Computing Eigenpairs of Two-Parameter Sturm-Liouville

Systems Using the Bivariate Sinc-Gauss Formula

R. M. Asharabi? and J. Prestin†

?Department of Mathematics, College of Arts and Sciences,Najran University, Najran, Saudi Arabia

E-mail: rashad1974@hotmail.com

†Institute of Mathematics, University of Lubeck, 23562 Lubeck, GermanyE-mail: prestin@math.uni-luebeck.de

Abstract

The use of sampling methods in computing eigenpairs of two-parameter boundary valueproblems is extremely rare. As far as we know, there are only two studies up to now usingthe bivariate version of the classical and regularized sampling series. These series have a slowconvergence rate. In this paper, we use the bivariate sinc-Gauss sampling formula that wasproposed in [6] to construct a new sampling method to compute eigenpairs of a two-parameterSturm-Liouville system. The convergence rate of this method will be of exponential order, i.e.

O(e−δN/

√N)

where δ is a positive number and N is the number of terms in the bivariate

sinc-Gaussian formula. We estimate the amplitude error associated to this formula, which givesus the possibility to establish the rigorous error analysis of this method. Numerical illustrativeexamples are presented to demonstrate our method in comparison with the results of the bi-variate classical sampling method.

AMS(2010): 34B05, 34B09, 65F18

Keywords: Sinc approximation, multiparameter spectral theory, eigencurve.

1 Introduction

The issue of computing eigenvalues of one-parameter eigenvalue problems using sinc methods hasattracted many researchers. During the period 1996-2018, six sampling methods have been developedto compute the eigenvalues of boundary value problems of various types. These are the classicalsinc (1996) [13], the regularized sinc (2005) [14], the sinc-Gaussian (2008) [2], the Hermite (2012)[3], the Hermite-Gauss (2016) [4], and the generalized sinc-Gaussian method (2018) [7]. The use ofsampling methods in computing eigenpairs of two-parameter boundary value problems is extremelyrare. As far as we know, there are only two studies, cf. [1, 15]. In [1], the authors used the bivariateclassical sampling series of Whittaker, Kotelnikov and Shannon (WKS) to find a representationfor the eigencurves of a two-parameter Sturm-Liouville eigenvalue system. The convergence rate of

this method is of order O(

ln(N)/√N)

. The authors of [15] used the regularized sampling method

to compute the eigenpairs of a two-parameter Sturm-Liouville eigenvalue problem with three-pointboundary conditions, without studying the error analysis. The convergence rate of the regularizedsampling series is not much better than that of the classical sampling method and it still is ofpolynomial order.

1

This paper is concerned with constructing a new sampling method to compute eigenpairs of a two-parameter Sturm-Liouville eigenvalue system with separate boundary conditions using a bivariatesinc-Gauss sampling formula. This formula is established in [6] and it is used for the first time tocompute eigenpairs of two-parameter eigenvalue problems. The convergence rate of this formulais of exponential order, so it will give us higher accuracy results. Indeed, we consider the regulartwo-parameter Sturm-Liouville eigenvalue system

y′′r (xr) +[λ2pr(xr) + µ2qr(xr) + wr(xr)

]yr(xr) = 0, xr ∈ [0, b], r = 1, 2, (1.1)

with the separated boundary conditions

cos(αr)yr(0)− sin(αr)y′r(0) = 0, αr ∈ [0, π), (1.2)

cos(βr)yr(b)− sin(βr)y′r(b) = 0, βr ∈ [0, π). (1.3)

The parameters λ and µ can be complex but will be assumed to be real in most of what follows. Weassume that

wr ∈ L1[0, b] and pr, qr ∈ C ′[0, b], with p′r, q′r ∈ AC[0, b], (1.4)

where AC[0, b] is the class of absolutely continuous functions. The solution of each equation in(1.1)-(1.3) exists and is an entire furcation in λ and µ for each fixed xr ∈ [0, b]. For more details onthe system (1.1)-(1.3) see e.g. [8, 11].

A pair (λ2n, µ

2n) is called an eigenpair of (1.1)-(1.3) if this system has a pair of nontrivial solutions,

yr(·) 6= 0 for r = 1, 2. Under the definiteness condition, cf. e.g. [12, Theorem 1.1] and [8, Theorem10.7.1],

δ(x1, x2) :=

∣∣∣∣ p1(x1) q1(x2)p2(x1) q2(x2)

∣∣∣∣ > 0, for all (x1, x2) ∈ [0, b]2, (1.5)

the system (1.1)-(1.3) has an infinite sequence of eigenpairs lying in R2. For a fixed real value ofµ, each Sturm-Liouville problem in (1.1)-(1.3) has a sequence of eigenvalues {λr,n(µ)}n≥1, r = 1, 2.From the one-parameter Sturm-Liouville theory it follows that

λr,1(µ) < λr,2(µ) < λr,3(µ) < . . . , r = 1, 2,

and these sequences converge to ±∞ as µ → ∓∞ if the functions pr and qr have the same sign in[0, b] and to ±∞ as µ → ±∞ if pr, qr have different signs, [8, pp. 110-111]. For each n, λr,n(µ)describes a curve in the λµ-plane and each λr,n(µ) will be monotonically decreasing or increasingdepending on the sign of pr and qr. The curves {λr,n(µ)}r=1,2 are called the n-th eigencurves of

system (1.1)-(1.3) and they are analytic in −∞ < µ < ∞, cf. e.g [8, 11]. Evidently, the eigenpair(λ2n, µ

2n) of system (1.1)-(1.3) is the intersection of the n-th eigencurves. The asymptotic analysis of

eigencurves for problem (1.1)-(1.3) is investigated in [9, 10]. A review and bibliography of eigencurvesfor two-parameter Sturm-Liouville equations can be found in [11].

