Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 81, pp. 1–15.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2023.81

INVERSE NODAL PROBLEMS FOR DIRAC OPERATORS AND

THEIR NUMERICAL APPROXIMATIONS

FEI SONG, YUPING WANG, SHAHRBANOO AKBARPOOR

Abstract. In this article, we consider an inverse nodal problem of Dirac oper-

ators and obtain approximate solution and its convergence based on the second
kind Chebyshev wavelet and Bernstein methods. We establish a uniqueness

theorem of this problem from parts of nodal points instead of a dense nodal

set. Numerical examples are carried out to illustrate our method.

1. Introduction

We consider the matrix equation

Lu := Bu′ +Q(x)u = λu, 0 ≤ x ≤ π, (1.1)

with the boundary conditions

u1(0, λ) sinα+ u2(0, λ) cosα = 0, (1.2)

u1(π, λ) sinβ + u2(π, λ) cosβ = 0, (1.3)

where

B =

(
0 1
−1 0

)
, Q(x) =

(
v(x) +m 0

0 v(x)−m

)
, u(x) =

(
u1(x)
u2(x)

)
,

and λ is the spectral parameter, v(x) is a real-valued, absolutely continuous func-
tion, and α, β ∈ [0, π), m > 0 is constant.
L := L(v,m, α, β) is called the Dirac operator which is the relativistic Schrödinger

operator in quantum physics [12]. Inverse nodal problems consist in recovering the
potential Q(x) and the coefficients α, β from the given subsets of the nodal points
(zeros of eigenfunctions). This class of inverse nodal problems has been studied for
the Sturm-Liouville operator [15], which showed that one set of nodal points can
determine the Sturm-Liouville operators uniquely. The solution of inverse nodal
problem was given by Hald and McLaughlin [10]. Some recent results on such
problems can be found in [3, 4, 6, 7, 8, 20, 21, 23, 29]. The stability of the inverse
nodal problem was explored in [5, 16]. Numerical solutions of the inverse nodal
problem were presented in [17, 22].

Basic and comprehensive results about Dirac operators were introduced in [12].
Inverse nodal problems for Dirac operators were studied in [1, 9, 11, 24, 27, 30].

2020 Mathematics Subject Classification. 34A55, 34B99, 34L05, 45C05.
Key words and phrases. Dirac operator; inverse nodal problem; Chebyshev wavelet;
Bernstein method; uniqueness.
©2023. This work is licensed under a CC BY 4.0 license.
Submitted May 30, 2023. Published December 6, 2023.

1



2 F. SONG, Y. WANG, SH. AKBARPOOR EJDE-2023/81

Inverse nodal problems of reconstructing the Dirac operator on a finite interval were
studied in [28], where it was proved that the operator L is determined uniquely by
specifying a dense set of nodal points. This article investigates the inverse nodal
problem of the Dirac operator (1.1)-(1.3). We establish a uniqueness theorem of the
inverse nodal problem for the operator L from parts of nodal points. Meanwhile, we
will show numerical solution of inverse nodal problem for the Dirac operator based
on the second kind Chebyshev wavelet (SCW) [26, 31, 32] and Bernstein methods
[14].

We first present preliminaries and give the asymptotic formulas of nodal points,
which plays an important role in the following analysis. We denote by u(x, λ) the
solution of (1.1), satisfying the initial conditions

u1(0, λ) = cosα, u2(0, λ) = − sinα.

Then u1(x, λ), u2(x, λ) have the asymptotic formulas [25], respectively,

u1(x, λ) = cos
(
λx− η(x)− α

)
+O

(eτx

λ

)
,

u2(x, λ) = sin
(
λx− η(x)− α

)
+O

(eτx

λ

)
,

where τ = | Imλ|, η(x) =
∫ x
0
v(t)dt.

According to [12], the asymptotic forms of u1(x, λ), u2(x, λ) can be written by
the method of successive approximations

u1(x, λ) = cos
(
λx− η(x)− α

)
+
U1(x, λ)

λ
+ o
(eτx

λ

)
, (1.4)

u2(x, λ) = sin
(
λx− η(x)− α

)
− U2(x, λ)

λ
+ o
(eτx

λ

)
(1.5)

for large |λ| and

U1(x, λ) = m sin
(
λx− η(x)

)
sinα+

m2x

2
sin
(
λx− η(x)− α

)
,

U2(x, λ) = m sin
(
λx− η(x)

)
cosα+

m2x

2
cos
(
λx− η(x)− α

)
.

All estimates are uniform with respect to x for x ∈ [0, π].
The characteristic function ∆(λ) of (1.1)-(1.3) is defined by

∆(λ) := u1(π, λ) sinβ + u2(π, λ) cosβ,

and all zeros λn, n ∈ Z of ∆(λ) coincide with the eigenvalues of L.
We use σ(L) := {λn : n ∈ Z} to denote all eigenvalues λn, which are real and

simple. By applying (1.4)-(1.5), eigenvalues λn satisfy the asymptotic formula [25]

λn = n+
ω1

π
+
ω2

n
+ o
( 1

n

)
,

where

ω1 = α− β + η(π), ω2 =
m(sin 2α− sin 2β) +m2π

2π cos2 ω1
.

