Relations between Spectrum Curves of Discrete Sturm-Liouville Problem with Nonlocal Boundary Conditions and Graph Theory. II

Jonas Vitkauskas; Artu¯ras Štikonas

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Abstract: In this paper, relations between discrete Sturm–Liouville problem with nonlo- cal integral boundary condition characteristics (poles, critical points, spectrum curves) and graphs characteristics (vertices, edges and faces) were found. The previous article was de- voted to the Sturm–Liouville problem in the case two-points nonlocal boundary conditions.

Keywords: Sturm–Liouville problem, spectrum curves, integral boundary condition, graphs.

Summary: Šiame straipsnyje pristatomos sąsajos tarp diskrečiojo Šturmo ir Liuvilio uždavinio su nelokaliąja integraline kraštine sąlyga (poliai, kritiniai taškai ir spektrine˙s kreive˙s) bei grafu˛ charakteristiku˛ (viršu¯ne˙s, briaunos ir veidai). Ankstesnis straipsnis buvo skirtas Sturmo ir Liuvilio uždaviniui su dvitaške˙mis nelokaliosiomis kraštine˙mis sąlygomis.

Keywords: Šturmo ir Liuvilio uždavinys, spektrine˙s kreive˙s, integraline˙ kraštine˙ sąlyga, grafai.

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Articles

Relations between Spectrum Curves of Discrete Sturm-Liouville Problem with Nonlocal Boundary Conditions and Graph Theory. II

Diskrečiojo Šturmo ir Liuvilio uždavinio su nelokaliosiomis kraštine˙mis sąlygomis spektriniu˛ kreiviu˛ ir grafu˛ teorijos sąsajos. II

Jonas Vitkauskas vitkauskas@mif.stud.vu.lt

Vilnius University, Lituania

Artu¯ras Štikonas arturas.stikonas@mif.vu.lt

Vilnius University, Lituania

Lietuvos matematikos rinkinys, vol. 62 Ser. A, pp. 1-8, 2021
Vilniaus Universitetas

Esta obra está bajo una Licencia Creative Commons Atribución 4.0 Internacional.

Recepción: 23 Julio 2021

Publicación: 01 Diciembre 2021

DOI: https://doi.org/10.15388/LMR.2021.25128

1 A discrete Sturm–Liouville Problem

In this paper, particular properties of the spectrum of a discrete Sturm–Liouville Problem (dSLP) [3, 4, 5] with Integral Boundary Condition were found using Euler’s charakteristic formula [2]. In previous article [5] we have found relations between spec- trum curve properties and graphs theory in the case of two-points nonlocal boundary conditions.

We introduce a uniform grid and we use notation ${\overline{ω}}^{h} = {t_{j} = j h, j = 0,$ $. ., n; n h = 1}$ for $2 < n \in ℕ$ and $ℕ^{h} ≔ (0, n) \cap ℕ, {\overline{ℕ}}^{h} = ℕ^{h} \cup {0, n}$ Also ,we make an assumption that $ξ_{1}$ and $ξ_{2}$ are located on the qrid $i . e ., ξ_{1} = m_{1} h = m_{1} h = m_{1} / n,$ $ξ_{2} = m_{2} h = m_{2} / n, m \in S_{ξ}^{h} ≔ {(m_{1}, m_{2}) :$ $: 0 \leq m_{1} < m_{2} \leq n, m_{1}, m_{2} \in {\overline{ℕ}}^{h}}$ Let $ξ = m / n, = (m_{1} / n, m_{2} / n), ξ = ξ_{1} / ξ_{2} =$ $m_{+} / n, ξ_{-} = ξ_{1} - ξ_{2} = m_{-} / n,$ here $m_{+} ≔ m_{1} + m_{2}, m_{-} = m_{2} - m_{1}$ We denote ${ℕ_{o}}_{d}_{d} = {k \in ℕ : k - o d d} {ℕ_{e v}}_{e}_{n} = {k \in ℕ : k - e v e n}$

Let us consider a dSLP (an approximation by Finite-Difference Scheme) [3, 5]

\frac{{U_{i}}_{+}_{1} - 2 U_{i} + {U_{i}}_{- 1}}{h^{2}} = ⋋ U_{i}, i = 1, . . ., n - 1

(1)

$⋋ \in ℂ$ with a classical discrete Dirichlet Boundary Condition (BC)

U_{0} = 0

(2)

and Integral Boundary Condition (approximated by trapezoidal formula):

U_{n} = γ h (\frac{U_{m_{1}} + U_{m_{2}}}{2} + \sum_{k = m_{1} + 1}^{m_{2 - 1}} U_{k}

