1. No category

FD-method for solving nonlinear Sturm-Liouville problems with
distribution potentials
6-th Europian
Congress of
Mathematics
Denys V. Dragunov ([email protected])
Krakow, Poland
July, 2-7
Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, Ukraine
1. What is the problem?
3.The FD approach in general
4. Getting back to the SLP
We consider the following nonlinear SturmLiouville problem (SLP): (N [u(x)]+λ)u(x) =
Let us denote by L(H) the space of linear operators acting on a Hilbert space H.
We replace the initial eigenvalue problem (EVP)
To apply the FD approach to SLP (1)–(5) we
assume that H = L2 (0, 1), D = {u ∈ D|u′ (0) = 1},
d2 u(x)
−
Q(x)
+
N
(u(x))
−
λ
u(x)
=
0,
(1)
=
dx2
def
u(x) ∈ D(N [·]) = D = u| u, u
˜ ∈ W21 (0, 1),
x
R
u
˜ = u′− Q(ξ)dξu, u(0) = u(1) = 0 , u′(0) = 1, (2)
0
Nδ
P
Q(x) =
i=1
βi δ(x − αi ) + q(x), αi ∈ (0, 1), (3)
βi ≥ 0, x ∈ (0, 1)\{α1, α2 , . . . , αNδ }, (4)
∞
P
ap up , ∀u ∈ R, q(x) ∈ L1 (0, 1), (5)
N (u) =
N [u]u + λu = 0,
τ ∈ [0, 1], M[·] : D → L(H), M[·] ∈ C∞ (D),
where M[·] “approximates” N [·] (in some sense).
Assuming that problem (7) is solvable and
u(τ ) =
∞
P
(i)
u τ i , λ(τ ) =
i=0
Simple
Matrix
Methods;
Variational
Methods;
Shooting and Profer
Methods;Pruess
Methods (see [2])
Piecewise Perturbation
Methods
(see [4]);
Functional-Discrete Methods
∞
P
i=0
lim k¯
qω (x, h) − q(x)k1 = 0, h = max (xi −xi−1 ).
h→0
(i)
i
τ
λ , τ ∈ [0, 1],
(0)
2u
d
(0) (0)
(0)
(x)
−(¯
qω (x)+N ([ u(x)]ω )−λ ) u(x) = 0, (11)
2
dx
αi +0
(0)
(0)′
(0)
(0)
(0)′
u (x)α −0 = βi u(αi ), u(0) = u(1) = 0, u (0) = 1,
i
(k)
(k)
unknown u and λ can be found as solutions
to the basic problem (which is supposed to be
“easier” then the initial problem (6))
(0)(0)
u∈D
M[ u ] u + λ u = 0,
whereas system (9) transforms into a recurrence
system of linear BVPs (with parameters)
(8)
(k)
2u
d
(0) (k)
(0)
(x)
u
u
−(¯
q
(x)+N
([
(x)]
)−
)
(x) = (12)
λ
ω
ω
2
dx
k (j)(k−j)
P
(p)
=
Aj (N (·); [ u(x)]ω , p = 0, j)−λ u (x)−
and the system of linear recurrence equations
(k) (i) (i)
Φ 0( u , λ , i
Analytical
methods
= 0, k) =
∞ (i)
∞ (i)
dk P
P
i
i
u τ ,
= kΦ
= 0.
λ τ dτ
τ =0
i=0
i=0
Variational
Iteration and Hmotopy
Perturbation Methods (see [3]); Adomian Decomposition
Method (see [5]);
j=1
(k−1−j)
(9)
(k)
find λ from the orthogonality condition
E
D(0) (k) (i) (i)
u , Φ 0( u , λ , i = 0, k) = 0.
j
X
d
p vp τ )
.
Aj (N (·); vp , p = 0, j) = j N (
dτ
τ =0
p=0
Conjecture 1. Suppose that the pair
hun (x), λn i is a solution to SLP (1)–(5).
Then for a sufficiently small h there exists
(13)
which has been proved to have a unique solution µ = µn on each interval [πn, π(n + 1)), n ∈ N. Once
µn is computed we can find the solutions to the basic problem (11) and BVPs (12) via the formulas
x
(k)
R