Let yr(·, λ, µ) denote the solution of (1.1) satisfying the initial conditions

yr(0, λ, µ) = sin(αr), y′r(0, λ, µ) = cos(αr).

By a theorem on analytic parameter dependence, the function

∆r(λ, µ) = cos(βr)yr(b, λ, µ)− sin(βr)y′r(b, λ, µ), r = 1, 2, (1.6)

is an entire function in λ and µ. The couples (λ2n, µ

2n) are the eigenpairs of system (1.1)-(1.3) if and

only if (λn, µn) are the common zeros of the equations (1.6), [8, p. 106]. Those zeros cannot becomputed exactly except for extremely rare cases. If {yr(·, λn, µn)}r=1,2 is a corresponding set of

simultaneous solutions of (1.1)-(1.3), then the product∏2j=2 yr(·, λn, µn) is an eigenfunction of this

system corresponding to the eigenpair (λ2n, µ

2n), and it is unique up to a multiplicative factor. Under

2

the condition (1.5), the set of eigenfunctions is complete in L2 over the region [0, b]2 with respect tothe weight function δ(x1, x2), cf. [8, Theorem 10.6.1].

The rest of the paper has been organized as follows: The next section is devoted to briefly describethe bivariate sinc-Gauss sampling formula. Since alternative samples will be used in our samplingformula, the amplitude error appears in our method. For this reason, we will derive estimates forthe amplitude error associated with the bivariate sinc-Gauss sampling formula, which gives us thepossibility to establish the rigorous error analysis of this method. The method is constructed inSection 3 and 4, where we also establish a rigorous error analysis associated with this method.Section 5 deals with illustrative examples and comparisons. Lastly, Section 6 concludes the paper.

2 Bivariate sinc-Gauss sampling formula

This section is devoted to describe briefly the bivariate sinc-Gauss sampling formula and investigatethe amplitude error associated with it. Let γ = (γ1, γ2), γr > 0, r = 1, 2. The Bernstein space,Bγ,∞(R2) is the class of entire functions of two variables satisfying the following growth condition

|f(z)| ≤ ‖f‖∞ exp

(2∑r=1

γr|=zr|

), z := (z1, z2) ∈ C2,

which belong to L∞(R2) when restricted to R2. For hr ∈ (0, π/γr], set δr := (π − γrhr)/2 and letE2 be the class of all entire functions on C2. For the Bernstein space B2

γ,∞, in [6] we defined alocalization operator Gh,N : Bσ,∞(R2)→ E2 ∩ Lp(R2) via

Gh,N [f ](z) :=∑

k∈Z2N (z)

f(k1h1, k2h2)

2∏r=1

sinc(πh−1

r zr − krπ)

exp

(−δr (zr − krhr)2

Nh2r

), (2.1)

where N is a positive integer, h := (h1, h2), k := (k1, k2), z ∈ C2 and

Z2N (z) :=

{k ∈ Z2 : |bh−1

r <zr + 1/2c − kr| ≤ N, r = 1, 2}. (2.2)

The sinc function is defined by

sinc (t) :=

sin t

t, t 6= 0,

1, t = 0.

This operator is generalized in [5] including samples from the function and its partial derivatives. Letus mention here that formula (2.1) is defined for wider classes than the Bernstein space Bγ,∞(R2),cf. [6], but here this space is sufficient to treat our problem. There, we estimated the absolute error|f(z)− Gh,N [f ](z)| where f ∈ Bγ,∞(R2) and bounds of exponential order were found. Since theeigenpairs of the system (1.1)-(1.3) are a subset of R2, we state a bound for |f(z)− Gh,N [f ](z)| onlyon real domain. If f ∈ Bγ,∞(R2), then for all x ∈ R2, we have [6, Corollary 3.4]

|f(x)− Gh,N [f ](x)| ≤ 2‖f‖∞Aδ,N (x)e−δN√πδN

, (2.3)

where δ := min{δ1, δ2} and Aδ,N is the real-valued function defined by

Aδ,N (x) :=

(1 +

2√πδN

+1

e2πN − 1

) 2∑r=1

∣∣sin(πh−1r xr)

∣∣+ 2

2∏r=1

∣∣sin(πh−1r xr)

∣∣ . (2.4)

As the function Aδ,N (x) is bounded for all x ∈ R2, it does not affect the convergence rate of theerror bound in (2.3). Formula (2.1) has a convergence rate of exponential order. Thus, it will

