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ISSN 2518-7929 (Print) ISSN 2663–5011(Online) Индексi 74618

Индекс 74618

MATHEMATICS Series

№ 3(99)/2020

July-August-September September, 30^th, 2020

Founded in 1996 Published 4 times a year

Karaganda, 2020

BULLETIN

OF THE KARAGANDA

UNIVERSITY

Main Editor

Candidate of Physics and Mathematics sciences

N.T. Orumbayeva

Responsible secretary PhD

M.T. Kosmakova

Editorial board

A. Ashyralyev, M.A. Sadybekov, M. Otelbayev, B.R. Rakishev, U.U. Umirbaev, T. Bekjan, B. Poizat, A.A. Shkalikov, A.S. Morozov, G. Akishev, A.T. Asanova, N.A. Bokaev, M.T. Jenaliyev, A.R. Yeshkeyev, K.T. Iskakov, L.K. Kusainova, E.D. Nursultanov, M.I. Ramazanov, E.S. Smailov,

Guest editor, Professor of Mathematics, Dr. of phys.-math. sciences, Near East University, Nicosia, TRNC, Mersin 10 (Turkey);

Guest editor, Corresponding member of NAS RK, Dr. of phys.-math. sciences, IMMM, Almaty (Kazakhstan);

Academician of NAS RK, Dr. of phys.-math. sciences, Gumilyov ENU, Nur-Sultan (Kazakhstan);

Academician of NAS RK, Dr. of techn. sciences, Turysov IGOM, Almaty (Kazakhstan);

Corresponding member of NAS RK, Dr. of phys.-math. sciences, Gumilyov ENU, Nur-Sultan (Kazakhstan);

Professor, Xinjiang University (China);

Professor, Universite Claude Bernard Lyon-1, Villeurbanne, (France);

Corresponding member of RAS RF, Dr. of phys.-math. sciences, Lomonosov Moscow State University (Russia);

Dr. of phys.-math. sciences, Sobolev Institute of Mathematics (Russia);

Dr. of phys.-math. sciences, Gumilyov ENU, Nur-Sultan (Kazakhstan);

Dr. of phys.-math. sciences, IMMM, Almaty (Kazakhstan);

Dr. of phys.-math. sciences, Gumilyov ENU, Nur-Sultan (Kazakhstan);

Dr. of phys.-math. sciences, IMMM, Almaty (Kazakhstan);

Dr. of phys.-math. sciences, Buketov KU (Kazakhstan);

Dr. of phys.-math. sciences, Gumilyov ENU, Nur-Sultan (Kazakhstan);

Dr. of phys.-math. sciences, Gumilyov ENU, Nur-Sultan (Kazakhstan);

Dr. of phys.-math. sciences, KB Lomonosov MSU, Nur-Sultan (Kazakhstan);

Dr. of phys.-math. sciences, Buketov KU (Kazakhstan);

Dr. of phys.-math. sciences, IAM, Karaganda (Kazakhstan)

Postal address: 28, University Str., 100024, Kazakhstan, Karaganda Теl.: (7212) 77-04-38 (add. 1026); fax: (7212) 35-63-98.

E-mail: [email protected]. Web-site: mathematics-vestnik.ksu.kz

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G.K. Kalel

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CONTENTS

MATHEMATICS

Preface . . . . Ashyralyyev C., Cay A. Numerical solution to elliptic inverse problem with Neumann-type integral condition and overdetermination . . . . Karwan H.F. Jwamer and Rando R.Q. Rasul. A Comparison between the fourth order linear differential equation with its boundary value problem . . . . Mardanov M.J., Sharifov Y.A., Ismayilova K.E. Existence and uniqueness of solutions for the system ofintegro-differential equations with three-point and nonlinear integral boundary conditions . . . . Dovletov D.M.Nonlocal boundary value problem with Poissons operator on a rectangle and its difference interpretation . . . . Hincal E., Mohammed S., Kaymakamzade B.Stability analysis of an ecoepidemiological model consisting of a prey and tow competing predators with Si-disease in prey and toxicant . . . . Akat M., Kosker R., Sirma A.On the numerical schemes for Langevin-type equations . . . . Ashyralyev A., Sozen Y., Hezenci F.A remark on elliptic differential equations on manifold . Hincal E., Kaymakamzade B., Gokbulut N. Basic reproduction number and effective reproduction number for North Cyprus for fighting covid-19 . . . . Ashyralyev A., Ashyralyyev C., Zvyagin V.G.A note on well-posedness of source identification elliptic problem in a Banach space . . . . Ashyralyev A., Turk K., Agirseven D. On the stable difference scheme for the time delay telegraph equation . . . . Ashyraliyev M., Ashyralyyeva M., Ashyralyev A. A note on the hyperbolic-parabolic identification problem with involution and Dirichlet boundary condition . . . . Ashyralyev A., Erdogan A.S., Sarsenbi A.A note on the parabolic identification problem with involution and Dirichlet condition . . . . АВТОРЛАР ТУРАЛЫ МӘЛIМЕТТЕР — СВЕДЕНИЯ ОБ АВТОРАХ — INFORMATION ABOUT AUTHORS . . . .