(3)

Let us consider a bijection (see [1])

λ = λ^{h} (q) ≔ \frac{4}{h^{2}} s i n^{2} \frac{π q h}{2}

(4)

between $ℂ_{λ} ≔ ℂ$ and $ℂ_{q}^{h}, ℂ_{q}^{h} ≔ ℝ_{q}^{h} \cup ℂ_{q}^{h -}, ℝ_{q}^{h} ≔ ℝ_{y}^{-} \cup {0} \cup ℝ_{x}^{h} \cup$ ${n} \cup ℝ_{y}^{h +}$ $, ℝ_{y}^{-} ≔ {q = i y : y > 0}, ℝ_{x}^{h} ≔ {q = x : 0 < x < n},$ $ℝ_{q}^{h +} ≔ {q = n + i y : y > 0}, ℂ_{q}^{h +} ≔ {q = x + i y : 0$ $< x < n, y > 0}, ℂ_{q}^{h -} ≔ {q = x + i y : 0 < x < n, y < 0} .$

The general solution U_j for a discrete equation (1) is equal to:

U_{j} = C_{1} s i n (π q t_{j}) (1 - h q)^{- 1} π^{- 1} q^{- 1} + C_{2} c o s (π q t_{j})

Then by using BCs (2) and (3) we get an equation:

ℤ^{h} (q) = γ ℙ_{ξ}^{h} (q), q \in ℂ_{q}^{h},

where functions $ℤ^{h} (q)$ and $ℙ_{ξ}^{h} (q)$ are as follows:

ℤ^{h} (q) = \frac{s i n (π q)}{π q} . \frac{s i n (\frac{π}{2} q h)}{π q h c o s (\frac{π}{2} q h)},_{}^{} (q) = \frac{s i n (\frac{π}{2} q (ξ_{2} - ξ_{1}))}{π q} . \frac{s i n (\frac{π}{2} q (ξ_{2} + ξ_{1}))}{π q}

Constant Eigenvalues. For any constant eigenvalue $λ \in ℂ λ$ there exists theConstant Eigenvalue Point (CEP) $q \in ℂ ℂ_{q}$ . CEP are roots of the system [1]:

ℤ^{h} (q) = 0, ℙ_{ξ}^{h} (q) = 0

For every CEP $c_{j}$ we define nonregular Spectrum Curve $ℕ_{j} = {c_{j}}$

Nonconstant eigenvalues. Let us consider Complex Characteristic Function:

γ_{c} (q; ξ) : = \frac{ℤ^{n} (q)}{ℙ_{ξ}^{h} (q)}, q \in ℂ_{q}^{h} .

(5)

All nonconstant eigenvalues (which depend on the parameter $γ$ ) are $γ$ -points of (Complex-Real) Characteristic Function (CF)[6]. CF $γ$ (q) is the restriction of Com- plex CF $γ_{c}$ (q) on a set $D_{ξ} : = {q \in ℂ_{q}^{h} :$ Im (see more in [5]). We call such

Fig. 1.
Spectrum Curves for n = 2[4]

Fig. 2.
Spectrum Curves for n = 3 [4].

Fig. 3.
Spectrum Curves for n = 4 [4].

Fig. 4.
Spectrum Curves for n = 5 [4].

There exist Pole Points (PP) of the second order. They are described as follows:

p_{k}^{12} = 2 n k / g c d (m_{+}, m_{-}, k = 1, . . ., [n / p_{1}^{12}] .

All PP of the second order are in $ℝ_{x}^{h} .$ . Note that there could be a PP of second order in Ramification Point $q = n,$ , so they become simple PP of the first order.

Two or more Spectrum Curves may intersect at CP. We denote a set $CP β : = {b_{i}, i = \overline{1, n_{b}}},$ , where $n_{b}$ is the number of CPs at $ℂ_{q}^{h} .$ The number of CPs at $ℝ_{q}^{h}$ and $ℂ_{q}^{h +}$ we denote as $n_{c r}$ ,and $n_{c r}^{+}$ , respectively. Note that the part of the spectrumdomain in set $ℂ_{q}^{h +}$ is symmetric to the part in set $ℂ_{q}^{h -} .$ . So, $n_{b} = n_{c r} + 2 n_{c r}^{+} .$ If $b \in β$ then $d e g^{+} (b)$ is one unit larger than the order of this CP.

The pole at $q = \infty$ is of

n_{\infty} = n - m_{2}

(6)

order. If $m_{2} < n, q = \infty$ is a PP. For $m_{2} = n, q = \infty$ is a Removable Singularity Point.