3

give us higher accuracy results when we use it in computing eigenpairs of a two-parameter Sturm-Liouville system. It is approximating the function from the Bernstein space Bγ,∞(R2) using only afinite number of samples of the function. However, sometimes these samples cannot be computedexplicitly. This is why the amplitude error associated with formula (2.1) appears. This amplitudeerror arises when the exact values f(k1h1, k2h2) of (2.1) are replaced by close approximate ones. Weassume that the approximated samples f(k1h1, k2h2) are close to the original samples f(k1h1, k2h2),i.e. there is ε > 0 sufficiently small such that

supk∈Z2

N (z)

∣∣∣f(k1h1, k2h2))− f(k1h1, k2h2)∣∣∣ < ε. (2.5)

The amplitude error associated with formula (2.1) is defined by Gh,N [f ](x)−Gh,N [f ](x), x ∈ R2. In

the following theorem, we will show a bound for |Gh,N [f ](x)−Gh,N [f ](x)| on the real domain, whichwill be used in the investigation of the error analysis of our method in Section 4.

Theorem 2.1. Let γr > 0, hr ∈ (0, π/γr] and δr = (π − γrhr)/2. Assume that (2.5) holds. Thenwe have for x ∈ R2

∣∣∣Gh,N [f ](x)− Gh,N [f ](x)∣∣∣ < 4 ε

2∏r=1

e−δr/4N(

1 +√N/δr

). (2.6)

Proof. Using condition (2.5) and the definition of Z2N (z) in (2.2), we obtain∣∣∣Gh,N [f ](x)− Gh,N [f ](x)

∣∣∣ < ε

2∏r=1

∑kr∈ZN (xr)

exp

(−δr (xr − krhr)2

Nh2r

), (2.7)

where we have used the fact |sinc(πh−1

r xr − krπ)| ≤ 1 for all xr ∈ R, and the index ZN (xr) is

defined as followsZN (xr) :=

{kr ∈ Z : |bh−1

r xr + 1/2c − kr| ≤ N}.

Let bh−1r xr + 1/2c − kr = lr. Then∑

kr∈ZN (xr)

exp

(−δr (xr − krhr)2

Nh2r

)≤

∑|lr|≤N

exp

(−δrN

(lr − 1/2)2

)

≤ 2e−δr/4N + 2

∫ N

0

e−δrN (tr+1/2)2dtr

≤ 2e−δr/4N + 2√N/2δr

∫ ∞√δr/2N

e−12 t

2rdtr. (2.8)

Combining Mills’ Ratio inequality, cf. [17],∫ ∞x

e−12 t

2

dt <4e−

12x

2

3x+√x2 + 8

, x > 0,

with (2.8) and (2.7) implies (2.6).

3 The method

This section is devoted to the construction of our method. The main concept of this approximationis to split the two simultaneous equations in (1.6) into two parts. The first is known and the second isunknown, but it belongs to the Bernstein space Bγ,∞(R2). Therefore, we approximate the unknownparts of both equations (1.6) using our bivariate sinc-Gauss sampling formula (2.1). We distinguishbetween the two cases αr = 0 and αr 6= 0 because the representation of solutions of system (1.1)-(1.3)is different in these cases, cf. [12, Theorem 6.1]. From now on, unless otherwise stated, r = 1, 2.

4

3.1 The case αr = 0

Let us start with the following result which we will use in the sequel.

Lemma 3.1. If a, b > 0 and λ, µ ∈ C, then we have

=(√

aλ2 + bµ2)≤√a(=λ)2 + b(=µ)2. (3.1)

Proof. Applying the identity (=√z)2 = 1

2 (|z| − <z) for z ∈ C twice, we obtain(=√aλ2 + bµ2

)2

=1

2

(|aλ2 + bµ2| − <(aλ2 + bµ2)

)≤ a

2

(|λ2| − <(λ2)

)+b

2

(|µ2| − <(µ2)

)= a

(=√λ2)2

+ b(=õ2)2

= a(=λ)2 + b(=µ)2,

from which the assertion follows immediately.

Denote

ρr(xr) =

∫ xr

0

σ2r(τr)dτr, where σr(τr) = (λ2pr(τr) + µ2qr(τr))

1/4, (3.2)

and let Sn be a family of non-empty subsets

Sn =

{(λ2, µ2) ∈ C2 : inf

xr∈[a,b]|σr(xr)| ≥ n, n ∈ N

}.

Under the assumptions (1.4), pr, qr > 0 on [0, b], and for (λ2, µ2) ∈ Sn as n → ∞ and αr 6= 0, thesolutions of the system (1.1)-(1.3) are given by, cf. [12, Theorem 6.1],

yr(xr, λ, µ) =1

σr(0)σr(xr)[sin(ρr(xr))

+

∫ xr

0

w∗r(τr)− wr(τr)σ2r(τr)

sin[ρr(xr)− ρr(τr)] sin(ρr(τr))dτr

]+O

(e|=ρr(xr)|

n2

),(3.3)

where σr, ρr, w∗r are functions of λ, µ and the function w∗r is defined as

w∗r(xr) =σ′′r (xr)

σr(xr)− 2

(σ′r(xr)

σr(xr)

)2

satisfing sup(λ,µ)∈Sn

‖w∗r‖L1 <∞. (3.4)

The derivative of the solutions yr(·, λ, µ) is also given by

y′r(xr, λ, µ) =σr(xr)

