x, y)
∥∥
1,D̄
≤ σ
∥∥(k)
F
∥∥, σ = max
{
σ1, σ2
√
X2 + Y 2
}
. (70)
Recalling the explicit formula for
(k)
F (x, y) (see (58)) we can proceed with the estimation of∥∥(k)
u
∥∥
1
as follows:
∥∥(k)
u
∥∥
1
≤ σ
(k−1∑
s=1
Ak−s
(
Ñ ;
∥∥(0)
u
∥∥, . . . ,∥∥(k−s)
u
∥∥)∥∥(s)
u
∥∥+
+
∥∥∥k−1∑
s=0
[
Ak−s−1
(
N ;
(0)
u⊥(x, y), . . . ,
(k−s−1)
u⊥ (x, y)
)
−
−Ak−s−1
(
N ;
(0)
u (x, y), . . . ,
(k−s−1)
u (x, y)
)] (s)
u (x, y)
∥∥∥
D̄
+
+Ak
(
Ñ ;
∥∥(0)
u
∥∥, . . . ,∥∥(k)
u
∥∥)∥∥(0)
u
∥∥− ∥∥(0)
u
∥∥∥∥(k)
u
∥∥Ñ ′(∥∥(0)
u
∥∥)), (71)
where
Ñ(u) =
∞∑
s=0
|νs|us.
To estimate the second term in the right-hand side of inequality (71) we need the lemma stated below.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10
1410 V. L. MAKAROV, D. V. DRAGUNOV, D. A. SEMBER
Lemma 3.∥∥∥As(N ;
(0)
u⊥(x, y), . . . ,
(s)
u⊥(x, y)
)
−As
(
N ;
(0)
u(x, y), . . . ,
(s)
u(x, y)
)∥∥∥
D̄
≤
≤ hAs
(
Ñ1;
∥∥(0)
u
∥∥
1
, . . . ,
∥∥(s)
u
∥∥
1
)
, Ñ1(u) = Ñ ′(u)u.
Lemma 3 is a partial case of Lemma 1 from [20].
Using Lemma 3 we can estimate the right-hand side of inequality (71) in the following way:
∥∥(k)
u
∥∥
1
≤ σ
(
k−1∑
s=0
Ak−s
(
Ñ ;
∥∥(0)
u
∥∥
1
, . . . ,
∥∥(k−s)
u
∥∥
1
)∥∥(s)
u
∥∥
1
+
+h
k−1∑
s=0
Ak−s−1
(
Ñ1;
∥∥(0)
u
∥∥
1
, . . . ,
∥∥(k−s−1)
u
∥∥
1
)∥∥(s)
u
∥∥
1
−
∥∥(0)
u
∥∥
1
∥∥(k)
u
∥∥
1
Ñ ′
(∥∥(0)
u
∥∥
1
))
. (72)
Let us consider the sequence of real numbers {vk}∞k=0 defined by the formulas
v0 =
∥∥(0)
u
∥∥
1
, vk = σ
(
k−1∑
s=0
Ak−s
(
Ñ ; v0, . . . , vk−s
)
vs+
+
k−1∑
s=0
Ak−s−1
(
Ñ1; v0, . . . , vk−s−1
)
vs − vkÑ ′
(
v0
)
v0
)
, k = 1, 2, . . . . (73)
It is easy to see that for the sequence {vk}∞k=0 defined above the inequalities
∥∥(k)
u
∥∥
1
≤ vkhk, k = 0, 1, . . . , (74)
hold true. Assuming that the series
g(z) =
∞∑
k=0
vkz
k (75)
has a nonzero convergence radius, say R > 0, and g(R) <∞, we immediately arrive at the inequality
vkR
k ≤ c
k1+ε
(76)
with some positive parameters c and ε. From inequality (76) it follows that the condition h ≤ R is
sufficient for the series
∑∞
k=0
∥∥(k)
u
∥∥
1
to converge, that is, for the FD-method to converge. Thus, to
prove that inequality (56) holds for a parameter h, chosen sufficiently small, we have to investigate
convergence of power series (75).
