FD-method for a nonlinear eigenvalue problem with discontinuous eigenfunctions
An algorithm for solution of a nonlinear eigenvalue problem with discontinuous eigenfunctions is developed. The numerical technique is based on a perturbation of the coefficients of differential equation combined with the Adomian decomposition method for the nonlinear term of the equation. The propo...
Saved in:
| Date: | 2007 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут математики НАН України
2007
|
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/7247 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | FD-method for a nonlinear eigenvalue problem with discontinuous eigenfunctions / V.L. Makarov, N.O. Rossokhata // Нелінійні коливання. — 2007. — Т. 10, № 1. — С. 126-143. — Бібліогр.: 12 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860192983786192896 |
|---|---|
| author | Makarov, V.L. Rossokhata, N.O. |
| author_facet | Makarov, V.L. Rossokhata, N.O. |
| citation_txt | FD-method for a nonlinear eigenvalue problem with discontinuous eigenfunctions / V.L. Makarov, N.O. Rossokhata // Нелінійні коливання. — 2007. — Т. 10, № 1. — С. 126-143. — Бібліогр.: 12 назв. — англ. |
| collection | DSpace DC |
| description | An algorithm for solution of a nonlinear eigenvalue problem with discontinuous eigenfunctions is developed. The numerical technique is based on a perturbation of the coefficients of differential equation combined with the Adomian decomposition method for the nonlinear term of the equation. The proposed approach provides an exponential convergence rate dependent on the index of the trial eigenvalue and on the transmission coefficient. Numerical examples support the theory.
Розроблено алгоритм для числового розв'язування нелінійних задач на власні значення з розривними власними функціями. В основі числового методу лежить збурення коефіцієнтів диференціального рівняння в поєднанні з методом декомпозиції Адомяна нелінійної частини рівняння. Запропонований підхід забезпечує експоненціальну швидкість збіжності, яка залежить від порядкового номера власного значення та коефіцієнта трансмісії. Наведені числові розрахунки підтверджують теоретичні висновки.
|
| first_indexed | 2025-12-07T18:07:29Z |
| format | Article |
| fulltext |
UDC 517 . 983 . 27
FD-METHOD FOR A NONLINEAR EIGENVALUE PROBLEM
WITH DISCONTINUOUS EIGENFUNCTIONS
FD-МЕТОД ДЛЯ НЕЛIНIЙНОЇ ЗАДАЧI НА ВЛАСНI ЗНАЧЕННЯ
З РОЗРИВНИМИ ВЛАСНИМИ ФУНКЦIЯМИ
V. L. Makarov, N. O. Rossokhata
Inst. Math. Nat. Acad. Sci. Ukraine
Tereshchenkivs’ka Str., 3, Kyiv 4, 01601, Ukraine
e-mail: makarov@imath.kiev.ua
nataross@gmail.com
An algorithm for solution of a nonlinear eigenvalue problem with discontinuous eigenfunctions is develo-
ped. The numerical technique is based on a perturbation of the coefficients of differential equation combi-
ned with the Adomian decomposition method for the nonlinear term of the equation. The proposed
approach provides an exponential convergence rate dependent on the index of the trial eigenvalue and
on the transmission coefficient. Numerical examples support the theory.
Розроблено алгоритм для чисельного розв’язування нелiнiйних задач на власнi значення з роз-
ривними власними функцiями. В основi чисельного методу лежить збурення коефiцiєнтiв дифе-
ренцiального рiвняння в поєднаннi з методом декомпозицiї Адомяна нелiнiйної частини рiвнян-
ня. Запропонований пiдхiд забезпечує експоненцiальну швидкiсть збiжностi, яка залежить вiд
порядкового номера власного значення та коефiцiєнта трансмiсiї. Наведенi чисельнi розрахун-
ки пiдтверджують теоретичнi висновки.
1. Introduction. A functional-discrete method (FD-method) to find numerical solution of boun-
dary-value problems for linear differential equations and, in particular eigenvalue prob-
lems, was proposed by V. L. Makarov, in [1 – 3]. The idea of the approach is to approximate
the original problem by a recursive sequences of problems with the same differential operator
with piecewise constant coefficients and varying right-hand part of the equation dependent
on the solutions of previous problems in the recursive sequence. Such a method provides an
exponential convergence rate which improves when the eigenvalue index increases. In [4 – 9]
this technique is developed for different kinds of boundary conditions. Particularly, in [8, 9]
the authors study a linear eigenvalue problem with discontinuous eigenfunctions, or in other
words, the eigenvalue transmission problem. Based on the numerical FD-method they estab-
lish a qualitative result about dependence of the eigenvalue arrangement on the transmission
conditions. In [10] the approach above described, combined with the Adomiane decomposition
method [11], is developed for a numerical solution of a nonlinear Sturm – Liouville problem,
for which a unique solvability result and basic properties of eigenfunctions are established in
[12] and the literature cited therein. In [10] it is shown that for a nonlinear eigenvalue problem,
the algorithm based on the FD-approach converges with the same (exponential) characteristics
as the algorithms for the linear problems.
In this paper we apply the approach from [10] to develop a numerical algorithm for the
nonlinear eigenvalue transmission problem. The technique also provides an exponential conver-
c© V. L. Makarov, N. O. Rossokhata, 2007
126 ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
FD-METHOD FOR A NONLINEAR EIGENVALUE PROBLEM . . . 127
gence rate. However, unlike Dirichlet boundary conditions [10], as the index of the eigenvalue
increases, it tends to a constant defined by the transmission coefficient. We also study the
influence of the transmission coefficient, matching point, and normalizing conditions on conver-
gence of the algorithm and an arrangement of the eigenvalues.