σr(0)

[cos(ρr(xr))−

σ′r(xr)

σ3r(xr)

sin(ρr(xr))

+

∫ xr

0

w∗r(τr)− wr(τr)σ2r(τr)

cos[ρr(xr)− ρr(τr)] sin(ρr(τr))dτr

]+O

(e|=ρr(xr)|

n2

).(3.5)

By substituting from (3.3) and (3.5) into (1.6), we obtain

∆r(λ, µ) =cos(βr) sin(ρr(b))

σr(0)σr(b)− sin(βr)

σr(b)

σr(0)

[cos(ρr(b))−

σ′r(b)

σ3r(b)

sin(ρr(b))

]+ cos(βr)

∫ b

0

w∗r(τr)− wr(τr)σ2r(τr)

sin[ρr(b)− ρr(τr)] sin(ρr(τr))dτr

− sin(βr)

∫ b

0

w∗r(τr)− wr(τr)σ2r(τr)

cos[ρr(b)− ρr(τr)] sin(ρr(τr))dτr +O

(e|=ρr(b)|

n2

).(3.6)

5

Observe that the simultaneous functions ∆r(λ, µ) can be split into two parts as follows

∆r(λ, µ) = Kr(λ, µ) + Ur(λ, µ), (3.7)

where Kr(λ, µ) is the known part

Kr(λ, µ) :=cos(βr) sin(ρr(b))

σr(0)σr(b)− sin(βr)

σr(b)

σr(0)

[cos(ρr(b))−

σ′r(b)

σ3r(b)

sin(ρr(b))

], (3.8)

and the second part, which will be denoted by Ur(λ, µ), consists of the remaining terms in (3.6), i.e.

Ur(λ, µ) := cos(βr)

∫ b

0

w∗r(τr)− wr(τr)σ2r(τr)

sin[ρr(b)− ρr(τr)] sin(ρr(τr))dτr

− sin(βr)

∫ b

0

w∗r(τr)− wr(τr)σ2r(τr)

cos[ρr(b)− ρr(τr)] sin(ρr(τr))dτr +O

(e|=ρr(b)|

n2

).(3.9)

In the following theorem, we will show that the unknown part Ur belongs to the Bernstein spaceBγr,∞(R2). That gives us the possibility to approximate Ur via the bivariate sinc-Gauss samplingformula (2.1).

Theorem 3.2. Assume that (1.4) holds. Let γr = (γr1, γr2) be such that

γr1 =

∫ b

0

√pr(t) dt, γr2 =

∫ b

0

√qr(t) dt. (3.10)

Then Ur ∈ Bγr,∞(R2).

Proof. Applying inequality (3.1) implies

=(√

λ2pr(τr) + µ2qr(τr))≤√pr(τr)(=λ)2 + qr(τr)(=µ)2 ≤

√pr(τr)|=λ|+

√qr(τr)|=µ|.

It is easy to see that|sin[ρr(b)− ρr(τr)] sin(ρr(τr))| = O

(e|=ρr(b)|) ,

|cos[ρr(b)− ρr(τr)] sin(ρr(τr))| = O(e|=ρr(b)|) , (3.11)

for all τr ∈ [a, b] where ρr is defined in (3.2). Combining (3.11) and (3.9), using the fact wr ∈ L1[0, b],σr(τr) ≥ n for all τr ∈ [0, b] and sup(λ,µ)∈Sn ‖w

∗r‖L1 < ∞, cf. (3.4), we get a positive constant A,

independent on λ, µ, such that

|Ur(λ, µ)| ≤ Aeγr1|=λ|+γr2|=µ|, (λ, µ) ∈ Sn, (3.12)

where γr1 and γr2 are defined above. It follows from (3.12) that Ur belongs to the Bernstein spaceBγr,∞(R2).

Since Ur ∈ Bγr,∞(R2), we can approximate it using the bivariate sinc-Gauss sampling formula(2.1), i.e. Ur(λ, µ) ≈ Gh,N [Ur](λ, µ) where the samples are given by

Ur(k1h1, k2h2) = cos(βr)yr(b, k1h1, k2h2)− sin(βr)y′r(b, k1h1, k2h2)−Kr(k1h1, k2h2), (3.13)

where k ∈ Z2N (λ, µ) and βr ∈ [0, π). Unfortunately, the samples Ur(k1h1, k2h2) cannot be determined

explicitly in the general case. That is why the amplitude error usually appears. Let Ur(k1h1, k2h2)be the approximation of the samples Ur(k1h1, k2h2) when the solution yr(b, k1h1, k2h2) and itsderivative y′(b, k1h1, k2h2) are computed numerically at the nodes (k1h1, k2h2), k ∈ Z2

N (λ, µ). Now,let us define the following interesting function

∆r,h,N (λ, µ) := Kr(λ, µ) + Gh,N [Ur](λ, µ), (3.14)

6

where Kr is defined in (3.8) and Gh,N [Ur] is the bivariate sinc-Gauss sampling formula (2.1) which is

constructed by the approximated samples Ur(k1h1, k2h2). The simultaneous functions ∆r,h,N (λ, µ)are determined explicitly and will be very close to the functions ∆r(λ, µ) which are defined in (1.6),as we will see in the next result. Therefore, the zeros of the simultaneous equations ∆r,h,N (λ, µ)will be very close to the desired zeros, which are precisely the eigenpairs of the system (1.1)-(1.3),of ∆r(λ, µ).