4 5 18

26 38 55 62 75 86 96 105 120 130

140

Preface

This issue is a collection of 12 selected papers. These papers are presented at the Fifth International Conference on Analysis and Applied Mathematics (ICAAM 2020) organized by Near East University, Lefkosa (Nicosia), Mersin 10, Turkey.

The meeting was held on September 23–30, 2020 in North Cyprus, Turkey. The main organizer of the conference is Near East University, Nicosia (Lefkosa), Mersin 10, Turkey. The conference was also supported by Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan and Analysis

&PDE Center, Ghent University, Belgium.

The conference is organized biannually. Previous conferences were held in Gumushane, Turkey in 2012; in Shymkent, Kazakhstan in 2014; in Almaty, Kazakhstan in 2016; in 2018 Lefkosa,Mersin 10, Turkey. The proceedings of ICAAM 2012, ICAAM 2014, ICAAM 2016, and ICAAM 2018 were published in AIP Conference Proceedings (American Institute of Physics) and in some rating scientific journals.

Near East University was pleased to host the fifth conference which was focused on various topics of analysis and its applications, applied mathematics and modeling. The main aim of the International Conferences on Analysis and Applied Mathematics (ICAAM) is to bring mathematicians working in the area of analysis and applied mathematics together to share new trends of applications of mathematics.

In mathematics, the developments in the field of applied mathematics open new research areas in analysis and vice versa. That is why, we planned to find the conference series to provide a forum for researches and scientists to communicate their recent developments and to present their original results in various fields of analysis and applied mathematics. This issue presents papers by authors from different countries: Azerbaijan, Iraq, Russia, Turkey, Turkmenistan, USA, Kazakhstan. Especially we are pleased with the fact that many articles are written by co-authors who work in different countries.

We are confident that such international integration provides an opportunity for a significant increase in the quality and quantity of scientific publications.

Finally, but not least, we would like to thank the Editorial board of the "Bulletin of the Karaganda University - Mathematics", who kindly provided an opportunity for the formation of this special issue.

July 2020

GUEST EDITORS:

Allaberen Ashyralyev

Department of Mathematics, Near East University, Nicosia, TRNC, Mersin 10, Turkey;

Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan;

Peoples’ Friendship University of Russia (RUDN University), Moscow, Russia;

[email protected]

Makhmud A. Sadybekov

Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan;

[email protected]

МАТЕМАТИКА MATHEMATICS

DOI 10.31489/2020M3/5-17 MSC 35N25, 65J22, 39A14

C. Ashyralyyev^1,2, A. Cay³

1Department of Mathematical Engineering, Gumushane University, Gumushane, 29100, Turkey

2Department of Computer Technology, TAU, Ashgabat, 744000, Turkmenistan

3Netas Head Office Yenisehir Mah. Osmanli Bulvari 11 34912 Kurtkoy-Pendik, Istanbul (E-mail: [email protected], [email protected])

Numerical solution to elliptic inverse problem

with Neumann-type integral condition and overdetermination

In modeling various real processes, an important role is played by methods of solution source identification problem for partial differential equation. The current paper is devoted to approximate of elliptic over determined problem with integral condition for derivatives. In the beginning, inverse problem is reduced to some auxiliary nonlocal boundary value problem with integral boundary condition for derivatives. The parameter of equation is defined after solving that auxiliary nonlocal problem. The second order of accuracy difference scheme for approximately solving abstract elliptic overdetermined problem is proposed. By using operator approach existence of solution difference problem is proved. For solution of constructed difference scheme stability and coercive stability estimates are established. Later, obtained abstract results are applied to get stability estimates for solution Neumann-type overdetermined elliptic multidimensional difference problems with integral conditions. Finally, by using MATLAB program, we present numerical results for two dimensional and three dimensional test examples with short explanation on realization on computer.

Keywords: difference scheme, inverse elliptic problem, overdetermination, source identification problem, stability, coercive stability, estimate.

Introduction

Methods of solutions and theory nonlocal boundary value problems (BVPs) for differential equations have been studied by numerous authors (see [1–5, 7–12, 14–16, 18, 19] and references herein).

Let us I is identity operator andA is a selfadjoint and positive definite operator (SAPDO) in an arbitrary Hilbert space H. It is known thatA > δI for some positive numberδ, and the operator C= ^τ₂A+

q

A+^τ2₄^A2) is also SAPDO.