Taking into account equalities (73) we arrive at the conclusion that function g(z) satisfies the
nonlinear functional equation(
g(z)− v0
)(
1 + Ñ ′
(
v0
)
v0
)
= σ
[(
Ñ
(
g(z)
)
− Ñ
(
v0
))
g(z) + zÑ ′
(
g(z)
)(
g(z)
)2]
(77)
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10
FD-METHOD FOR SOLVING THE NONLINEAR KLEIN – GORDON EQUATION 1411
for all z ∈ (−R,R). To prove that the radius of convergence of power series (75) is nonzero, i.e.,
R > 0, we have to consider the inverse function z = g−1. From equation (77) we can easily derive
the explicit formula for z = z(g) :
z(g) =
(
g − v0
)(
1 + Ñ ′
(
v0
)
v0
)
−
(
Ñ
(
g
)
− Ñ
(
v0
))
gσ
σÑ ′
(
g
)
g2
. (78)
Taking into account that z(v0) = 0 we can easily find the value of z′(v0) :
z′(v0) = lim
g→v0
z(g)− z(v0)
g − v0
=
1
σÑ ′
(
v0
)
v2
0
. (79)
Since function z(g) (78) is holomorphic in some open neighborhood of the point g = v0 and z′(v0) >
> 0, we can conclude that there exists an inverse function z−1 = g which is holomorphic in some
open interval (−R,R) (see [15]). Supposing that g(R) =∞ we get the contradiction (see (77))
1 + Ñ ′(v0)v0 = lim
z→+∞
(
v0(1 + Ñ ′(v0)v0)
g(z)
+ σ
[(
Ñ
(
g(z)
)
− Ñ
(
v0
))
+ zÑ ′
(
g(z)
)
g(z)
])
= +∞.
(80)
This contradiction proves inequality (76) for some positive constants c and ε which depend on Ñ(u)
and v0 only. Thus, we have that condition h ≤ R provides the validity of inequalities (56) and
∥∥(k)
u
∥∥
1
≤ c
k1+ε
(
h
R
)k
. (81)
Assume that h ≤ R. Then inequality (56) allows us to consider the function
∞
u(x, y) =
∞∑
k=0
(k)
u (x, y) ∈ C(D̄).
Furthermore, from equation (10) it follows that
(k)
u (x, y) ∈ C1,1(Pi,j) and
∥∥∥∂2
(k)
u (x, y)
∂x∂y
∥∥∥
Pi,j
≤
(
MN +M ′NMu
)∥∥(k)
u
∥∥
1
+
∥∥(k)
F
∥∥, i ∈ 1, N1, j ∈ 1, N2. (82)
Inequality (82) together with (81) imply that
∞
u(x, y) ∈ C1,1(Pi,j) and
∂2 ∞u(x, y)
∂x∂y
=
∞∑
k=0
∂2
(k)
u (x, y)
∂x∂y
(x, y) ∈ Pi,j , i ∈ 1, N1, j ∈ 1, N2.
The latter fact allows us to sum up equations (7) and (10) over k from 1 to ∞ and this, taking into
account the obvious equality
N
(∞
u(x, y)
)
= N
(∞
u(x, y)
) ∞
u(x, y) =
∞∑
k=0
k∑
s=0
Ak−s
(
N ;
(0)
u (x, y), . . . ,
(k−s)
u (x, y)
) (s)
u (x, y),
results in the equality
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10
1412 V. L. MAKAROV, D. V. DRAGUNOV, D. A. SEMBER
∂2 ∞u(x, y)
∂x∂y
+ N(
∞
u(x, y)) = f(x, y), (83)
(x, y) ∈ D̄ ∩
{
(x, y) |x 6= xi, y 6= yj , i ∈ 0, N1, j ∈ 0, N2
}
.
Hence, we see that equality (83) formally coincides with equation (2). To obtain the identity
∞
u(x, y) ≡ u(x, y), (x, y) ∈ D̄
it is enough to remind that
∞
u(0, y) ≡ u(0, y) ≡ φ(y),
∞
u(x, 0) ≡ u(x, 0) ≡ ψ(y) and to mention the
fact that the solution u(x, y) to problem (2), (4) is unique on D.
Thereby, we have proved the following theorem.