3.2 The case αr 6= 0

In this case, we describe our method briefly. Under the assumptions (1.4), pr, qr > 0 on [0, b] andfor (λ2, µ2) ∈ Sn as n → ∞ and αr 6= 0, the solutions of system (1.1)-(1.3) are given by, cf. [12,Theorem 6.1],

yr(xr, λ, µ) =σr(0) sin(αr)

σr(xr)

[cos(ρr(xr)) +

(cot(αr)

σ2r(0)

+σ′r(0)

σ3r(0)

)sin(ρr(xr))

+

∫ xr

0

w∗r(τr)− wr(τr)σ2r(τr)

sin[ρr(xr)− ρr(τr)] cos(ρr(τr))dτr

]+O

(e|=ρr(xr)|

n2

),

(3.15)

where the function w∗r is defined in (3.4). In this case, the derivative of the solution yr(·, λ, µ) is alsogiven by

y′r(xr, λ, µ) = σr(xr)σr(0) sin(αr)

[− sin(ρr(xr)) +

(cot(αr)

σ2r(0)

+σ′r(0)

σ3r(0)

− σ′r(xr)

σ3r(xr)

)cos(ρr(xr))

+

∫ xr

0

w∗r(τr)− wr(τr)σ2r(τr)

cos[ρr(xr)− ρr(τr)] cos(ρr(τr))dτr

]+O

(e|=ρr(xr)|

n2

).(3.16)

Combining (3.15), (3.16) and (1.6) and splitting the function ∆r into two parts, as we have done inthe last case, the known part will be

Kr,αr (λ, µ) = cos(βr)σr(0) sin(αr)

σr(b)

[cos(ρr(b)) +

(cot(αr)

σ2r(0)

+σ′r(0)

σ3r(0)

)sin(ρr(b))

]− sin(βr)σr(b)σr(0) sin(αr)

[− sin(ρr(b) +

(cot(αr)

σ2r(0)

+σ′r(0)

σ3r(0)

− σ′r(b)

σ3r(b)

)cos(ρr(b))

],

(3.17)

and the unknown part will be

Ur,αr (λ, µ) = cos(βr)

∫ b

0

w∗r(τr)− wr(τr)σ2r(τr)

sin[ρr(b)− ρr(τr)] cos(ρr(τr))dτr

− sin(βr)

∫ b

0

w∗r(τr)− wr(τr)σ2r(τr)

cos[ρr(b)− ρr(τr)] cos(ρr(τr))dτr +O

(e|=ρr(b)|

n2

).

(3.18)

Applying the same technique as in the proof of Theorem 3.2, we can prove the following result.

Theorem 3.3. Assume that (1.4) holds. Then Ur,αr belongs to the Bernstein space Bγr,∞(R2)where γr = (γr1, γr2) and γrs is defined in (3.10).

Since Ur,αr ∈ Bγr,∞(R2), we can approximate it using the bivariate sinc-Gauss sampling formula(2.1). Here, we complete the method as we have done in the same way as in the last case.

Remark 3.4. If α1 = 0 and α2 6= 0, the solutions y1 and y2 will be as given in (3.3) and (3.15),respectively. The case α1 6= 0 and α2 = 0 is similar. In these cases, the method will be completedas indicated above.

7

4 Error analysis

In this section, we show that the function ∆r,h,N is very close to the function ∆r. Furthermore, wefind a bound for the standard Euclidean norm ‖(λ∗, µ∗) − (λε,N , µε,N )‖R2 , where

((λ∗)2, (µ∗)2

)is

an eigenpair of the system (1.1)-(1.3) and(λ2ε,N , µ

2ε,N

)is its desired approximation.

Theorem 4.1. For (λ, µ) ∈ R2 and N ∈ N, we have∣∣∣∆r(λ, µ)− ∆r,h,N (λ, µ)∣∣∣ < Tr,h,N (λ, µ) +Aε,r,N , (4.1)

where ∆r and ∆r,h,N are given in (1.6) and (3.14), respectively. The functions Tr,h,N and Aε,r,Nare defined by

Tr,h,N (λ, µ) := 2τrAδr,N (λ, µ)e−δrN√πδrN

, (4.2)

Aε,r,N := 4 ε

2∏r=1

e−δr/4N(

1 +√N/δr

), (4.3)

where δr = min{δrs, δrs}, δrs = (π − γshs)/2, hs ∈ (0, π/γs], s = 1, 2 and

τr :=

‖Ur‖∞, αr = 0,

‖Ur,αr‖∞ αr 6= 0.

The functions Aδr,N , Ur and Ur,αr are given in (2.4), (3.9) and (3.18), respectively. Moreover,

∆r,h,N −→ ∆r holds uniformly on R2 for ε→ 0 and N →∞.