If cosω1 = 0, we replace ω2 by 0. Let u(x, λn) = (u1(x, λn), u2(x, λn))T be
the eigenfunction corresponding to the λn of L. xjn is called the nodal point of
u1(x, λn), i.e. u1(xjn, λn) = 0. And for sufficiently large |n|, u1(x, λn) has |n| nodal
points in (0, π) (see [28]),

0 < x1n < x2n < · · · < xnn < π, for n > 0,



EJDE-2023/81 INVERSE NODAL PROBLEMS FOR DIRAC OPERATORS 3

0 < x−1n < x−2n < · · · < xn+1
n < π, for n < 0.

When |n| → ∞, xjn satisfies the following asymptotic formula by using (1.4)-
(1.5),

xjn =
(j − 1

2 )π + α+ η(xjn)

n
−
(
(j − 1

2 )π + α+ η(xjn)
)
ω1

n2π

+
(−1)j+1

(
m sin 2α+m2xjn

)
2n2

+O
( 1

n3

)
,

(1.6)

which is uniform with respect to j ∈ Z.
We denote the nodal set by Xn := {xjn}

n+j0
j=j0+1 for some j0 and n > 0, j0 ∈ Z.

Clearly, X := ∪nXn is a dense nodal set on [0, π]. For simplicity, we assume j0 = 0
in this paper.

This article is organized as follows. In Section 2, the approximation of solution
and convergence of the inverse nodal problem of the Dirac operator are studied
based on the second kind Chebyshev wavelet and Bernstein methods. In Section
3, we establish a uniqueness theorem of the inverse nodal problem for the Dirac
operator from parts of nodal points instead of a dense nodal set. Section 4 is
devoted to showing numerical examples to demonstrate the efficiency of our method.
Conclusion is given in Section 5.
Approximate solution. For each fixed n, n� 1, we obtain a nodal set {xjn}nj=1

which satisfies (1.6) together with the coefficients
(
m,α, β, η(π)

)
, and consequently

give the numerical solution of v(x). It follows from [28, Theorem 2.2].

Theorem 1.1 ([28, Theorem 2.2]). The function v(x) on [0, π] and the coefficient
m,α, β can be uniquely determined by the dense nodal set X.

IP1 Inverse Problem 1: Reconstruct v(x),m, α, β by parts of nodal set X and
η(π).

2. Approximation of solution and its convergence

In this section, we give the approximate solution of v(x) by the nodal points xjn
for n > 0 and j = 1, n together with coefficients

(
m,α, β, η(π)

)
for sufficiently large

n. It follows from (1.4) that∫ xjn

0

v(t)dt ∼= (n+
ω1

π
+
ω2

n
)xjn − α− (j − 1

2
)π +

(−1)j(m sin 2α+m2xjn)

2n
. (2.1)

The above integral equation is called the first type of Fredholm integral equation.
By using the SCW and Bernstein methods, we convert the 1-st type of Fredholm
integral equation into linear equation systems. Since the solution of (2.1) is also a
solution of v(t), we obtain an approximation of the potential v(x) by the SCW and
Bernstein methods.

Consider the first type of Fredholm integral equation∫ π

0

f(s)k(t, s)ds = g(t), 0 ≤ t ≤ π, (2.2)

where the functions g(t) is continuous on the intervals [0, π], k(t, s) is continuous
on [0, π]× [0, π], and the function f(s) is unknown.

In recent years, some numerical methods have been studied to obtain the ap-
proximate solution of the equation (2.2) (see for example [2, 18, 19]). In this article,
we use the SCW and Bernstein methods to calculate the approximate solution of



4 F. SONG, Y. WANG, SH. AKBARPOOR EJDE-2023/81

1-st type of Fredholm integral equation and consequently compute the approximate
solution of v(x).

For the convenience of readers, we briefly present the SCW and Bernstein meth-
ods.

Second kind Chebyshev wavelet method (SCW method). The second kind
Chebyshev wavelets ψl,m(t) = ψ(k, l,m, t) are a family of four-parameters functions
which are defined on the interval [0, 1) (see [26, 31, 32]),

ψl,m(t) =

{
2k/2Ũm(2kt− 2l + 1), l−1

2k−1 ≤ t < l
2k−1 ,

0, otherwise,

where k can be any positive integer, l = 1, 2, . . . , 2k−1,

Ũm(t) :=
( 2

π

)1/2
Um(t),

Um(t) is of the second kind Chebyshev polynomial for m = 0, 1, 2, . . . . We note
that {Um(t)} is a sequence of orthogonal polynomial on the interval [−1, 1] with

respect to the weight function w(t) =
√

1− t2 and can be defined by the recursive
formula

U0(t) = 1, U1(t) = 2t,

Um+1(t) = 2tUm(t)− Um−1(t), m = 1, 2, . . . .