Assume that given function f ∈ C¹([0, T], H), elements φ, η, ζ ∈ H, number λ0 ∈ [0,1]. Denote by [0,1]_τ ={t_i =iτ, i= 1,· · · , N, τ N =T} the uniform grid space with step size τ >0,whereN is a fixed integer number. Let β be known scalar continuous function satisfying condition

N

X

j=1

β

t_j−1 2

τ <1. (1)

C.Ashyralyyev, A. Cay

In the study [10] established well-possedness of elliptic inverse problem with Neumann-type over- determination and integral condition for obtaining a functionu∈C²([0, T], H)∩C([0, T], D(A))and an element p∈H such that







−u⁰⁰(t) + Au(t) = f(t) + p, t∈(0, T), u⁰(0) =φ, u⁰(T) =

T

R

0

β(λ)u⁰(λ)dλ+η, u(λ0) =ζ. (2) Moreover, in [10], the stability inequalities for solution of inverse problem (2) were applied to investigate the following source identificating problem (SIP) for multi dimensional elliptic partial differential equation















−u_tt(t, x) −

n

P

r=1

(ar(x)uxr(t, x))xr+σu(t, x) = f(t, x) + p(x), (t, x)∈(0, T)×Ω, u_t(0, x) =φ(x), u_t(T, x) =

T

R

0

β(γ)u_γ(γ, x)d γ+η(x), u(λ₀, x) =ζ(x), x∈Ω, u(t, x) = 0, (t, x) ∈[0, T]×S.

(3)

HereΩ = (0, T)ⁿ is open cube inRⁿwith boundaryS, Ω = Ω∪S;a_r, ζ, φ, η , f are given sufficiently smooth functions;∀x∈Ω, a_r(x)≥a₀ >0;σ >0,0< λ₀ < T are known numbers.

We denote byR, P, and D,the corresponding operatorsR= (I+τ C)⁻¹, P = (I−R^2N)⁻¹, D= (I+τ C)(2I +τ C)⁻¹C⁻¹.

Now, let us to give some lemmas that will be used in further.

Lemma 1. [8] The following estimates hold:

kR^kk_H→H≤M(δ) (1 +δ¹²τ)^−k, kCR^kk_H→H≤ 1

kτM(δ), k≥1,kP k_H→H≤M(δ), δ >0. (4) Lemma 2.

Suppose that inequality (1) is satisfied, then the operator

G2 =

−3(I −R^2N) + 4 R−R^2N−1

− R²−R^2N−2h 3−τ β

t_N−³

2

(I−R^2N) +

−4−τ β t_N−5

2

R−R^2N−1 +

1−τ β t_N−7

2

+τ β t_N−3

2

R²−R^2N−2 +τ β

t3 2

R^N−1−R^N+1 +

N−3

P

i=2

τh β

t_i+1 2

−β t_i−3

2

i

R^N−i−R^N+i

−

R^N−1−R^N+1−R^N−2+R^N+2h

−

4 +τ β t_N−5

2

R^N−1−R^N+1 +

1−τ β t_N−7

2

+τ β t_N−3

2

R^N−2−R^N+2 +

N−3

P

i=2

τh β

t_i+1 2

−β t_i−3

2

i

Rⁱ−R^2N−i +τ β

t³

2

R−R^2N−1 +τ β

t¹

2

(I−R^2N) i

(5) has an inverse G⁻¹₂ and its norm is bounded, i.e.

kG⁻¹₂ k_H→H≤M(δ). (6) In the paper [8] , for givenv₀ andv_N,the solution of difference scheme

−τ⁻²(v_i+1−2v_i+vi−1) +Av_i=f_i, 1≤i≤N −1 (7) was represented by formula

v_i =P

Rⁱ−R^2N−i

v₀+ R^N−i−R^N+i v_N

−P R^N−i−R^N+i D

×

N−1

P

j=1

R^N−j−R^N+j

fjτ +D

N−1

P

j=1

R^|i−j|−R^i+j

fjτ , 1≤i≤N −1. (8)

Numerical solution to elliptic ...

Letα∈(0,1)is a given number. Introduce notations forC_τ(H), C_τ^α(H),andCτ^α,α(H),the Banach spaces ofH-valued grid functions w_τ ={w_k}^N−1_k=1 with the corresponding norms,

kw_τk_C

τ(H) = max

1≤k≤N−1kw_kk_H, kw_τk_Cα

τ(H) = sup

1≤k<k+n≤N−1

(nτ)^−αkw_k+n−w_kk_H+kw_τk_C

τ(H), kw_τk_C^α,α

τ (H)=kw_τk_C

τ(H)+ sup

1≤k<k+n≤N−1

(1−kτ)^α(nτ)^−α(kτ +nτ)^αkw_k+n−w_kk_H.