Proof. According to (3.7) and (3.14), we have for all (λ, µ) ∈ R2∣∣∣∆r(λ, µ)− ∆r,h,N

∣∣∣ ≤ |Ur(λ, µ)−Gh,N [Ur](λ, µ)|+∣∣∣Gh,N [Ur](λ, µ)−Gh,N [Ur](λ, µ)

∣∣∣ . (4.4)

Since Ur ∈ Bγr,∞(R2), cf. Theorem 3.2, we have cf. (2.3)

|Ur(λ, µ)−Gh,N [Ur](λ, µ)| ≤ Tr,h,N (λ, µ), (4.5)

where Tr,h,N is defined as above. Assume that ε be sufficiently small such that condition (2.5) holds,then we have, cf. Theorem 2.1,∣∣∣Gh,N [Ur](λ, µ)−Gh,N [Ur](λ, µ)

∣∣∣ < Aε,r,N , (4.6)

where Aε,r,N is defined as above. Combining (4.6), (4.5) and (4.4), we obtain (4.1) in the caseαr = 0. The proof of the case αr 6= 0 is similar. In view of (4.2) and (4.3), the right-hand side of(4.1) goes to zero uniformly when ε → 0 and N → ∞, and therefore ∆r,h,N −→ ∆r uniformly onR2.

Denote by J∆ the Jacobian matrix

J∆(λ, µ) =

(∂∂ζ∆1(λ, µ) ∂

∂η∆1(λ, µ)∂∂ζ∆2(λ, µ) ∂

∂η∆2(λ, µ)

).

If (λ∗, µ∗) is a zero of both equations in (1.6), then the Jacobian determinant |J∆(λ∗, µ∗)| is nonzero,cf. [16, Lemma 4.1]. This fact will be used in the proof of the following theorem.

8

Theorem 4.2. Let((λ∗)2, (µ∗)2

)be an eigenpair of system (1.1)-(1.3) and denote by

(λ2ε,N , µ

2ε,N

)the corresponding approximation. In the notation of (4.2) and (4.3), we have the following estimate

‖(λ∗, µ∗)− (λε,N , µε,N )‖R2 < Dε,N

2∑r=1

(Tr,h,N (λε,N , µε,N ) +Aε,r,N ) , (4.7)

where Dε,N := max(ζ1,ζ2)∈Nε,N (λ∗,µ∗)

‖J−1∆ (ζ1, ζ2)‖, J−1

∆ is the inverse Jacobian matrix and Nε,N (λ∗, µ∗)

is a neighborhood of (λ∗, µ∗) depending on ε, N . Furthermore, ‖(λ∗, µ∗)−(λε,N , µε,N )‖R2 approaches0 as ε→ 0 and N →∞.

Proof. Replacing (λ, µ) by (λε,N , µε,N ) in (4.1) implies

|∆r(λε,N , µε,N )| < Tr,h,N (λε,N , µε,N ) +Aε,r,N . (4.8)

Expanding the function ∆r using bivariate Taylor expansion at the point (λ∗, µ∗) and replacing(λ, µ) by (λε,N , µε,N ), we obtain

∆r(λε,N , µε,N ) = (λε,N − λ∗)∂

∂λ∆r(ζ1, ζ2) + (µε,N − µ∗)

∂

∂µ∆r(ζ1, ζ2), (4.9)

where (ζ1, ζ2) is sufficiently close to the point (λ∗, µ∗). Indeed, (ζ1, ζ2) is an unknown point on theline joining (λε,N , µε,N ) and (λ∗, µ∗). System (4.9) can be represented as(

∆1(λε,N , µε,N )∆2(λε,N , µε,N )

)= J∆(ζ1, ζ2)

(λε,N − λ∗µε,N − µ∗

). (4.10)

Since the Jacobian determinant is an entire function and |J∆(λ∗, µ∗)| 6= 0, for a sufficiently largeN and a sufficiently small ε we can find a neighborhood Nε,N (λ∗, µ∗) such that the inverse of theJacobian matrix J−1

∆ (ζ1, ζ2) exists for all points in this neighborhood. Now (4.10) implies that(λε,N − λ∗µε,N − µ∗

)= J−1

∆ (ζ1, ζ2)

(∆1(λε,N , µε,N )∆2(λε,N , µε,N )

).

Therefore

‖(λ∗, µ∗)− (λε,N , µε,N )‖R2 ≤ ‖J−1∆ (ζ1, ζ2)‖

2∑r=1

|∆r(λε,N , µε,N )| . (4.11)

Combining (4.11) and (4.8) yields (4.7). The right-hand side of (4.7) approaches 0 as ε → 0 andN →∞, so ‖(λ∗, µ∗)− (λε,N , µε,N )‖R2 −→ 0.

5 Numerical Illustrations

This section is devoted to the presentation of three numerical examples. In all examples, our resultsare compared with the results for the bivariate WKS sampling method, cf. [1]. Our method producesresults which are more accurate than the bivariate WKS method because its convergence rate ishigher than that of the bivariate WKS method. All numerical computations were carried out usingMathematica 12 on a personal computer. For the sake of simplicity, we set νr := max

r=1,2{γr1, γr2} and

h := h1 = h2. Denote by ‖(λ∗k, µ∗k) − (λk,ε,N , µk,ε,N )‖R2 the norm of the error where (λ∗k, µ∗k) is an

exact zero of system (1.1)-(1.3) and (λk,ε,N , µk,ε,N ) is its desired approximation. In Examples 5.1and 5.2, we choose αr = βr = 0, while in Example 5.3 αr = π/2 and βr = 0.