The paper is organized in the following way. In Section 2 we describe a numerical techni-
que for nonlinear eigenvalue transmission problem with an additional differential normalizing
condition (vanishing the first derivative). We prove a convergence theorem which shows the
exponential convergence rate. Section 3 is devoted to numerical results. We analyze dependence
of the convergence rate on the eigenvalue index, matching point, and the transmission coeffi-
cient. In Section 4 we study the nonlinear eigenvalue transmission problem with an additional
integral normalizing condition. We obtain a convergence result similar to the result from Secti-
on 2, and illustrate the algorithm with numerical examples which confirm the theoretical ones.
2. A differential normalizing condition. Let us consider the following eigenvalue transmi-
ssion problem:
d2ui(x)
dx2
+ λui(x)−Ni(ui(x)) = 0, x ∈ Ωi, (1)
with the Dirichlet boundary conditions
u1(0) = u2(1) = 0, (2)
the matching conditions
dui(x(1))
dx
= r[u(x(1))], r > 0, i = 1, 2, (3)
and the normalizing condition
du1(0)
dx
= 1, (4)
where Ω1 = (0, x(1)), Ω2 = (x(1), 1), [u(x(1))] = u2(x(1))− u1(x(1)) is a jump of the function
at the matching point x(1), Ni(u) : <1 → <1 is an analytic function with respect to ui such that
N1(0) = 0, |N (k)
i,u (ui)| =
∣∣∣∣dkNi(ui)
duk
i
∣∣∣∣ ≤ N
(k)
i,u (|ui|), ui ∈ R, (5)
and N i(ui) is an analytic function with nonnegative derivatives for u ≥ 0.
According to the FD-approach, we write the numerical solution of problem (1) – (4) as
truncated series:
m
λn =
m∑
j=0
λ(j)
n ,
m
uni =
m∑
j=0
u
(j)
ni , i = 1, 2, (6)
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
128 V. L. MAKAROV, N. O. ROSSOKHATA
and we find the terms of the truncated series as follows.
The zero approximation (λ(0)
n , u
(0)
ni (x), i = 1, 2) is a solution of the so-called basic nonpertur-
bed eigenvalue transmission problem
d2u
(0)
ni (x)
dx2
+ λ(0)
n u
(0)
ni (x) = 0, x ∈ Ωi, i = 1, 2,
u
(0)
n1 (0) = u
(0)
n2 (1) = 0,
du
(0)
n1 (0)
dx
= 1,
du
(0)
ni (x(1))
dx
= r[u(0)
n (x(1))].
In [8] it is shown that depending on the transmission point x(1) there can exist the following
two kinds of eigenvalues λ
(0)
n :
1) λ
(0)
n is defined by the formula
λ(0)
n =
π2(2k + 1)2
4(x(1))2
=
π2(2n + 1)2
4(1− x(1))2
(7)
for n, k ∈ {0} ∪N such that
x(1)
1− x(1)
=
2k + 1
2n + 1
,
2) λ
(0)
n is a solution of the transcendental equation
r
(
tg
(√
λ
(0)
n x(1)
)
+ tg
(√
λ
(0)
n (1− x(1))
))
+
√
λ
(0)
n = 0, (8)
if
x(1)
1− x(1)
6= 2k + 1
2n + 1
.
The corresponding eigenfunctions are the following:
u
(0)
n1 (x) =
sin
(√
λ
(0)
n x
)
√
λ
(0)
n
, u
(0)
n2 (x) = ϕn
sin
(√
λ
(0)
n (1− x)
)
√
λ
(0)
n
, (9)
where
ϕn =
(−1)n−k,
x(1)
1− x(1)
=
2k + 1
2n + 1
,
−
cos
(√
λ
(0)
n x(1)
)
cos
(√
λ
(0)
n (1− x(1))
) ,
x(1)
1− x(1)
6= 2k + 1
2n + 1
.
The corrections u
(j+1)
ni (x), j = 0, 1, 2, . . . , are solutions of the system of transmission problems
for linear nonhomogenous differential equations
d2u
(j+1)
ni (x)
dx2
+ λ(0)
n u
(j+1)
ni (x) = −F
(j+1)
ni (x), x ∈ Ωi, i = 1, 2, (10)
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
FD-METHOD FOR A NONLINEAR EIGENVALUE PROBLEM . . . 129
u
(j+1)
n1 (0) = u
(j+1)
n2 (1) = 0,
du
(j+1)
n1 (0)
dx
= 0,
du
(j+1)
ni (x(1))
dx
= r[u(j+1)
n (x(1))],
where
F
(j+1)
ni (x) = −
j∑
s=0
λ(j+1−s)
n u
(s)
ni (x) + A
(j)
ni (u(0)
ni , u
(1)
ni , . . . , u
(j)
ni )
and the Adomian polynomials A
(j)
ni (u(0)
ni , . . . , u
(j)
ni ) are defined in the following way:
A
(j)
ni (u(0)
ni , . . . , u
(j)
ni ) =
=
∑
α1+···+αj=j
N
(α1)
iui
(u(0)
ni (x))
[u(1)
ni (x)]α1−α2
(α1 − α2)!
· · ·
[u(j−1)
ni (x)]αj−1−αj
(αj−1 − αj)!
[u(j)
ni (x)]αj
(αj)!
, j > 0,
A
(0)
ni (u(0)
ni ) = N(u(0)
ni ).