Then, the SCW defined on the interval [0, π) can be written in the form

ψl,m(t) =

{
1
π2k/2Ũm( 2k

π t− 2l + 1), (l−1)π
2k−1 ≤ t < lπ

2k−1 ,

0, otherwise.

When v(t) ∈ L2[0, π), it can be approximated by the SCW method,

v(t) =

2k−1∑
l=1

∞∑
m=0

cl,mψl,m(t),

where

cl,m = 〈v(t), ψl,m(t)〉L2
w[0,π) =

∫ π

0

v(t)ψl,m(t)w
(2k

π
t− 2l + 1

)
dt,

and 〈·, ·〉L2
w[0,π) is the inner product in L2

w[0, π).
By truncating the above infinite series, we obtain

v(t) ∼=
2k−1∑
l=1

M−1∑
m=0

cl,mψl,m(t) = CTΨ(t), (2.3)

where

C = [c1,0, c1,1, . . . , c1,M−1, c2,0, . . . , c2,M−1, . . . , c2k−1,0, . . . , c2k−1,M−1]T ,

Ψ(t) =
[
ψ10(t), . . . , ψ1(M−1)(t), ψ20(t), . . . , ψ2(M−1)(t), . . . , ψ2k−10(t),

. . . , ψ2k−1(M−1)(t)
]T
.

We denote by ‖v‖2,w the norm of v(t) in L2
w[0, π). Modifying the proof in [31],

we have the following theorem to investigate the convergence of SCW method.

Theorem 2.1. For each fixed n = 2k−1M , 2k−1,M � 1, we have



EJDE-2023/81 INVERSE NODAL PROBLEMS FOR DIRAC OPERATORS 5

(i) The potential function v(x) can be written as an infinite sum of the second
kind Chebyshev wavelets and this series converges to v(x), that is

v(x) =

2k−1∑
l=1

∞∑
m=0

cl,mψl,m(x)

and

‖v − vn‖22,w =

∫ π

0

(v(x)− vm(x))2w(x)dx =

2k−1∑
l=1

∞∑
m=M

c2l,m,

where the approximation of the potential function v(x) is

vn(x) =

2k−1∑
l=1

M−1∑
m=0

cl,mψl,m(x).

(ii) Let v(x) be a differentiable function on [0, π) and v′(x) satisfies the Lipschitz
condition. Then, v(x) can be expanded as an infinite sum of the second kind
Chebyshev wavelets and the series converges to v(x) uniformly on [0, π).
Moreover

‖v − vn‖22,w =

2k−1∑
l=1

∞∑
m=M

c2l,m ≤
B2π

2k+6

∞∑
m=M

1

m4

for some B > 0.

When n� 1, taking (2.3) into (2.1), we have

2k−1∑
l=1

M−1∑
m=0

cl,m

(∫ xjn

0

ψl,m(t)dt
)
∼=
(
n+

ω1

π
+
ω2

n

)
xjn − α− (j − 1

2
)π

+
(−1)j

(
m sin 2α+m2xjn

)
2n

,

(2.4)

for j = 1, n and n = 2k−1M . Consequently, the potential function v(t) can be
created by using the following numerical algorithm 1.

Numerical algorithm 1: (1) For n � 1, choose the values M , k, α, β, m and
set n = 2k−1M . We obtain the numerical values of nodal points {xjn}nj=1 through
formula (1.6) together with η(π).

(2) Use formula (2.4) to compute the vector Y by the linear system

AY = B,

A = [A1,A2, . . . ,A2k−1 ],

A1 =


∫ x1

n

0
ψl,0(t)dt

∫ x1
n

0
ψl,1(t)dt . . .

∫ x1
n

0
ψl,M−1(t)dt∫ x2

n

0
ψl,0(t)dt

∫ x2
n

0
ψl,1(t)dt . . .

∫ x2
n

0
ψl,M−1(t)dt

· · · · · · · · ·∫ xnn
0

ψl,0(t)dt
∫ xnn
0

ψl,1(t)dt . . .
∫ xnn
0

ψl,M−1(t)dt

 l = 1, 2k−1.



6 F. SONG, Y. WANG, SH. AKBARPOOR EJDE-2023/81

Matrix B is written in the form

B =


(n+ ω1

π + ω2

n )x1n − α− π
2 −

(m sin 2α+m2x1
n)

2n

(n+ ω1

π + ω2

n )x2n − α− 3π
2 +

(m sin 2α+m2x2
n)

2n
· · ·

(n+ ω1

π + ω2

n )xnn − α− (n− 1
2 )π +

(−1)n(m sin 2α+m2xnn)
2n


and Y = [y1, y2, . . . , yn]T .

(3) Calculate the values v(ti) by the formula

[v(ti)] = YTΦ,

where

ti =
(2i− 1)π

2kM
, i = 1, 2k−1M, Φ =

[
Ψ(

π

2n
),Ψ(

3π

2n
), . . . ,Ψ(

(2n− 1)π

2n
)
]
.