For poles and CP $d e g^{+} (q), q \in P \cup B∪ {\infty},$ corresponds to the number of outgoing Spectrum Curves at that point. Note that incoming Spectrum Curves alternate with outgoing, so $d e g^{+} (q) = d e g^{-} (q)$

2 Relations between dSLP and graphs properties in the case of integral BC

It is possible to define relations between properties of dSLP and graph theory. Poles or CPs refer to vertices of a certain graph and parts of Spectrum Curves could be interpreted as edges. In our case, we have a simple balanced weakly connected digraph. Definitions and notations in graphs theory are described in [5].

Fig. 5.
Spectrum Curves for n = 6 [4].

2.1 Properties of Spectrum Curves

Poles, CPs, regular and nonregular Spectrum Curves, CEPs were found by A. Skučaite˙ [4].

There is $n - 1$ Spectrum Curves for every $n \in ℕ, n \geq 2 .$ Nonregular Spectrum Curves are CEPs and belong to $ℝ_{x}^{h} = (0, n) .$ . The number of such Spectrum Curves is equal to

\begin{matrix} n_{c e} = [\frac{n - 1}{2 n} d b d (2 n; m_{+}) + [\frac{n - 1}{2 n} d b d (2 n; m_{-})] \\ - [\frac{n - 2}{2 n} d b d (2 n; m_{+}, m_{-})] \end{matrix}

(7)

Number of regular Spectrum Curves $n_{n c e} = n - 1 - n_{c e}$ . The poles of CF belong to $ℝ_{x}^{h} \cup {0} \cup {n} \cup {\infty}$ and $n_{p} + n_{\infty} = n_{n c e} .$ So, we have formula

n_{p} + n_{c e} = m_{2} - 1

(8)

Let us denote

d e g_{r}^{+} : = \sum_{b \inB \cap ℝ_{q}^{h}}^{\infty} d e g^{+} (b), d e g_{c}^{+} : = \sum_{b \inB \cap ℂ_{q}^{h \pm}}^{\infty} d e g^{+} (b) = 2 \sum_{b \in \cap ℂ_{q}^{h +}}^{\infty} d e g^{+} (b) .

Let $n_{c}$ is the number of Spectrum Curves parts in $ℂ_{q}^{h +}$ between two CP (including $q = 0$ and $q = n$ ).

2.2 Spectrum domain as a graph

We consider Spectrum domain as graph on sphere (Riemann sphere C¯ ) because Ch S2. The poles and CPs of the CF are the vertices of this graph. The point is the pole or CP.

Lemma 1. The number of vertices is

\begin{matrix} u = n_{p} - n_{2 p} + n_{b} + 1 - [m_{2} / n] \\ = n_{p} - n_{2 p} + n_{c r} + 2 n_{c r}^{+} + 1 - [m_{2} / n] \\ = m_{2} - n_{c e} - n_{2 p} + n_{c r} + 2 n_{c r}^{+} - [m_{2} / n] \end{matrix}

(9)

Lemma 2. The number of edges is

e = n_{p} + d e g_{r}^{+} + d e g_{c}^{+} + n_{\infty .}

(10)

Lemma 3. The number of faces is

\begin{matrix} f = 2 (n_{\infty} + n_{2 p} + n_{c} - n_{c r}^{+} + ⌊ m_{2} / n ⌋) \\ = 2 (n - m_{2} + n_{2 p} + n_{c} - n_{c r}^{+} + ⌊ m_{2} / n ⌋) \end{matrix}

(11)

This lemma is valid for $n_{c} = n_{c r}^{+} = 0$ . Each part of spectrum curve between two CPs $b_{1}, b_{2} \in ℝ_{q}^{h}$ increases the number of faces by one. So, this formula is valid for the case $n_{c r}^{+} = 0$ . Each additional CP $CP b \in C_{q}^{h +}$ increases the number of faces by 2( $d e g^{+} (b) - 1$ )and number parts of Spectrum Curves between this CP and other CPs by $2 d e g^{+} (b) - 1$ .

Numbers of spectrum vertices, edges and faces, expressed by the formulas above, inserted to the Euler’s characteristic’s formula of sphere $u - e + f = 2$ give new relation.