The solvability condition for equation (10) yields
λ(j+1)
n =
=
− j∑
p=1
λ
(j+1−p)
n
2∑
i=1
∫
Ωi
u
(p)
ni (ξ)u(0)
ni (ξ)dξ +
2∑
i=1
∫
Ωi
A
(j)
ni (u(0)
ni (ξ), . . . , u(j)
ni (ξ))u(0)
ni (ξ)dξ
‖u(0)
n ‖2
,
(11)
where ‖un‖ =
√∑2
i=1
∫
Ωi
u2
ni(x)dx is the L2(Ω1 × Ω2)-norm and
‖u(0)
n ‖ =
√
2(1− x(1))
π(2n + 1)
=
√
2x(1)
π(2k + 1)
,
x(1)
1− x(1)
=
2k + 1
2n + 1
,
| cos
√
λ
(0)
n x(1)|√
2λ
(0)
n
√√√√ x(1)
cos2
√
λ
(0)
n x(1)
+
1− x(1)
cos2
√
λ
(0)
n (1− x(1))
+
1
r
,
x(1)
1− x(1)
6= 2k + 1
2n + 1
.
Then we can write the solution of the nonhomogeneous problem as
u
(j+1)
ni (x) = û
(j+1)
ni (x), (12)
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
130 V. L. MAKAROV, N. O. ROSSOKHATA
where
û
(j+1)
n1 (x) =
=
x∫
0
sin (
√
λ
(0)
n (x− ξ))√
λ
(0)
n
−
j∑
p=0
λ(j+1−p)
n u
(p)
n1 (ξ) + A
(j)
n1 (u(0)
n1 (ξ), u(1)
n1 (ξ), . . . , u(j)
n1 (ξ))
dξ,
(13)
x ∈ (0, x(1)),
û
(j+1)
n2 (x) =
x(1)∫
0
sin (
√
λ
(0)
n (x− ξ))√
λ
(0)
n
+
cos (
√
λ
(0)
n (x(1) − x)) cos (
√
λ
(0)
n (x(1) − ξ))
r
×
×
−
j∑
p=0
λ(j+1−p)
n u
(p)
n1 (ξ) + A
(j)
n1 (u(0)
n1 (ξ), u(1)
n1 (ξ), . . . , u(j)
n1 (ξ))
dξ+
+
x∫
x(1)
sin (
√
λ
(0)
n (x− ξ))√
λ
(0)
n
−
j∑
p=0
λ(j+1−p)
n u
(p)
n2 (ξ) + A
(j)
n2 (u(0)
n2 (ξ), u(1)
n2 (ξ), . . . , u(j)
n2 (ξ))
dξ,
(14)
x ∈ (x(1), 1).
Hence, the numerical algorithm for the nonlinear eigenvalue transmission problem with di-
fferential normalizing condition (1) – (4) consists of finding eigenvalues of the basic eigenvalue
transmission problem according to (7) (or (8)), (9), evaluating (11) – (14) for j = 0, 1, 2, . . . ,m
and (6).
The error of the algorithm can be estimated as
‖un −
m
un‖∞ ≤
∞∑
j=m+1
‖u(j)
n ‖∞,
(15)
|λn −
m
λn| ≤
∞∑
j=m+1
|λ(j)
n |.
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
FD-METHOD FOR A NONLINEAR EIGENVALUE PROBLEM . . . 131
From (13), (14), (11) we receive the estimates
‖u(j+1)
n ‖∞ ≤
1√
λ
(0)
n
+
1
r
j∑
p=0
|λ(j+1−p)
n |‖u(p)
n ‖∞+
+
∑
α1+...+αj=j
N
(α1)(‖u(0)
n ‖∞)
‖u(1)
n ‖α1−α2
∞
(α1 − α2)!
. . .
‖u(j−1)
n ‖αj−1−αj
∞
(αj−1 − αj)!
‖u(j)
n ‖αj
∞
αj !
,
|λ(j+1)
n | ≤
j∑
p=1
|λ(j+1−p)
n |‖u(p)
n ‖∞+
+
∑
α1+...+αj=j
N
(α1)(‖u(0)
n ‖∞)
‖u(1)
n ‖α1−α2
∞
(α1 − α2)!
. . .
‖u(j−1)
n ‖αj−1−αj
∞
(αj−1 − αj)!
‖u(j)
n ‖αj
∞
αj !
/‖u(0)
n ‖.
Introducing the new variables
uj+1 =
‖u(j+1)
n ‖∞
‖u(0)
n ‖
1√
λ
(0)
n
+
1
r
−(j+1)
, µj+1 =
|λ(j+1)
n |
‖u(0)
n ‖
1√
λ
(0)
n
+
1
r
−j
and their numerical majorants
uj+1 ≤ uj+1, u0 =
‖u(0)
n ‖∞
‖u(0)
n ‖
, µj+1 ≤ µj+1.
we obtain the following majorazing system of equations:
uj+1 =
j∑
p=0
µj+1−pup + A
(j)(N, u1, . . . , uj),
µj+1 = uj+1 − µj+1u0, j = 0, 1, . . . ,
which leads to
uj+1 =
j∑
p=1
uj+1−pup + (1 + u0)A
(j)(N, u0, . . . , uj).
Analogously to [10] introducing the generating function for the sequence {uj} by
f(z) =
∞∑
j=0
zjuj , (16)
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
132 V. L. MAKAROV, N. O. ROSSOKHATA
from the last equation we get
(f(z)− u0)2 − (f(z)− u0) + z(1 + u0)N(f(z)) = 0.