9

Example 5.1. Consider the following system

y′′1 (x) +(2λ2 + µ2 + 1

)y1(x) = 0, x ∈ [0, 1], (5.1)

y′′2 (x) +(λ2 + 2µ2 + 2

)y2(x) = 0, x ∈ [0, 1], (5.2)

yr(0) = 0, yr(1) = 0, r = 1, 2. (5.3)

It is easily checked that conditions (1.4)-(1.5) hold. In this example, αr = βr = 0, Ur,0 ∈B(ν1,ν2),∞(R2) where ν1 = ν2 =

√2, and it is a simple task to compute the functions ∆r

∆1(λ, µ) = sinc(√

2λ2 + µ2 + 1), ∆2(λ, µ) = sinc

(√λ2 + 2µ2 + 2

).

The function Kr, which is defined in (3.8), is computed as K1(λ, µ) = sinc(√

2λ2 + µ2)

and

K2(λ, µ) = sinc(√

λ2 + 2µ2)

. Since the functions ∆r are given explicitly, the amplitude error

does not appear in this example, i.e. ε = 0. Figure 1 shows the eigencurves of system (5.1)-(5.3)in the region [−15, 15]2. Their intersections are the eigenpairs of the system and we can computefew of them using our method described above. Note that eigencurves for the two equations insystem (5.1)-(5.3) are shaped like ellipses. Table 1 compares results of our method and results of thebivariate WKS sampling method. In Table 2, we show the norm error ‖(λ∗k, µ∗k)− (λk,ε,N , µk,ε,N )‖R2

associated with the two methods. In Figures 2 and 3, we illustrate the behaviour of our methodwith respect to h and N .

-15 -10 -5 0 5 10 15

-15

-10

-5

0

5

10

15

Figure 1: The eigencurves in Example 1.

10

Table 1: Approximation of eigenpairs with h = 1

kBivariate WKS sampling Bivariate sinc-Gauss sampling

λk,0,15 µk,0,15 λk,0,15 µk,0,15

1 1.813797507802172 1.513239555736101 1.813799364683959 1.5132310236649422 3.627597850186581 3.487076569018237 3.627598728958227 3.4870435231679273 5.441403076170987 5.348799863878748 5.441398093112097 5.3487207073547804 7.255208077727408 7.186073252549332 7.255197457187725 7.1859508863485645 9.069006371615362 9.013757735420938 7.255197457187725 9.0136953211371816 10.88280054723645 10.836602869705539 10.882796185506988 10.83675471755794

Table 2: The norm error ‖(λ∗k, µ∗k)− (λk,0,15, µk,0,15)‖R2

k Bivariate WKS samplingBivariate sinc-Gauss sampling

h = 1 h = 0.51 8.73082×10−6 1.00333×10−9 6.94810×10−12

2 3.30572×10−5 5.91952×10−10 1.23606×10−11

3 7.93134×10−5 4.50801×10−10 3.57188×10−11

4 1.22826×10−4 2.82605×10−10 2.22197×10−11

5 6.31407×10−5 2.54946×10−10 2.27273×10−12

6 1.51911×10−4 2.33596×10−10 7.64573×10−12

Example 5.2. Consider the following system

y′′1 (x) +(2(x+ 1)λ2 + (x+ 1)µ2 + x

)y1(x) = 0, x ∈ [0, 1], (5.4)

y′′2 (x) +(λ2 + 2µ2 + 3

)y2(x) = 0, x ∈ [0, 1], (5.5)

yr(0) = 0, yr(1) = 0, r = 1, 2. (5.6)

This system is a special case of system (1.1)–(1.3) and satisfies the conditions (1.4)-(1.5). In thisexample, Ur,0 ∈ B(ν1,ν2),∞(R2) where ν1 = ν2 = 2(4−

√2)/3, and the function ∆1 can be expressed

in terms of Airy functions Ai, Bi and their first derivatives as

∆1(λ, µ) =Ai(

3√−1(2+4λ2+2µ)

ω(λ,µ)

)Bi(

3√−1(2λ2+µ)ω(λ,µ)

)−Ai

(3√−1(2λ2+µ)ω(λ,µ)

)Bi(

3√−1(2+4λ2+2µ)

ω(λ,µ)

)ω(λ, µ)

(Ai′(

3√−1(2λ2+µ)ω(λ,µ)

)Bi(

3√−1(2λ2+µ)ω(λ,µ)

)−Ai

(3√−1(2λ2+µ)ω(λ,µ)

)Bi′(

3√−1(2λ2+µ)ω(λ,µ)

)) ,with ω(λ, µ) = (1 + 2λ2 + µ)2/3 while the function ∆2 is given as the explicit form

∆2(λ, µ) = sinc(√

λ2 + 2µ2 + 3).

In this example, K1(λ, µ) = ν123/4 sinc

(ν1√

2

√2λ2 + µ2

)and K2(λ, µ) = sinc

(√λ2 + 2µ2

). To com-

pute the norm error in this example, the exact common zeros of equations (5.2) and (5.2) arecomputed sufficiently exactly with Mathematica. We apply our method and summarize the result inTables 3 and 4. Figure 2 demonstrates the eigencurves of system (5.1)-(5.5) in the region [−15, 15]2.Here, too, the eigencurves are shaped like ellipses.