Bernstein method. The N−th degree Bernstein basis polynomials on [0, 1] are
defined as (see [14])

Bk,N (t) =

(
N
k

)
tk(1− t)N−k, k = 0, N.

An arbitrary function f(t) defined on [0, 1] can be approximated by Bernstein

polynomials BfN (t) as

f(t) ∼= BfN (t) :=

N∑
k=0

f
( k
N

)
Bk,N (t).

Suppose that the function f(t) is defined on [0, π]. Thus, using the variable t
π

instead of t, we can write

Bk,N
( t
π

)
=

(
N
k

)( t
π

)k
(1− t

π
)N−k, k = 0, N.

And then

f(t) ∼= BfN (t) :=

N∑
k=0

f
(kπ
N

)
Bk,N (

t

π
).

Theorem 2.2 ([14]). For the function f(t) bounded on [0, 1], the relation

lim
N→∞

BfN (t) = f(t) (2.5)

holds at each point of continuity t of f(t); and the relation holds uniformly on [0, 1]
if f(t) is continuous on this interval.

Consequently, if f(t) is continuous on interval [0, π], then the relation (2.5) holds
uniformly on [0, π].



EJDE-2023/81 INVERSE NODAL PROBLEMS FOR DIRAC OPERATORS 7

Now, let the function f(t) be integrable on [0, π] and take F (t) =
∫ t
0
f(s)ds.

Then, by using the same process in [14], we can write

P fN (t) =
d

dt
BFN+1(t)

=

N∑
k=0

N + 1

π

(
N
k

)( t
π

)k(
1− t

π

)N−k ∫ (k+1)π
N+1

kπ
N+1

f(s)ds

=

N∑
k=0

ckBk,N (
t

π
),

(2.6)

where ck = N+1
π

∫ (k+1)π
N+1
kπ
N+1

f(s)ds. Therefore, if the function f(t) is integrable on

[0, π], we have

P fN (t) =

N∑
k=0

ckBk,N
( t
π

)
. (2.7)

Theorem 2.3 ([14]). For t ∈ [0, π], if f(t) is the derivative of its indefinite integral,
then

lim
N→∞

P fN (t) = f(t)

holds almost everywhere.

Theorem 2.4 ([14]). Suppose the kernel KN (t, s) is measurable in the square a ≤
t ≤ b, a ≤ s ≤ b and∫ b

a

|KN (t, s)|ds ≤M,

∫ b

a

|KN (t, s)|dt ≤M,

with a constant M for all N = 1, 2, . . . and almost all t or s, respectively. Then for
f ∈ Lp, p > 1, the singular integral

Fn(t) =

∫ b

a

KN (t, s)f(s)ds

exists for almost all t ∈ [a, b] and belongs to the class Lp. In addition, for an
everywhere dense H in Lp, if Fn → f ∈ H strongly, it is also true for any f ∈ Lp:

‖f − Fn‖ :=
(∫ b

a

|f(t)− Fn(t)|pdt
)1/p

→ 0.

We have the following convergence theorem for the Bernstein method.

Theorem 2.5. Suppose that f ∈ Lp[0, π] and

P fN (t) =

N∑
k=0

ckBk,N
( t
π

)
,

where ck = N+1
π

∫ (k+1)π
N+1
kπ
N+1

f(s)ds. Then it holds almost everywhere that

lim
N→∞

P fN (t) = f(t).



8 F. SONG, Y. WANG, SH. AKBARPOOR EJDE-2023/81

Proof. According to [14], the formula (2.6) can be regarded as

P fN (t) =

N∑
k=0

∫ (k+1)π
N+1

kπ
N+1

KN (t, s)f(s)ds =

∫ π

0

KN (t, s)f(s)ds, (2.8)

where for 0 ≤ t ≤ π,

KN (t, s) =
N + 1

π

(
N
k

)
(
t

π
)k(1− t

π
)N−k,

kπ

N + 1
< s ≤ (k + 1)π

N + 1
,

for k = 0, N . Then, we have∫ π

0

|KN (t, s)|ds =

N∑
k=0

∫ (k+1)π
N+1

kπ
N+1

N + 1

π

(
N
k

)
(
t

π
)k(1− t

π
)N−kds

=

N∑
k=0

(
N
k

)
(
t

π
)k(1− t

π
)N−k

=

N∑
k=0

Bk,N (
t

π
) = 1,

and ∫ π

0

|KN (t, s)|dt =

∫ π

0

N + 1

π

(
N
k

)
(
t

π
)k(1− t

π
)N−kdt

=
N + 1

π

(
N
k

)
(N − k)!

N(N − 1) . . . (k + 2)(k + 1)

∫ π

0

(
t

π
)Ndt

=
[
(
t

π
)N+1

]π
0
dt = 1.