Considering z as a variable depending on f , we obtain the following expression:
z =
(f − u0)(1 + u0 − f)
(1 + u0)N(f)
.
Since z ≥ 0, we have u0 ≤ f ≤ 1 + u0. It is easy to see that on the interval (u0, 1 + u0) there
exists a unique extremum for z(f),
zmax = R = z(fmax), (17)
where fmax satisfies z′(fmax) = 0.
Thus series (16) converges, that is, there exists a positive generating function and
Rjuj ≤
c
j1+ε
,
where c and ε are some positive constants.
From this we receive
‖u(j+1)
n ‖∞ ≤ c
(j + 1)1+ε
1
R
1√
λ
(0)
n
+
1
r
j+1
.
Analogously we obtain
|λ(j+1)
n | ≤ c
R(j + 1)1+ε
1
R
1√
λ
(0)
n
+
1
r
j
.
Hence, series (6) converge and (15) implies the estimates
‖un −
m
un‖∞ ≤ c
(m + 1)1+ε
1
R
1√
λ
(0)
n
+
1
r
m+1
,
(18)
|λn −
m
λn| ≤
c
R(m + 1)1+ε
1
R
1√
λ
(0)
n
+
1
r
m
provided that
1
R
1√
λ
(0)
n
+
1
r
< 1. (19)
Thus, we have proven the following convergence result.
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
FD-METHOD FOR A NONLINEAR EIGENVALUE PROBLEM . . . 133
Theorem 1. Let condition (5) hold and let the solution of the basic eigenvalue problem, λ
(0)
n ,
satisfy (19), where R is defined by (17). Under these assumptions, the numerical algorithm (6),
(7) (or (8)), (9), (11) – (14) converges exponentially to (1) – (4) with estimates (18).
Corollary 1. Since, according to [8],
√
λ
(0)
n =
π(2n + 1)
2(1− x(1))
,
x(1)
1− x(1)
=
2k + 1
2n + 1
,
π(2n + 1)
2(1− x(1))
+
2r
π(2n + 1)
+ O
(
n−2
)
,
x(1)
1− x(1)
6= 2k + 1
2n + 1
,
the convergence improves with the increase of the index of the trial eigenpair and tends to a
constant defined by the transmission coefficient r.
3. Numerical results. We consider the problem
d2ui(x)
dx2
+ λui(x)− u3
i (x) = 0, x ∈ Ωi, i = 1, 2,
u1(0) = u2(1) = 0,
du1(0)
dx
= 1,
dui(x(1))
dx
= r[u(x(1))].
To estimate the error of the algorithm, we compare the numerical solution obtained by the
FD-method,
m
λn, with the solution obtained by the bisection method using the Maple procedure
dverc 78, λex
n , with the accuracy 10−8.
1. Firstly, let us consider the case x(1) = 1/3. According to our theory all roots of the basic
problem depend on r. The results of calculations for λ1, λ3 and λ6 for r = 1 are given in Tables
1 – 3. The error for the eigenvalues with different indexes in the logarithmic scale is depicted
in Fig. 1. From Fig. 1 we note that the absolute value of the deviation of the approximate ei-
genvalue from the exact one, δn(m) = |
m
λn − λex
n |, obey the following functional dependence:
log δn(m) ≈ αnm + c,
which confirms the exponential convergence rate. From Tables and Fig. 1 we can also see that
the convergence rate improves with the increase of the index of the eigenvalue and tends
to a constant. Dependence of the error on the transmission coefficient for λ6 is depicted in
Table 1
m
m
λ1 |
m
λ1 − λex∗
1 |
0 7,4165088739016625 6, 3420 · 10−1
1 8,0958853583842360 4, 5174 · 10−2
2 8,0462992047046045 4, 4126 · 10−3
3 8,0512031756222008 4, 9133 · 10−4
4 8,0506527365158858 5, 9109 · 10−5
∗λex
1 = 8, 05071184559989 . . .
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
134 V. L. MAKAROV, N. O. ROSSOKHATA
Table 2
m
m
λ3 |
m
λ3 − λex∗
3 |
0 53,2835555483183284 3, 3503 · 10−1
1 53,6275661821205418 6, 3441 · 10−3
2 53,6337909108476428 1, 1941 · 10−4
3 53,6339077207744245 2, 6050 · 10−6
∗λex
3 = 53, 6339103257738684 . . .
Table 3
m
m
λ6 |
m
λ6 − λex∗
6 |
0 452,5368547373059060 3, 9380 · 10−1
1 452,9320089941100867 2, 3104 · 10−3
2 452,9296847981662003 1, 3732 · 10−5
3 452,9296985957231688 6, 5348 · 10−8
∗λex
6 = 452, 9296985303755375 . . .
Fig. 1. Dependence of the absolute error on the discretization
parameter m for eigenvalues with different indexes: δn(m) =
= ln (|
m
λn − λex
n |), x(1) = 1/3, r = 1. The curves are shown
for different values of n, n = 1 (1), n = 3 (2), n = 8 (3),
n = 5 (4).
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
FD-METHOD FOR A NONLINEAR EIGENVALUE PROBLEM . . . 135
Fig. 2. Dependence of the absolute error on the discretization
parameter m for λ6 with different values of the transmissi-
on coefficient: δr
6(m) = ln (|
m
λ6 − λex
6 |), x(1) = 1/3. The
curves are shown for different values of r, r = 1/5 (1), r =
= 1/2 (2), r = 1 (3).