11

0.4 0.6 0.8 1.0 1.2 1.4 1.6h

-14

-12

-10

-8

log error

Figure 2: The logarithm of the normerror ‖(λ∗k, µ∗k)− (λk,0,20, µk,0,20)‖R2 fork = 1, . . . , 6 in Example 1.

0.4 0.6 0.8 1.0 1.2 1.4 1.6h

-18

-16

-14

-12

-10

-8

-6

log error

Figure 3: The logarithm of the normerror ‖(λ∗3, µ∗3) − (λ3,0,N , µ3,0,N )‖R2 forN = 10, 15, 20, 25 in Example 1.

-15 -10 -5 0 5 10 15

-15

-10

-5

0

5

10

15

Figure 4: The eigencurves in Example 2.

Table 3: Approximation of eigenpairs with h = 1

kBivariate WKS sampling Bivariate sinc-Gauss sampling

λk,0,15 µk,0,15 λk,0,15 µk,0,15

1 1.359821568195881 1.584365124779384 1.359811348447286 1.5843796115688472 2.294869574533618 3.950449753437753 2.294859272608290 3.9504454470616103 5.235270180088456 2.129614898841256 5.235258286227501 2.1296112553831384 6.477462750322421 4.683407147326487 6.477390507161662 4.6834736855276055 7.667060430768335 6.932768257873039 7.666946848415846 6.9329501689639266 8.825101408478323 9.106626719982193 8.825213824551480 9.106473269299752

Example 5.3. The following system

y′′1 (x) +(3λ2 + 2µ2 +

√x)y1(x) = 0, x ∈ [0, 1], (5.7)

12

Table 4: The norm error ‖(λ∗k, µ∗k)− (λk,0,15, µk,0,15)‖R2 with h = 1

k Bivariate WKS sampling Bivariate sinc-Gauss sampling1 1.77230×10−5 6.16607×10−9

2 1.11696×10−5 8.90886×10−9

3 1.24318×10−5 7.90168×10−9

4 9.82270×10−5 1.60926×10−8

5 2.14480×10−4 2.13712×10−8

6 1.90214×10−4 8.33743×10−9

y′′2 (x) +(2λ2 + 3µ2 + x

√x)y2(x) = 0, x ∈ [0, 1], (5.8)

y′r(0) = 0, y′r(1) = 0, r = 1, 2, (5.9)

is a special case of system (1.1)-(1.3). Here, we cannot compute the function ∆r and then theeigenpairs can not be computed exactly. To find the samples Ur(nh,mh), (n,m) ∈ Z2

N (λ, µ) inthis example, we compute the solution of system (1.1)-(1.3), i.e. yr(1, nh,mh), numerically at thenodes (n,m) ∈ Z2

N (λ, µ) and then use (3.13). Therefore, we approximate the function ∆r as in

(3.14). Here, K1(λ, µ) = cos(√

3λ2 + 2µ2)

and K2(λ, µ) = cos(√

2λ2 + 3µ2)

. Few eigenpairs of

this system are given in Figure 5 and the eigencurves are shown in Figure 5, as well.

-15 -10 -5 0 5 10 15

-15

-10

-5

0

5

10

15

Figure 5: The eigencurves in Example 3.

6 Conclusions

This work is devoted to constructing a new sampling method to compute eigenpairs of a two-parameter Sturm-Liouville eigenvalue system with separate boundary conditions. This method isbuilt by using the bivariate sinc-Gauss sampling formula which was established by the authors in2016. Due to the exponential order of convergence, the method described here, i.e. the bivariatesinc-Gauss method, is superior to comparable methods for computing the eigenpairs of the boundary

13

Table 5: Approximation of eigenpairs with h = 1 and ε = 10−8

kBivariate WKS sampling Bivariate sinc-Gauss sampling

λk,ε,15 µk,ε,15 λk,ε,15 µk,ε,15

1 0.515656277786066 0.762177530812667 0.515671212590693 0.7621736040880732 2.051784932534724 2.114071060975086 2.051799490194234 2.1140685145914093 3.478736174942723 3.516120586250193 3.478721556147922 3.5161236053567944 4.893280857549082 4.919999452785837 4.893200190982574 4.9200111118799445 6.303637655607843 6.324376642474745 6.303486384409280 6.3243924197918606 7.712016456380302 7.728978003793143 7.711851697945461 7.7289863149878917 9.119198704295004 9.133705129467295 9.119176716920121 9.1336934546890878 10.525522861700562 10.538515646451458 10.525875454418745 10.538468065160703

value system. The accuracy of the method increases without additional cost when the parameter Nis fixed and h is decreasing, except that the function is approximated on a smaller domain. Examplesare provided to illustrate the effectiveness of the approximation. This method can be applied tocompute eigenpairs for various types of two-parameter boundary eigenvalue systems. This will bestudied in future works.

Funding. The first author gratefully acknowledges the support by the Alexander von Humboldtfoundation under grant 3.4-YEM/1142916.

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