(

Sn (x−ξ)Fn(ξ)dξ,
x ∈ [0, α1 ] ,

S
(x)/µ
,
x
∈
[0,
α
]
,
(0)
(k)
n
n
1
un(x)= (0)
un (x) = 0
1
(k)
R
(k)

cnSn (x−1), x ∈ (α1 , 1] ,

 cn Sn (1−x)− Sn (x−ξ)Fn(ξ)dξ, x ∈ (α1 , 1] ,
λn =
3
(0) ′
′
cn= µn Sn (1)+β1 Sn (α1 )Sn (1−α1) /µn ,
Sn (x) = sin(µn x),
k (j) (k−j)
P
(k)
R1
(k)
cn Sn (1−α1 ) = Sn (α1 −ξ)Fn(ξ)dξ,
0
(k)
cn Sn′ (1−α1 )
α
R1
(k)
R1
(14)
|λn −
m
m
kun (x)− un(x)k∞ = O(am+1 rnm+1 ), |λn − λn
| = O(am+1 rnm ),
(15)
(k)
(k)
√
where am = 1/((2m − 1) πm) and the pairs hun(x), λn i are computed according to formulas (14).
p
Theorem 2. Suppose that Nδ = 1 and rn = (1+ 1+(1+β1/µn )2 /(µn Rn ) < 1, where Rn > 0 denotes
∞
P
z j v¯j that satisfies the functional equation
the convergence radius of the power series f (z) =
j=1
2
(0)
′
e1 (f (z))+ N
e1 (0)v 0,n − N
e1 (0) , N
e1 (u) = N
e (u+k unk∞ ),
f (z) = f (z) +z(1+v0,n ) kqk1 f (z)+v0,n + N
∞
P
e (u) =
N
|ap |up+1 . Then asymptotical equalities (15) holds true with am = 1/mε for some ε > 0.
k=0
m (k)
P
λn| = O
k=0
[1]
A. M. Savchuk, A. A. Shkalikov, On the eigenvalues of the SturmLiouville operator with potentials in Sobolev spaces, Mat. Zametki,V.80, 6, P.864–884.
[2]
J.D. Pryce, Numerical solution of Sturm-Liouville problems,
Monographs on Numerical Analysis, New York, 1993, p. xiv+322.
[3]
A. Neamaty and R. Darzi,Comparison between the variational iteration method and the homotopy perturbation method for the
Sturm-Liouville differential equation, Bound. Value Probl. 2010.
[4]
V. Ledoux, M. Van Daele, On CP, LP and other piecewise perturbation methods for the numerical solution of the Schrödinger
equation, Z. Angew. Math. Phys., V.62, 6, P.993–1011.
[5]
S. Somali, G. Gokmen, Adomian decomposition method for nonlinear Sturm-Liouville problems, Surv. Math. Appl., V.2, P.11–20.
[6]
V. Seng, K. Abbaoui, Y. Cherruault, Adomian’s Polynomials for
Nonlinear Operators, Math. Comput. Modelling, V.24, 1, P.59–65.
[7]
V. Makarov, N. Rossokhata, D. Dragunov, Exponentially convergent functional-discrete method for solving SLPs with potentials
that include the Dirac δ-function. arXiv:1112.2540v1.
I want to acknowledge with thanks the support of my academic father
Prof. Volodymyr Makarov, whose brilliant mind and mathematical
intuition I will always admire.
6. “I must at once [. . . ] make you a witness of the things we’ve told you.”
u3(x)
u’1(x)
-
-
-5
r
-15
-10
where H(x) denotes the Heaviside function. The numerical data presented in the table
below and on the figures to the right were obtained via a C++ implementation of the
algorithm that is available at www.sourceforge.net/projects/imathsoft/files/.
-25
-20
n
1
2
3
4
5
m
10
10
10
10
10
m
λn
23.437363200234028176
50.879953432153777724
102.294039773949565868
167.932111361326104363
261.703789042290324125
k
(m)
un (x)k∞
1.5e-11
7.5e-12
3.0e-13
2.2e-16
2.5e-17
(m)
| λn |
7.6e-10
2.4e-10
5.7e-11
1.2e-13
6.8e-16
m
ρn
2.5e-11
2.0e-11
1.9e-13
1.1e-16
3.3e-17
n
6
7
8
9
10
m
10
10
10
10
10
u3’(x)
u3’(x)
0
0
u3(x)
-
8
and N (u) = u . Since the exact eigensolutions to this problem are unknown to estimate
the approximation errors we will use the functional
i
ξ h
1
(m)
R
R
m
m
m
m
d un(ξ)
q(ξ1 ) −λn + N (un(ξ1 )) un(ξ1 ) dξ1 dξ,
ρn = 1 − dξ + H(ξ − α1 )β1un(α1 ) +
(J.B.Moliere, Tartuffe.)
u1(x)
8
m rn
m+1 , η(h) = max{h, kqω −qk1 },
8. Acknowledgements