Fig. 2. As it follows from Fig. 2, convergence rate improves with the increase of the transmission
coefficient r.
2. Let us consider the case x(1) = 1/4. According to our theory there are two kinds of the
eigenvalues λ
(0)
n for the basic eigenvalue transmission problem:
2.1) eigenvalues dependent on r with n = 4k − 2, k = 1, 2, . . .;
2.2) eigenvalues not dependent on r with n = 4k − 3, 4k − 1, 4k, k = 1, 2, . . . .
For r = 1 numerical results are presented in Tables 4 – 6. The error of the algorithm for
eigenvalues with different indexes in the logarithmic scale is depicted in Fig. 3. From Tables and
Fig. 3 we conclude that the convergence rate is exponential. For the eigenvalues with dependent
on r zero approximation λ
(0)
n , n = 1, 4, 9, it improves with the increase of the index of ei-
genvalue and tends to a constant. For the eigenvalues with not dependent on r zero approxi-
mation λ
(0)
n , n = 2, 6, 10, it also improves with the increase of the index of the eigenvalue, but
does not tends to a constant. We note that the numerical eigenvalues
m
λn of the original problem
with zero approximation not dependent on r also do not depend on r. The reason for this is the
type of nonlinearity, that is, in this case the term on 1/r for û
(j+1)
n2 (x) identically equals to zero,
Table 4
m
m
λ1 |
m
λ1 − λex∗
1 |
0 6,2315467060814359 9, 1749 · 10−1
1 7,1939908990839667 4, 4960 · 10−2
2 7,1463227453492218 2, 7085 · 10−3
3 7,1492041124602485 1, 7288 · 10−4
∗λex
1 = 7, 1490312331582256 . . .
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
136 V. L. MAKAROV, N. O. ROSSOKHATA
Table 5
m
m
λ2 |
m
λ2 − λex∗
2 |
0 39,4784176043544344 1, 8995 · 10−2
1 39,4974153262903728 2,6658 · 10−6
2 39,4974126598608147 6, 5919 · 10−10
3 39,49741266053293244 1, 2932 · 10−11
∗λex
2 = 39, 4974126605298376 . . .
Table 6
m
m
λ4 |
m
λ4 − λex∗
4 |
0 112,4437999204786364 5, 3781 · 10−1
1 112,9769575287073761 4, 6492 · 10−3
2 112,9815829113849929 2, 3778 · 10−5
3 112,9816068711603011 1, 8141 · 10−7
4 112,9816066967542689 7, 0034 · 10−9
∗λex
4 = 112, 9816066897508219 . . .
and hence, the expected convergence is O
(
1/
√
λ
(0)
n
)
= O (1/n). Dependence of the error on
the transmission coefficient for λ1 is depicted in Fig. 4. As it follows from Fig. 4 the convergence
rate improves with the increase of the transmission coefficient r.
Hence, we can conclude that numerical results confirm the theoretical ones.
Fig. 3. Dependence of the absolute error on the discretization
parameter m for eigenvalues with different indexes: δn(m) =
= ln (|
m
λn − λex
n |), x(1) = 1/4, r = 1. The curves are shown
for different values of n, n = 1 (1), n = 4 (2), n = 9 (3),
n = 2 (4), n = 6 (5), n = 10 (6).
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
FD-METHOD FOR A NONLINEAR EIGENVALUE PROBLEM . . . 137
Fig. 4. Dependence of the absolute error on the discretization
parameter m for λ1 with different values of the transmissi-
on coefficient: δr
1(m) = ln (|
m
λ1 − λex
1 |), x(1) = 1/4. The
curves are shown for different values of r, r = 1/2 (1), r =
= 1 (2), r = 100 (3).
4. An integral normalizing condition. We consider problem (1) – (3) with integral normali-
zing condition
‖u‖2 =
2∑
i=1
∫
Ωi
u2
ni(x)dx = M. (20)
The solution of the corresponding basic linear eigenvalue transmission problem,
d2u
(0)
ni (x)
dx2
+ λ(0)
n u
(0)
ni (x) = 0, x ∈ Ωi, i = 1, 2,
u
(0)
n1 (0) = u
(0)
n2 (1) = 0,
du
(0)
ni (x(1))
dx
= r[u(0)
n (x(1))],
‖u(0)
n ‖ =
√
M
is the following:
λ(0)
n =
π2(2k + 1)2
4(x(1))2
=
π2(2n + 1)2
4(1− x(1))2
,
x(1)
1− x(1)
=
2k + 1
2n + 1
,
λ
(0)
n : r(tg(
√
λ
(0)
n x(1)) + tg(
√
λ
(0)
n (1− x(1)))) +
√
λ
(0)
n = 0,
x(1)
1− x(1)
6= 2k + 1
2n + 1
,
(21)
k, n ∈ {0} ∪N ,
u
(0)
n1 (x) =
√
2Mϕ1 sin(
√
λ
(0)
n x), u
(0)
n2 (x) =
√
2Mϕ2 sin(
√
λ
(0)
n (1− x)), (22)
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
138 V. L. MAKAROV, N. O. ROSSOKHATA
where
ϕ1 =
1,
x(1)
1− x(1)
=
2k + 1
2n + 1
,
1
cos(
√
λ
(0)
n x(1))
x(1)
cos2(
√
λ
(0)
n x(1))
+
+
1− x(1)
cos2(
√
λ
(0)
n (1− x(1)))
+
1
r
−1/2
,
x(1)
1− x(1)
6= 2k + 1
2n + 1
,
ϕ2 =
(−1)n−k,
x(1)
1− x(1)
=
2k + 1
2n + 1
,
− 1
cos(
√
λ
(0)
n x(1))
x(1)
cos2(
√
λ
(0)
n x(1))
+
+
1− x(1)
cos2(
√
λ
(0)
n (1− x(1)))
+
1
r
−1/2
,
x(1)
1− x(1)
6= 2k + 1
2n + 1
.