Take M = 1. Assume that f ∈ H and H is the set of continuous functions in
Lp, p > 1. According to Theorem 2.1 and Theorem 2.3, it can be written that

P fN → f . Since H is an everywhere dense set in Lp (see [14]), P fN → f in Lp

by using Theorem 2.4. Thus, for any f ∈ Lp, the polynomial P fN (t) is strongly
convergent to f . �

Therefore, the approximate solution of potential function v(t) ∈ L2([0, π]) and
the solution of integral equation (2.1) can be obtained by the Bernstein method.
Indeed, according to (2.7), we have

v(t) ∼=
N∑
k=0

ckBk,N (
t

π
) = C1

Tφ(t), (2.9)

where

C1 = [c0, c1, . . . , cN ]T , φ(t) = [B0,N (
t

π
), B1,N (

t

π
), . . . , BN,N (

t

π
)]T .

For n� 1, taking (2.9) into (2.1), we arrive at

N∑
k=0

ck

(∫ xjn

0

Bk,N (
t

π
)dt
)

∼= (n+
ω1

π
+
ω2

n
)xjn − α− (j − 1

2
)π +

(−1)j(m sin 2α+m2xjn)

2n
,

(2.10)



EJDE-2023/81 INVERSE NODAL PROBLEMS FOR DIRAC OPERATORS 9

for j = 1, n and n = N+1. Consequently, the potential function v(t) can be created
by using the following numerical algorithm.

Numerical algorithm 2: (1) For n � 1, choose the values N , α, β, m and set
n = N+1. We obtain the numerical values of nodal points {xjn}nj=1 through formula
(1.6) together with η(π).

(2) Use formula (2.10) to compute the vector Y2 by the linear system

A0C1 = B, (2.11)

where

A0 =


∫ x1

n

0
B0,N ( tπ )dt

∫ x1
n

0
B1,N ( tπ )dt · · ·

∫ x1
n

0
BN,N ( tπ )dt∫ x2

n

0
B0,N ( tπ )dt

∫ x2
n

0
B1,N ( tπ )dt · · ·

∫ x2
n

0
BN,N ( tπ )dt

· · ·∫ xnn
0

B0,N ( tπ )dt
∫ xnn
0

B1,N ( tπ )dt · · ·
∫ xnn
0

BN,N ( tπ )dt

 ,

and the matrix B and C1 are as defined above.
(3) Calculate the values v(ti) by the formula

[v(ti)] = CT
1 Φ,

where

ti =
iπ

N
, i = 0, N, Φ = [φ(t0),φ(t1), . . . ,φ(tN )].

3. Uniqueness

In this section, we present a solution for IP1 and give the uniqueness theorem
of the inverse nodal problem for the Dirac operator. For sufficiently large k, let
nr = 2k−1Mr, where Mr is a strictly increasing sequence and M1 is sufficiently
large.

We denote by δγ the error of γ, then γ + δγ is the approximate solution of γ.
Equation (2.11) can be rewritten as

(A0 + δA0)(C1 + δC1) = B + δB. (3.1)

Now, we study the absolute error between the approximate solution C1 + δC1 and
exact solution C1 of (3.1). Similar to proof of [13, Theorems 23 ] on error estimates
(pages 206-207), one can easily prove the following theorem.

Theorem 3.1. The absolute error between the approximate solution C1 +δC1 and
exact solution C1 of (3.1) satisfies

‖δC1‖ = O
( 1

n

)
(3.2)

for sufficiently large n .

Theorem 3.2 (Uniqueness). Given the nodal point subset
⋃∞
r=1Xnr of L, where

xjnr satisfies (1.6) for each fixed nr with its mean value η(π), then
(
v(x),m, α, β

)
can be uniquely determined by

⋃∞
r=1Xnr .

Proof. (1) Find α and β. Choose x1nr and xnrnr , then

lim
nr→∞

x1nr = 0 and lim
nr→∞

xnrnr = π. (3.3)



10 F. SONG, Y. WANG, SH. AKBARPOOR EJDE-2023/81

It follows from (1.6) and (3.3) that

α = lim
nr→∞

(
nrx

1
nr −

π

2

)
and β = lim

nr→∞

[
nrx

nr
nr − (nr −

1

2
)π
]
. (3.4)

(2) Reconstruct m. If α 6= 0, then

m =
1

sin 2α
lim

nr→∞

(
n2rx

1
nr − nr(

π

2
+ α) +

ω1

π
(
π

2
+ α)

)
. (3.5)

If α = 0, then

m2 = − 1

π
lim

nr→∞

(
n2rx

nr
nr − nr(

π

2
+ α+ η(π)) +

ω1

π
(
π

2
+ α+ η(π))

)
. (3.6)

We reconstruct m from (3.5) or (3.6).
(3) Find v(x). For each nr, from (2.9), we have the approximation of solution

vnr (x) of vnr (x),

vnr,0(x) =

nr∑
k=0

ck,0Bk,nr
(x
π

)
. (3.7)

It follows from (3.2) and (3.7) that

‖vnr,0 − vnr‖ = O
( 1

n

)
. (3.8)

Using Theorem 2.5 and (3.8), we find

v(x) := lim
nr→∞

vnr (x) = lim
nr→∞

vnr,0(x). (3.9)

The proof is complete. �

From Theorem 3.1, we obtain the following theorem by parts of nodal points
instead of dense nodal set.