Theorem 1. The Euler’s characteristic’s formula is equivalent to

\begin{matrix} \underset{}{\sum_{b \inB}^{} d e g^{+} (b) = d e g_{c}^{+} + d e g_{c}^{+} = n_{\infty} + n_{2 p} + 2 n_{c} + n_{c r} - 1 + ⌊ m_{2} / n ⌋} \\ = n - m_{2} + n_{2 n} + 2 n_{c} + n_{c r} - 1 + ⌊ m_{2} / n ⌋ \end{matrix}

(12)

This formula was derived in [4] when are not CPs in $ℂ_{q}^{h \pm} (d e g_{c}^{+} = 0)$ ,all CPs are of the first order $(d e g_{r}^{+} = 2 n_{c r})$ and $n_{c} = 0$ .Then it can be rewritten as

n_{c r} = n_{2 p} + n_{\infty} - 1 + ⌊ m_{2} / n ⌋ = n_{2 n} + n - m_{2} - 1 + ⌊ m_{2} / n ⌋ .

Corollary 1. The number of edges is

\begin{matrix} e = 2 n_{\infty} + n_{p} + n_{2 p} + n_{c r} + 2 n_{c} - 1 + ⌊ m_{2} / n ⌋ \\ = 2 n - m_{2} - 2 - n_{c e} + n_{2 p} + n_{c r} + 2 n_{c} + ⌊ m_{2} / n ⌋ \end{matrix}

(13)

Remark 1. In the case m2 < n the formulas (9)–(13) are

\begin{matrix} u = m_{2} - n_{c e} - n_{2 p} + n_{c r} + 2 n_{c r}^{+}, \\ e = 2 n - m_{2} - 2 - n_{c e} + n_{2 p} + n_{c r} + 2 n_{c} \\ f = 2 (n - m_{2} + n_{2 p} + n_{c} - n_{c r}^{+}), d e g_{r}^{+} + d e g_{c}^{+} = n - m_{2} - 1 + n_{2 p} + 2 n_{c} + n_{c r}, \end{matrix}

where nce is defined by (7).

In the case $m_{2} = n$ we have $n_{\infty} = 0, n_{p} = n - 1 - n_{c e}$ and $n_{2 p} = 0, n_{c r} = 0$ $(s o, d e g_{r}^{+} = 0)$ (see [4]). In this case we get $d e g_{c}^{+} = 2 n_{c} = 0$ and $n_{c r}^{+} = 0$ . Thus, for are valid. Note, that in this case we have Cycle digraph $m_{2} = n$ the following formulas

\begin{matrix} u = n - 1 n_{c e} \\ e = n - 1 n_{c e} \\ f = 2 \end{matrix}

are valid .Note ,that in this cas we have Cycle digraph

Remark 2. If $n_{c r}^{+} = 0 (d e g_{c}^{+} = 0)$ then $d e g_{r}^{+} = n_{c r} = 2 n_{c} + n - m_{2} + n_{2 p} - 1 > n_{c r}$ shows that there are exist CPs in $ℝ_{q}^{h}$ of the second or the higher order.

Material suplementario

References

1.K. Bingele˙, A. Bankauskiene˙, A. Štikonas. Spectrum curves for a discrete Sturm– Liouville problem with one integral boundary condition. Nonlinear Anal. Model. Control, 24(5):755–774, 2019. https://doi.org/10.15388/NA.2019.5.5.

2 E. Manstavičius. Analizine˙ ir tikimybine˙ kombinatorika. TEV, Vilnius, 2007.

3.M. Sapagovas. Diferencialiniu˛ lygčiu˛ kraštiniai uždaviniai su nelokaliosiomis sąlygomis. Mokslo aidai, Vilnius, 2007.

4.A. Skučaite˙. Investigation of the spectrum for Sturm–Liouville problem with a nonlocal integral boundary condition. PhD thesis, Vilnius University, 2016.

5.J.Vitkauskas, A. Štikonas. Relations between spectrum curves and dis- crete sturm–liouville problem with nonlocal boundary conditions and graph theory. Proc. of the Lithuanian Mathematical Society, 61:1–6, 2020. https://doi.org/10.15388/LMR.2020.22474.

6.A. Štikonas, O. Štikoniene˙. Characteristic functions for Sturm–Liouville prob- lems with nonlocal boundary conditions. Math. Model. Anal., 14(2):229–246, 2009.https://doi.org/10.3846/1392-6292.2009.14.229-246.

Notas

Fig. 1.
Spectrum Curves for n = 2[4]

Fig. 2.
Spectrum Curves for n = 3 [4].

Fig. 3.
Spectrum Curves for n = 4 [4].

Fig. 4.
Spectrum Curves for n = 5 [4].

Fig. 5.
Spectrum Curves for n = 6 [4].