We find the terms u
(j+1)
ni (x), j = 0, 1, 2, . . ., from the recursive system of transmission
problems for the linear nonhomogeneous differential equations:
d2u
(j+1)
ni
dx2
+ λ(0)
n u
(j+1)
ni = −F
(j+1)
ni (x), x ∈ Ωi, i = 1, 2, (23)
u
(j+1)
n1 (0) = u
(j+1)
n2 (1) = 0,
du
(j+1)
ni (x(1))
dx
= r[u(j+1)
n (x(1))],
2∑
i=1
∫
Ωi
u
(0)
ni (x)u(j+1)
ni (x)dx = −1
2
2∑
i=1
j∑
p=1
∫
Ωi
u
(p)
ni (x)u(j+1−p)
ni (x)dx, (24)
where F
(j+1)
ni (x) is the same as in the case of the differential normalizing condition.
The solvability condition for nonhomogeneous equation (23) leads to the expression for
λ
(j+1)
n ,
λ(j+1)
n =
−
j∑
p=1
λ(j+1−p)
n
2∑
i=1
∫
Ωi
u
(p)
ni (ξ)u(0)
ni (ξ)dξ +
2∑
i=1
∫
Ωi
A
(j)
ni (u(0)
ni (ξ), . . . , u(j)
ni (ξ))u(0)
ni (ξ)dξ
M
.
(25)
So, the numerical algorithm for the eigenvalue transmission problem with integral normali-
zing condition (1) – (3), (20) consists of finding a solution of the basic problem according to
(21), (22), finding a solution of the nonlinear problems (23), (24), and calculating (25) for j =
= 0, 1, . . . ,m, and then calculating (6).
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
FD-METHOD FOR A NONLINEAR EIGENVALUE PROBLEM . . . 139
In order to estimate the convergence rate of the algorithm according to formulas (15), let
us write the solution of the nonhomogeneous problem (23) as follows:
u
(j+1)
ni (x) = B(j+1)
n u
(0)
ni (x) + û
(j+1)
ni (x), i = 1, 2, (26)
where u
(0)
ni (x) is defined by (22), and û
(j+1)
ni (x) is defined according to (13) or (14), correspondi-
ngly.
Substitution (26) in (24) leads to
MB(j+1)
n = −
2∑
i=1
∫
Ωi
u
(0)
ni (x)û(j+1)
ni (x)dx− 1
2
j∑
p=1
2∑
i=1
∫
Ωi
u
(p)
ni (x)u(j+1−p)
ni (x)dx, (27)
from which
|B(j+1)
n | ≤ ‖u(0)
n ‖
M
‖û(j+1)
n ‖∞ +
1
2M
j∑
p=1
‖u(p)
n ‖‖u(j+1−p)
n ‖.
Taking into account the last expression, from (26) we get
‖u(j+1)
n ‖∞ ≤
(
1 +
‖u(0)
n ‖∞√
M
)
‖û(j+1)
n ‖∞ +
‖u(0)
n ‖∞
2M
j∑
p=1
‖u(p)
n ‖∞‖u(j+1−p)
n ‖∞
and, together with (13) and (14), we receive the estimate
‖u(j+1)
n ‖∞ ≤
1√
λ
(0)
n
+
1
r
(1 +
‖u(0)
n ‖∞√
M
)
j∑
p=0
|λ(j+1−p)
n |‖u(p)
n ‖∞ + ‖A(j)
n ‖
+
+
‖u(0)
n ‖∞
2M
j∑
p=1
‖u(p)
n ‖∞‖u(j+1−p)
n ‖∞. (28)
From (25) we get that
|λ(j+1)
n | ≤ 1√
M
j∑
p=1
|λ(j+1−p)
n |‖u(p)
n ‖∞ + ‖A(j)
n ‖
. (29)
Then, the recursive system of inequalities (28), (29) leads to the majorating system of equations,
uj+1 =
1√
λ
(0)
n
+
1
r
(1 +
u0√
M
)
j∑
p=0
µj+1−p up + ‖A(j)
n ‖
+
u0
2M
j∑
p=1
up uj+1−p,
(30)
µj+1 =
1√
M
j∑
p=0
µj+1−p up + ‖A(j)
n ‖ − µj+1u0
, j = 0, 1, . . . ,
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
140 V. L. MAKAROV, N. O. ROSSOKHATA
with
‖u(j)
n ‖∞ ≤ uj , u0 = ‖u(0)
n ‖∞, and |λ(j)
n | ≤ µj .
Substituting
Uj =
1√
λ
(0)
n
+
1
r
j
uj , Mj =
1√
λ
(0)
n
+
1
r
j−1
µj (31)
from (30) we obtain
Uj+1 =
(
1 +
u0√
M
)
j∑
p=0
Mj+1−pUp + ‖A(j)
n ‖
+
u0
2M
j∑
p=1
UpUj+1−p,
(32)
Mj+1 =
1√
M + u0
j∑
p=0
Mj+1−pUp + ‖A(j)
n ‖
, j = 0, 1, . . . ,
with U0 = u0 = ‖u(0)
n ‖∞.