Theorem 3.3. If Xnr = X̃nr for all nr, r ∈ N and η(π) = η̃(π), then

v(x) = ṽ(x) on [0, π], α = α̃, β = β̃ m = m̃.

Proof. From Theorem 3.1, we reconstruct α, β and m by (3.3), (3.4) and (3.5),

or (3.6), respectively. It follows from (2.10), Xnr = X̃nr for each r ∈ N and
η(π) = η̃(π) that

Al = Ãl +O
( 1

n

)
and Bl = B̃l +O

( 1

n

)
. (3.10)

Using (3.2) and (3.10), we have

vnr,0 − ṽnr,0 = o(1) (3.11)

for r > N . Therefore, we obtain two sequences of functions {vnr,0(x)}∞r=1 and
{ṽnr,0(x)}∞r=1.

From Theorem 3.1 and (3.11), we obtain

lim
r→∞

vnr,0(x) = lim
r→∞

ṽnr,0(x). (3.12)

It follows from (3.11) and (3.12) that

ṽ(x) = v(x) on [0, π].

the proof is complete. �



EJDE-2023/81 INVERSE NODAL PROBLEMS FOR DIRAC OPERATORS 11

4. Numerical examples

In this section, the SCW and Bernstein methods are used to compute the ap-
proximate solution of the inverse problem and the accuracy of presented methods
is shown by providing two numerical examples.

Example 4.1. For the function v(x) = cos(3x) + sin(x), we take M = k = 3 in
the SCW method and N = 11 in the Bernstein method. Set α = π/6, β = π/3,
m = 0.4, then we obtain

η(π) =

∫ π

0

v(t)dt =

∫ π

0

(cos(3t) + sin(t))dt = 2.

The numerical values of nodal points xjn, j = 1, n = 1, 12 from formula (1.6) are
shown in Table 1.

Table 1. Numerical values of xjn with α = π/6, β = π/3 and
m = 0.4 obtained by (1.6) in Example 4.1.

j 1 2 3 4 5 6
xjn 0.16853351 0.42000759 0.66835934 0.90097215 1.13794007 1.37503758
j 7 8 9 10 11 12
xjn 1.63434058 1.89791120 2.17430407 2.43222644 2.68290530 2.90441560

The above nodal points xjn and the values of α, β and m are used to calculate
the approximation of the function v. We draw the exact solution and the numerical
approximations with n = 12, 24, 48, 96 by using the SCW and Bernstein methods
for α = π/6, β = π/3 and m = 0.4, which are shown in Figure 1 and Figure 2.

Example 4.2. Suppose that the function v(x) = (t+1)/
√
t2 + 1. Take M = k = 3

in the SCW method and N = 11 in the Bernstein method. Set α = π/10, β = π/4,
m = 0.01, then we have

η(π) =

∫ π

0

v(t)dt =

∫ π

0

t+ 1√
t2 + 1

dt = arcsinh(π) +
√
π2 + 1− 1 ∼= 4.15920405.

Using formula (1.6), we calculate the numerical values of nodal points xjn, j =
1, n = 1, 12 are shown in Table 2.

Table 2. Numerical values of xjn with α = π/10, β = π/4 and
m = 0.01 obtained by (1.6) in Example 4.2.

j 1 2 3 4 5 6
xjn 0.15380336 0.41665952 0.68218220 0.94872939 1.21557194 1.48210054
j 7 8 9 10 11 12
xjn 1.74829765 2.01394010 2.27919981 2.54393599 2.80835527 3.0723250

The above nodal points xjn and the values of α, β and m are used to compute
the approximation of function v. We draw the exact solution and the numerical
approximations with n = 12, 24, 48, 96 by using the SCW and Bernstein methods
for α = π/10, β = π/4 and m = 0.01, which are shown in Figure 3 and Figure 4.



12 F. SONG, Y. WANG, SH. AKBARPOOR EJDE-2023/81

0 0.5 1 1.5 2 2.5 3

x

-1

-0.5

0

0.5

1

1.5

2

(x
)

Exact solution

Approximate solution

0 0.5 1 1.5 2 2.5 3

x

-1

-0.5

0

0.5

1

1.5

2

(x
)

Exact solution

Approximate solution

n = 12 or M = k = 3 n = 24 or M = 3, k = 4

0 0.5 1 1.5 2 2.5 3

x

-1

-0.5

0

0.5

1

1.5

2

(x
)

Exact solution

Approximate solution

0 0.5 1 1.5 2 2.5 3

x

-1

-0.5

0

0.5

1

1.5

2

(x
)

Exact solution

Approximate solution

n = 48 or M = 3, k = 5 n = 96 or M = 3, k = 6

Figure 1. Exact and approximate solutions with α = π/6, β =
π/3 and m = 0.4 obtained by SCW method in Example 4.1.