p=1
Let us apply the algorithm described by formulas (13), (14) to SLP (1), (2) with Nδ = 1,
p
p
p
p
α1 = 1/2, β1 = 2, q(x) = 1/ |0.7 − x| + 1/ |0.1 − x| + 1/ |0.3 − x| + 1/ |0.4 − 2x|,
(k)
(k)
hun (x), λn i,
7. References
(k)
m (k)
m (k)
m
P
P
m
un(x) = un(x), λn = λn ,
k=0
k=0
(x), λn i that satisfies EVP (11)
where the constant θ is independent of h, n, m.
Sn (ξ− α1 )Fn(ξ)dξ− Sn′ (α1 −ξ)Fn(ξ)dξ.
0
0
p
Theorem 1. Suppose that Nδ = 1, N (u) ≡ 0 and rn = (1 + 1+(1+β1/µn )2 /(µn Rn ) < 1, where
q
(0) 2
(0)2
2
Rn = (1+2¯
v0,n −2 v¯0,n + v¯0,n )/((1 + v¯0,n )kqk1 ), v¯0,n = k unk∞ /k un k1 . Then
= β1
(0)
and uniquely determines the pairs
k ∈ N that satisfy BVPs (12) and the following
asymptotical equalities:
m (k)
m+1 P
rn
kun − unk∞ = O m+1 ,
rn = θη(h)/n,
(k−1)
(p)
Fn(x) = q(x) un (x)− λn un (x)+Ak−1 (N1 (·); un(x), p = 0, k−1) /µn , N1 (u) = N (u)u,
j=1
(k)
k−1
R1 (0) P (j) (k−j)
(0)2
(p)
(k−1)
λn = un(ξ) q(ξ) un (ξ)− λn un (ξ)+Ak−1 (N1 (·); un(ξ), p = 0, k−1) dξ/k un k1
j=1
0
(k)
where parameters cn (k ∈ N) can be found from the following two linear equations:
(k)
(0)
hun
a pair
x
2
µn ,
(k)
j
Let us consider a simple case when function Q(x) (3) has a unique δ-singularity on [0, 1] (Nδ = 1)
and the operator M[·] is defined by formula (10) with q¯ω (x) ≡ 0, [·]ω ≡ 0, as it was done in [7].
In this case the basic problem (11) can be reduced to the nonlinear equation
(0)
j=0
(p)
u (x) ∈ C[0, 1], u (x) ∈ C 1 [xi−1 , xi ], i = 1, K,
k = 0, 1, . . . . Here Aj (·) denotes an Adomian
polynomial of order j (see [6] for more detail):
(A. Einstein)
µ sin(µ) = −β1 sin(µα1 ) sin(µ(1 − α1 )),
(p)
Aj (N (·); [ u(x)]ω , p = 0, j)−
(k)
H
5. “. . . as simple as possible, but not simpler.”
(x)
k−1
P
(k−1)
−Aj (N (·); u (x), p = 0, j) +(q(x)−qω (x)) u (x),
αi +0
(k)′
(k)
(k)
(k)
(k)′
u (x)α −0 = βi u(αi ), u (0) = u (1) = u (0) = 0,
i
If M′ [ u ] = 0 and M[ u ] = M[ u ]∗ then we can
Functional-Discrete
Homotopy
(FD) Methods (see ≈ Pruess
+ Perturbation
Methods
[7] and ref. therein)
Methods
u
−
(0)
(0)
(0)
i=1,K
Then (1)–(5) ⇔ (6) and we get the basic problem (8) in the form of an EVP
we immediately get that u = u(1), λ = λ(1). The
2. Methods for solving SLPs
Numerical-Analytical methods
where ω = {0 = x0 < x1 < . . . < xK = 1}, K ∈ N,
q¯ω (x) = q¯ω (x, h) = q¯i = const, [f (x)]ω = f (xi−1 ),
∀x ∈ [xi−1 , xi ), i = 1, K, ∀f (x) ∈ C[0, 1] and
Φ(u(τ ), λ(τ )) = M[u(τ )]u(τ )+λ(τ )u(τ ) = (7)
= τ (M[u(τ )]−N [u(τ )])u(τ ), u(τ ) ∈ D,
(0) (0)
Numerical
methods
(6)
N [·] : D → L(H), N [·] ∈ C∞ (D) (quasi-differentiable by Dieudonne) with the perturbed one
p=1
where δ(x) denotes the Dirac δ-function. Such
problems are of great interest in quantum mechanics (see [1] and the references therein).
In what follows we assume that k · k∞ is the
Chebyshev norm on [0, 1] and that k · k1 denotes
the norm from L1 (0, 1).
u ∈ D ⊆ H,
Nδ
d2 P
N [·] = 2 −
βi δ(x−αi )−q(x)−N (·),
dx i=1
Nδ
d2 P
M[·] = 2 − βi δ(x−αi )− q¯ω (x)−N([·]ω), (10)
dx i=1
-10
-20
-30
-35
2
4
m
6
-30
m
8 10
λn
365.290665054662412777
497.311217072847814939
642.305601325675356973
813.233561353244869046
995.761252385458344653
2
k
4
6
(m)
un (x)k∞
1.5e-17
6.3e-19
6.7e-19
7.3e-21
3.7e-21
8
10
(m)
| λn |
9.5e-16
1.2e-16
4.6e-17
1.7e-17
2.9e-18
m
2
4
6
m
ρn
3.7e-18
1.1e-19
1.3e-19
9.1e-22
4.4e-22
8
10