Theorem 2. Suppose that the Goursat problem (2), (4) satisfies the following conditions:
1) N(u) = u
∑∞
k=0
νku
k, νk ∈ R, u ∈ R;
2) ψ(x) ∈ C(1) (D1) ∩ C
(
D̄1
)
, φ(y) ∈ C(1) (D2) ∩ C
(
D̄2
)
, f(x, y) ∈ C(D̄).
Then the FD-method described by formulas (5) – (8), (10), (11) converges superexponentially to the
exact solution u(x, y) of the problem, i.e., the inequalities
∥∥u(x, y)−
(m)
u (x, y)
∥∥
1,D̄
≤ cR
(m+ 1)1+ε(R− h)
(
h
R
)m+1
, m ∈ N ∪ {0}, (84)
holds true, provided that h < R, where positive real constants c,R, ε depend on the input data of
problem (2), (4) only.
6. Numerical example. Let us consider the Goursat problem
∂2u(x, y)
∂x∂y
= e2u(x,y), (x, y) ∈ D,
u(x, 0) =
x
2
− ln(1 + ex), u(0, y) =
y
2
− ln(1 + ey),
(85)
where D =
{
(x, y) | 0 < x < 4, 0 < y < 4
}
. Obviously, this problem satisfies the conditions of
Theorem 2. It is not hard to verify that the exact solution to problem (85) is
u∗(x, y) =
x+ y
2
− ln(ex + ey).
Using the FD-method described above we approximate the exact solution to problem (85) by a
finite sum (5) with the terms
(k)
u(x, y) satisfying the following recurrence system of linear Goursat
problems:
∂2
(0)
u(x, y)
∂x∂y
+
1− exp(2
(0)
u(xi−1, yj−1))
(0)
u(xi−1, yj−1)
(0)
u(x, y) = 1,
(0)
u(xi−1 + 0, y) =
(0)
u(xi−1 − 0, y),
(0)
u(x, yj−1 + 0) =
(0)
u(x, yj−1 − 0),
(0)
u (x, 0) =
x
2
− ln(1 + ex),
(0)
u (0, y) =
y
2
− ln(1 + ey),
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10
FD-METHOD FOR SOLVING THE NONLINEAR KLEIN – GORDON EQUATION 1413
∂2
(k)
u(x, y)
∂x∂y
+
1− exp(2
(0)
u (xi−1, yj−1))
(0)
u(xi−1, yj−1)
(k)
u(x, y) =
=
(
1− exp(2
(0)
u(xi−1, yj−1))((0)
u(xi−1, yj−1)
)2 +
2 exp(2
(0)
u(xi−1, yj−1))
(0)
u(xi−1, yj−1)
)
(k)
u(xi−1, yj−1)
(0)
u(x, y)+
+
(k)
F(x, y), x ∈ (xi−1, xi), y ∈ (yj−1, yj),
(k)
u(xi−1 + 0, y) =
(k)
u(xi−1 − 0, y),
(k)
u(x, yj−1 + 0) =
(k)
u(x, yj−1 − 0),
i ∈ 1, N1, j ∈ 1, N2,
(k)
u(x, 0) = 0,
(k)
u(0, y) = 0, k = 1, 2, . . . ,
where function
(k)
F (x, y) is defined by formula (58) and xi, yj are defined according to (6) with
X = Y = 4, N1 = N2 = 4; 20; 40; 80 (in terms of h1, h2 we have that h1 = h2 = 0.5; 0.2; 0.1; 0.05
respectively). To monitor the accuracy of the method we use the function
δ(h1, h2,m) =
∥∥(m)
u (x, y, h1, h2)− u∗(x, y)
∥∥
D̄
.