To solve system (32), we use the method of generating functions,
f(z) =
∞∑
j=0
zjUj , g(z) =
∞∑
j=0
zjMj−1.
Relations (32) lead to the following system:
f(z)− u0 =
(
1 +
u0√
M
)
z[g(z)f(z) + N(f(z))] +
u0
2M
[f(z)− u0]2,
g(z) =
1√
M + u0
[g(z)f(z) + N(f(z))].
From this,
z = z(f) =
√
Mf
(
1− u0
2M
f
)
(
√
M − f)
(
√
M + u0)2N(f + u0)
,
where f = f(z)− u0.
Since z = z(f) is a positive continuous function on the interval l =
(
0,min
{√
M,
2M
u0
})
and z is equal to zero at the ends of the interval l, there exists
zmax = R = z(fmax), (33)
where fmax satisfies the equation z′(f) = 0. The value zmax = R is obviously the convergence
radius for the series f(z). Hence,
RjUj <
c
j1+ε
, RjMj <
c
j1+ε
,
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
FD-METHOD FOR A NONLINEAR EIGENVALUE PROBLEM . . . 141
where c and ε are positive constants.
Taking into account (31) we obtain the estimates
‖u(j+1)
n ‖∞ ≤ c
(j + 1)1+ε
1
R
1√
λ
(0)
n
+
1
r
j+1
,
|λ(j+1)
n | ≤ c
R(j + 1)1+ε
1
R
1√
λ
(0)
n
+
1
r
j
.
Thus, series (6) converge and from (15) we receive the estimates
‖un −
m
un‖∞ ≤ c
(m + 1)1+ε
1
R
1√
λ
(0)
n
+
1
r
m+1
,
(34)
|λn −
m
λn| ≤
c
R(m + 1)1+ε
1
R
1√
λ
(0)
n
+
1
r
m
,
provided that
1
R
1√
λ
(0)
n
+
1
r
< 1. (35)
Thus, we have proven the following convergence result.
Theorem 2. Let conditions (5) and (35) be satisfied. Then the numerical algorithm (21) – (25)
converges exponentially to a solution of problem (1) – (3), (20) with estimates (34).
Corollary 2. The convergence rate of algorithm (21) – (25) improves with an increase of the
index of the trial eigenvalue and tends to a constant defined by the transmission coefficient r.
Corollary 3. As it follows from (35), the convergence rate of the algorithm improves with an
increase of the transmission coefficient r.
Example:
d2ui(x)
dx2
+ λui(x)− u3
i (x) = 0, x ∈ Ωi, i = 1, 2,
u1(0) = u2(1) = 0,
dui(x(1))
dx
= r[u(x(1))], ‖u‖2 = M.
We consider the case with two kinds of zero approximations of eigenvalues, that is, λ
(0)
n
dependent on the transmission coefficient r and λ
(0)
n not dependent on r. Let us set x(1) = 1/4,
M = 1. Using solver Maple 9, we find R < 0, 016, that provid convergence of our algorithm
for large enough n and r. However, requirement (35) is sufficient and our algorithm converges
also for much smaller indexes of the eigenvalues and the transmission coefficient. As in previous
calculations, to estimate the error of algorithm, we compare the numerical solution obtained by
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
142 V. L. MAKAROV, N. O. ROSSOKHATA
Fig. 5. Dependence of the absolute error on the discretization
parameter m for eigenvalues with different indexes: δn(m) =
= ln (|
m
λn − λex
n |), x(1) = 1/4, r = 1, M = 1. The curves are
shown for different values of n, n = 1 (1), n = 2 (2), n =
= 4 (3).
Table 7
m
m
λ1 |
m
λ1 − λex∗
1 |
0 6,2315467060814359 1,8193
1 8,0735525671370393 2, 2683 · 10−2
2 8,0495411885163239 1, 3281 · 10−3
3 8,0509697134406525 1, 0047 · 10−4
∗λex
1 = 8, 05086923780985 . . .
Table 8
m
m
λ2 |
m
λ2 − λex∗
2 |
0 39,4784176043544344 1,4976
1 40,9784176043544344 2, 3596 · 10−3
2 40,9760428891158172 1, 5094 · 10−5
3 40,9760579271757605 5, 5591 · 10−8
∗λex
2 = 40, 97605799284510 . . .
Table 9
m
m
λ4 |
m
λ4 − λex∗
4 |
0 112,4437999204786364 1,9720
1 114,4165149249466765 6, 9386 · 10−4
2 114,4158464832084226 2, 5428 · 10−5
3 114,4158184347494351 2, 6207 · 10−7
∗λex
4 = 112, 9816066897508219 . . .
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
FD-METHOD FOR A NONLINEAR EIGENVALUE PROBLEM . . . 143
FD-method with the solution obtained by bisection method using the Maple procedure dverc
78. For r = 1, the numerical results are given in Tables 7 – 9 and depicted in Fig. 5. From
Tables and Fig. 5 we note that the convergence rate is exponential. It improves with the increase
of the index of the eigenvalue and tends to a constant. The method converges better for the
eigenvalues with zero approximation not dependent on r.
1. Makarov V. L. About functional-discrete method of arbitrary order of accuracy for solving Sturm – Liouville
problem with piecewise smooth coefficients // Sov. DAN SSSR. — 1991. — 320, № 1. — P. 34 – 39.
2. Makarov V. L. FD-method: the exponential rate of convergence // J. Math. Sci. — 1997. — 104, № 6. —
P. 1648 – 1653.
3. Makarov V. L., Ukhanev O. L. FD-method for Sturm – Liouville problems. Exponential rate of convergence
// Appl. Math. and Inform. — 1997. — 2. — P. 1 – 19.