0 0.5 1 1.5 2 2.5 3

x

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

(x
)

Exact solution

Approximate solution

0 0.5 1 1.5 2 2.5 3

x

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

(x
)

Exact solution

Approximate solution

n = 12 or M = k = 3 n = 24 or M = 3, k = 4

0 0.5 1 1.5 2 2.5 3

x

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

(x
)

Exact solution

Approximate solution

0 0.5 1 1.5 2 2.5 3

x

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

(x
)

Exact solution

Approximate solution

n = 48 or M = 3, k = 5 n = 96 or M = 3, k = 6

Figure 2. Exact and approximate solutions with α = π/6, β =
π/3 and m = 0.4 obtained by Bernstein method in Example 4.1.

For the given examples, it can be seen that the SCW method gives a better
approximation than Bernstein method for the large values of n. Meanwhile, we can



EJDE-2023/81 INVERSE NODAL PROBLEMS FOR DIRAC OPERATORS 13

see the approximation result obtained by Bernstein method is also well except at
the beginning and end of the interval.

0 0.5 1 1.5 2 2.5 3

x

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

(x
)

Exact solution

Approximate solution

0 0.5 1 1.5 2 2.5 3

x

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

(x
)

Exact solution

Approximate solution

n = 12 or M = k = 3 n = 24 or M = 3, k = 4

0 0.5 1 1.5 2 2.5 3

x

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

(x
)

Exact solution

Approximate solution

0 0.5 1 1.5 2 2.5 3

x

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

(x
)

Exact solution

Approximate solution

n = 48 or M = 3, k = 5 n = 96 or M = 3, k = 6

Figure 3. Exact and approximate solutions with α = π/10, β =
π/4 and m = 0.01 obtained by SCW method in Example 4.2.

0 0.5 1 1.5 2 2.5 3

x

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

(x
)

Exact solution

Approximate solution

0 0.5 1 1.5 2 2.5 3

x

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

(x
)

Exact solution

Approximate solution

n = 12 or M = k = 3 n = 24 or M = 3, k = 4

0 0.5 1 1.5 2 2.5 3

x

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

(x
)

Exact solution

Approximate solution

0 0.5 1 1.5 2 2.5 3

x

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

(x
)

Exact solution

Approximate solution

n = 48 or M = 3, k = 5 n = 96 or M = 3, k = 6

Figure 4. Exact and approximate solutions with α = π/10, β =
π/4 and m = 0.01 obtained by Bernstein method in Example 4.2.



14 F. SONG, Y. WANG, SH. AKBARPOOR EJDE-2023/81

5. Conclusion

In this article, we consider the inverse nodal problem for Dirac operators. The
asymptotic form of the eigenfunctions and nodal points are presented. The unique-
ness theorem for solution of inverse problem by a dense subset of nodal points is
proved. We offer the second kind Chebyshev wavelets and Bernstein collocation
methods to obtain the approximate solution and finally give two numerical exam-
ples to demonstrate their efficiency.

Acknowledgments. This research was partially supported by the National Nat-
ural Science Foundation of China Grant 12201303, and by the China Postdoctoral
Science Foundation 2021M691528.

References

[1] S. Albeverio, R. Hryniv, Ya. Mykytyuk; Reconstruction of radial Dirac and Schrödinger

operators from two spectra, J. Math. Anal. Appl., 339 (2008), 45-57.

[2] E. Babolian, L. M. Delves; An augmented Galerkin method for first kind Fredholm equations,
IMA J. Appl. Math., 24 (1979), 157-174.

[3] P. J. Browne, B. D. Sleeman; Inverse nodal problem for Sturm-Liouville equation with eigen-

parameter dependent boundary conditions, Inverse Probl., 12 (1996), 377-381.
[4] S. A. Buterin, C.-T. Shieh; Incomplete inverse spectral and nodal problems for differential

pencils, Results Math., 62 (2012), 167-179.

[5] X. F. Chen, Y. H. Cheng, C. K. Law; Reconstructing potentials from zeros of one eigenfunc-
tion, Trans. Amer. Math. Soc., 363 (2011), 4831-4851.

[6] Y. H. Cheng, C. K. Law, J. Tsay; Remarks on a new inverse nodal problem, J. Math. Anal.

Appl., 248 (2000), 145-155.
[7] S. Currie, B. A. Watson; Inverse nodal problems for Sturm-Liouville equations on graphs,

Inverse Probl., 23 (2007), 2029-2040.
[8] Y. X. Guo, G. S. Wei; Inverse problems: Dense nodal subset on an interior subinterval, J.

Differential Equations, 255 (2013), 2002-2017.

[9] Y. X. Guo, G. S. Wei; Inverse Nodal Problem for Dirac Equations with Boundary Conditions
Polynomially Dependent on the Spectral Parameter, Results Math., 67 (2015), 95-110.