The error of the FD-method
as a function of the rank (m) and the mesh size (h)
δ(0.5, 0.5,m) δ(0.2, 0.2,m)
m = 0 1.0584498110834e-1 1.3629587830264e-2
m = 1 2.0875237867244e-2 5.6023714534399e-3
m = 2 1.5876742122176e-2 1.7852756996399e-3
m = 3 8.7851563393853e-3 1.7609349110392e-4
m = 4 2.0883887112112e-3 1.3084812991115e-5
m = 5 2.9359063800745e-4 5.1600423756071e-7
m = 6 1.7966735715635e-5 4.4543002859311e-9
m = 7 1.1298645650193e-6 3.4087147338034e-10
δ(0.1, 0.1,m) δ(0.05, 0.05,m)
m = 0 4.3352923963359e-3 1.0590305089182e-3
m = 1 2.0412391759766e-3 3.6571985266298e-4
m = 2 3.1676334086428e-4 4.3855534642367e-5
m = 3 2.0298330526525e-5 1.5101132648798e-6
m = 4 1.1360343226132e-6 4.6417353405381e-8
m = 5 5.6241341916952e-8 2.7419476073821e-9
m = 6 4.2864933415766e-10 5.3844093415473e-11
m = 7 6.4824133982673e-11 3.7704350835172e-12
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10
1414 V. L. MAKAROV, D. V. DRAGUNOV, D. A. SEMBER
The values of function δ(h1, h2,m) presented in table show that the convergence rate of the FD-
method increases as the mesh size decreases. Using the numerical data presented in the table it is
not hard to verify that the function δ(h1, h2,m), as a function of m, (i.e., for the fixed values of
parameters h1, h2) decreases exponentially as the rank m of the FD-method increases. This is in good
agreement with Theorem 2.
7. Conclusions. In the paper we have developed a numerical-analytic method for solving the
Goursat problem for nonlinear Klein – Gordon equation. Under relatively general assumptions, we
have proved that the method converges superexponentially, provided that the mesh size (h) is suffi-
ciently small. From Theorem 2 it follows that the accuracy of the FD-method can be increased either
by increasing the rank m of the method or by decreasing the mesh size h. The latter conclusion has
been confirmed by the results of the numerical example included in the paper.
Though not discussed in the paper, the question of developing an efficient software implemen-
tation for the proposed method is of great interest from both theoretical and practical point of view.
We leave this question to a subsequent paper.
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Received 16.05.12
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10
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| id | umjimathkievua-article-2667 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| language | English |
| last_indexed | 2026-03-24T02:27:57Z |
| publishDate | 2012 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/18/21cd54ced8c3e89bad9c10a0f379dc18.pdf |
| spelling | umjimathkievua-article-26672020-03-18T19:32:22Z FD-method for solving the nonlinear Klein - Gordon equation FD-метод розв’язування нелiнiйного рiвняння Кляйна – Гордона Dragunov, D. V. Makarov, V. L. Sember, D. A. Драгунов, Д. В. Макаров, В. Л. Сембер, Д. А. We propose a functional-discrete method for solving the Goursat problem for the nonlinear Klein-Gordon equation. Sufficient conditions for the superexponential convergence of this method are obtained. The obtained theoretical results are illustrated by a numerical example. Запропоновано функцiонально-дискретний метод розв’язування задачi Гурса для нелiнiйного рiвняння Кляйна – Гордона. Отримано достатнi умови, якi забезпечують суперекспоненцiальну швидкiсть збiжностi методу. Одержанi теоретичнi результати проiлюстровано на числовому прикладi. Institute of Mathematics, NAS of Ukraine 2012-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2667 Ukrains’kyi Matematychnyi Zhurnal; Vol. 64 No. 10 (2012); 1394-1415 Український математичний журнал; Том 64 № 10 (2012); 1394-1415 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2667/2093 https://umj.imath.kiev.ua/index.php/umj/article/view/2667/2094 Copyright (c) 2012 Dragunov D. V.; Makarov V. L.; Sember D. A. |
| spellingShingle | Dragunov, D. V. Makarov, V. L. Sember, D. A. Драгунов, Д. В. Макаров, В. Л. Сембер, Д. А. FD-method for solving the nonlinear Klein - Gordon equation |
| title | FD-method for solving the nonlinear Klein - Gordon equation |
| title_alt | FD-метод розв’язування нелiнiйного рiвняння Кляйна – Гордона |
| title_full | FD-method for solving the nonlinear Klein - Gordon equation |
| title_fullStr | FD-method for solving the nonlinear Klein - Gordon equation |
| title_full_unstemmed | FD-method for solving the nonlinear Klein - Gordon equation |
| title_short | FD-method for solving the nonlinear Klein - Gordon equation |
| title_sort | fd-method for solving the nonlinear klein - gordon equation |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2667 |
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