4. Bandyrskii B. I., Makarov V. L., Ukhanev O. L. Sufficient conditions for the convergence of non-classical
asymptotic expansions for Sturm – Liouville problems with periodic conditions // Different. Equat. — 1999.
— 35, № 3. — P. 369 – 381.
5. Bandyrskii B. I., Makarov V. L. Sufficient conditions for eigenvalues of the operator with Ionkin – Samarskii
conditions to be real-valued // Comput. Math. and Math. Phys. — 2000. — 40, № 12. — P. 1715 – 1728.
6. Bandyrskii B. I., Lazurchak I. I., Makarov V. L. A functional-discrete method for solving Sturm – Liouville
problems with an eigenvalue parameter in the boundary conditions // Ibid. — 2002. — 42, № 5. — P. 646 – 659.
7. Bandyrskii B. I., Gavrilyuk I. P., Lazurchak I. I., Makarov V. L. Functional-discrete method (FD-method) for
matrix Sturm – Liouville problems // Comput. Meth. Appl. Math. — 2005. — 5, № 4. — P. 1 – 25.
8. Bandyrskii B. I., Makarov V. L., Rossokhata N. O. Functional-discrete method with a high order of accuracy
for the eigenvalue transmission problems // Ibid. — 2004. — 4, № 3. — P. 324 – 349.
9. Bandyrskii B. I., Makarov V. L., Rossokhata N. O. Functional-discrete method for an eigenvalue transmission
problem with periodic boundary conditions // Ibid. — 2005. — 5, № 2. — P. 201 – 220.
10. Gavrilyuk I. P., Klimenko A. V., Makarov V. L., Rossokhata N. O. Exponentially convergent parallel algorithm
for nonlinear eigenvalue problems // IMA J. Numer. Anal. — 2007. — 27, № 2.
11. Abbaoui K., Pujol M. J., Cherruault Y., Himoun N., Grimalat P. A new formulation of Adomian method.
Convergence result // Kybernetes. — 2001. — 30, № 9/10. — P. 1183 – 1191.
12. Zhidkov P. E. Basis properties of eigenfunctions of nonlinear Sturm – Liouville problems // El. J. Different.
Equat. — 2000. — № 28. — P. 1 – 13.
Received 19.09.2006
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
|
| id | nasplib_isofts_kiev_ua-123456789-7247 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-3076 |
| language | English |
| last_indexed | 2025-12-07T18:07:29Z |
| publishDate | 2007 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Makarov, V.L. Rossokhata, N.O. 2010-03-26T10:09:02Z 2010-03-26T10:09:02Z 2007 FD-method for a nonlinear eigenvalue problem with discontinuous eigenfunctions / V.L. Makarov, N.O. Rossokhata // Нелінійні коливання. — 2007. — Т. 10, № 1. — С. 126-143. — Бібліогр.: 12 назв. — англ. 1562-3076 https://nasplib.isofts.kiev.ua/handle/123456789/7247 517.983.27 An algorithm for solution of a nonlinear eigenvalue problem with discontinuous eigenfunctions is developed. The numerical technique is based on a perturbation of the coefficients of differential equation combined with the Adomian decomposition method for the nonlinear term of the equation. The proposed approach provides an exponential convergence rate dependent on the index of the trial eigenvalue and on the transmission coefficient. Numerical examples support the theory. Розроблено алгоритм для числового розв'язування нелінійних задач на власні значення з розривними власними функціями. В основі числового методу лежить збурення коефіцієнтів диференціального рівняння в поєднанні з методом декомпозиції Адомяна нелінійної частини рівняння. Запропонований підхід забезпечує експоненціальну швидкість збіжності, яка залежить від порядкового номера власного значення та коефіцієнта трансмісії. Наведені числові розрахунки підтверджують теоретичні висновки. en Інститут математики НАН України FD-method for a nonlinear eigenvalue problem with discontinuous eigenfunctions FD-метод для нелінійної задачі на власні значення з розривними власними функціями Article published earlier |
| spellingShingle | FD-method for a nonlinear eigenvalue problem with discontinuous eigenfunctions Makarov, V.L. Rossokhata, N.O. |
| title | FD-method for a nonlinear eigenvalue problem with discontinuous eigenfunctions |
| title_alt | FD-метод для нелінійної задачі на власні значення з розривними власними функціями |
| title_full | FD-method for a nonlinear eigenvalue problem with discontinuous eigenfunctions |
| title_fullStr | FD-method for a nonlinear eigenvalue problem with discontinuous eigenfunctions |
| title_full_unstemmed | FD-method for a nonlinear eigenvalue problem with discontinuous eigenfunctions |
| title_short | FD-method for a nonlinear eigenvalue problem with discontinuous eigenfunctions |
| title_sort | fd-method for a nonlinear eigenvalue problem with discontinuous eigenfunctions |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/7247 |
| work_keys_str_mv | AT makarovvl fdmethodforanonlineareigenvalueproblemwithdiscontinuouseigenfunctions AT rossokhatano fdmethodforanonlineareigenvalueproblemwithdiscontinuouseigenfunctions AT makarovvl fdmetoddlânelíníinoízadačínavlasníznačennâzrozrivnimivlasnimifunkcíâmi AT rossokhatano fdmetoddlânelíníinoízadačínavlasníznačennâzrozrivnimivlasnimifunkcíâmi |