[10] O. H. Hald, J. R. McLaughlin; Solutions of inverse nodal problems, Inverse Probl., 5 (1989),

307-347.
[11] M. Horváth; On the inverse spectral theory of Schrödinger and Dirac operators, Trans. Amer.

Math. Soc., 353 (2001), 4155-4171.
[12] B. M. Levitan, I. S. Sargsjam; Sturm-Liouville and Dirac Operators, London: Kluwer Aca-

demic Publishers, 1991.

[13] Q. Y. Li, N. C. Wang, D. Y. Yi; Numerical analysis (4th edition, in Chinese), Tsinghua

University Press and Springer Press: Beijing, 2001.
[14] G. G. Lorentz; Bernstein polynomials, AMS Chelsia Publishing, 1986.

[15] J. R. McLaughlin; Inverse spectral theory using nodal points as data-a uniqueness result, J.
Differential Equations, 73 (1988), 354-362.

[16] S. Mosazadeh, H. Koyunbakan; On the stability of the solution of the inverse problem for

Dirac operator, Appl. Math. Lett., 102 (2020), 106118.
[17] A. Neamaty, Sh. Akbarpoor; Numerical solution of inverse nodal problem with an eigenvalue

in the boundary condition, Inverse Probl. Sci. Eng., 25 (2017), 978-994.

[18] M. Rabbani, K. Maleknejad, N. Aghazadeh, R. Mollapourasl; Computational projection
methods for solving Fredholm integral equation, Appl. Math. Comput., 191 (2007), 140-143.

[19] M. T. Rashed; Numerical solutions of the integral equations of the first kind, Appl. Math.

Comput., 145 (2003), 413-420.
[20] C. L. Shen; On the nodal sets of the eigenfunctions of the string equations, SIAM J. Math.

Anal., 19 (1988), 1419-1424.

[21] C.-T. Shieh, V. A. Yurko; Inverse nodal and inverse spectral problems for discontinuous
boundary value problems, J. Math. Anal. Appl., 347 (2008), 266-272.



EJDE-2023/81 INVERSE NODAL PROBLEMS FOR DIRAC OPERATORS 15

[22] Y. P. Wang, E. Yilmaz, Sh. Akbarpoor; The numerical solution of inverse nodal problem

for integro-differential operator by Legendre wavelet method, Int. J. Comput. Math., 100

(2023), 219-232.
[23] Y. P. Wang, V. A. Yurko; On the inverse nodal problems for discontinuous Sturm-Liouville

operators, J. Differential Equations, 260 (2016), 4086-4109.

[24] Y. P. Wang, V. A. Yurko; On the missing eigenvalue problem for Dirac operators, Appl.
Math. Lett., 80 (2018), 41-47.

[25] Y. P. Wang, V. A. Yurko, C.-T. Shieh; The uniqueness in inverse problems for Dirac operators

with the interior twin-dense nodal subset, J. Math. Anal. Appl., 479 (2019), 1383-1402.
[26] Y. X. Wang, L. Zhu; SCW method for solving the fractional integro-differential equations

with a weakly singular kernel, Appl. Math. Comput., 275 (2016), 72-80.

[27] Z. Y. Wei, Y. X. Guo, G. S. Wei; Incomplete inverse spectral and nodal problems for Dirac
operator, Adv. Differential Equations, 2015 (2015), 188.

[28] C. F. Yang, Z. Y. Huang; Reconstruction of the Dirac operator from nodal data, Int. Equ.
Oper. Theory, 66 (2010), 539-551.

[29] X. F. Yang; A new inverse nodal problem, J. Differential Equations, 169 (2001), 633-653.

[30] V. A. Yurko; Inverse spectral problems for differential systems on a finite interval, Results
Math., 48 (2005), 371-386.

[31] F. Y. Zhou, X. Y. Xu; Numerical solution of the convection diffusion equations by the second

kind Chebyshev wavelets, Appl. Math. Comput., 247 (2014), 353-367.
[32] L. Zhu, Q. B. Fan; Numerical solution of nonlinear fractional-order Volterra integro-

differential equations by SCW, Commun. Nonlinear Sci. Numer. Simulat., 18 (2013), 1203-

1213.

Fei Song
School of Information Management, Nanjing University, Jiangsu, 210023, China.

College of Science, Nanjing Forestry University, Jiangsu, 210037, China
Email address: songfei@njfu.edu.cn

Yuping Wang (corresponding author)

College of Science, Nanjing Forestry University, Jiangsu, 210037, China
Email address: ypwang@njfu.com.cn

Shahrbanoo Akbarpoor
Department of Mathematics, Jouybar Branch, Islamic Azad University, Jouybar, Iran

Email address: akbarpoor.kiasary@yahoo.com


	1. Introduction
	2. Approximation of solution and its convergence
	Second kind Chebyshev wavelet method (SCW method)
	Numerical algorithm 1:
	Bernstein method
	Numerical algorithm 2:

	3. Uniqueness
	4. Numerical examples
	5. Conclusion
	Acknowledgments

	References