In the current study, we construct the second order accuracy difference scheme (ADS) for approximately solution of inverse problem (2) and study well-posedness of difference problem. Then, we discuss the second order ADS for SIP (3).

The second order of ADS for SIP (3) Now, we study second order of ADS











−τ⁻²(u_k+1−2u_k+uk−1) +Au_k =f_k+p, f_k=f(t_k),1≤k≤N−1,

−3u₀+ 4u₁−u₂ = 2τ φ,3u_N−4u_N−1+uN−2 =

N−1

P

i=1

τ β t_i−1

2

(u_i+1−ui−1) + 2τ η, u_l+µ(u_l+1−u_l) =ζ

µ= ^λ_τ⁰ −l

(9)

for approximate solution inverse problem (2).

Theorem 1. Let us φ, η, ζ ∈D(A),and f_τ ∈C_τ(H) and inequality (1) is satisfied. Then, solution

{u_k}^N_k=1⁻¹, p

of difference problem (9) exists inCτ(H)×Hand the next stability estimates for solution

{u_k}^N−1_k=1

Cτ(H) ≤ M(δ)

kφk_H+kζk_H +kηk_H+kf_τk_C

τ(H)

, (10)

A⁻¹p

H ≤ M(δ)

kφk_H+kζk_H +kηk_H+kf_τk_C

τ(H)

(11) are fulfilled.

Proof. Firstly, by using

uk=vk+A⁻¹p, (12)

we get auxiliary difference problem for unknowns {v_k}^N_k=0 :



















−τ⁻²(v_k+1−2v_k+vk−1) +Av_k=f_k , 1≤k≤N −1,

−3v₀+ 4v1−v2 = 2τ φ,

3−τ β

t_N−³

2

v_N +

−4−τ β

t_N−⁵

2

vN−1

+

1−τ β

t_N−⁷

2

+τ β

t_N−³

2

vN−2+

N−3

P

i=2

τ h

β

t_i+¹

2

−β

t_i−³

2

i vi

+τ β t³

2

v₁ +τ β t¹

2

v₀ = 2τ η.

(13)

We seek solution of (13) by (8). By using (8), from first condition of difference problem (13), we get equation

−3(I−R^2N) + 4 R−R^2N−1

− R²−R^2N−2 v0

+

4 R^N−1−R^N+1

− R^N−2−R^N+2

v_N =F₁, (14)

for unknowns v0 and vN,where

F1= 2τ(I−R^2N)φ+ 4 R^N−1−R^N+1 D

N−1

P

j=1

R^N−j−R^N+j

fjτ −4(I−R^2N)D

×

N−1

P

j=1

R^|1−j|−R^1+j

fjτ − R^N−2−R^N+2 D

N−1

P

j=1

R^N−j−R^N+j fjτ +(I−R^2N)D

N−1

P

j=1

R^|2−j|−R^2+j fjτ.

C.Ashyralyyev, A. Cay

From integral condition follows the next equation

3−τ β t_N−³

2

(I −R^2N)v_N +

−4−τ β t_N−⁵

2

R^N−1−R^N+1

v0+ R−R^2N−1 v_N +

1−τ β t_N−7

2

+τ β t_N−3

2

R^N−2−R^N+2

v₀+ R²−R^2N−2 v_N +

N−3

P

i=2

τh β

t_i+1 2

−β t_i−3

2

i

Rⁱ−R^2N−i

v₀+ R^N−i−R^N+i v_N +τ β

t³

2

R−R^2N−1

v0+ R^N−1−R^N+1 vN

+τ β

t¹

2

(I−R^2N)v0 =F2

(15) for unknowns v₀ and v_N,where

F₂ =

−4−τ β t_N−⁵

2

"

R−R^2N−1 D

N−1

P

j=1

R^N−j −R^N+j

f_jτ −(I −R^2N)D

×^N

−1

P

j=1

R^|N−1−j|−R^N−1+j f_jτ

# +

1−τ β t_N−⁷

2

+τ β t_N−³

2

×

"

R²−R^2N−2 D

N−1

P

j=1

R^N−j−R^N+j

f_jτ−(I−R^2N)D

N−1

P

j=1

R^|N−2−j|−R^N−2+j f_jτ

#

−^N

−3

P

i=2

τh β

t_i+1 2

−β t_i−3

2

i

"

R^N−i−R^N+i D

N−1

P

j=1

R^N−j−R^N+j f_jτ

−(I−R^2N)D

N−1

P

j=1

R^|i−j|−R^i+j f_jτ

#

−τ β t3

2

R^N−1−R^N+1 D

×

N−1

P

j=1

R^N−j−R^N+j

fjτ−(I−R^2N)D

N−1

P

j=1

R^|1−j|−R^1+j

fjτ + 2τ(I −R^2N)η

# .

Thus, determinant operatorG₂of linear system equatıon (14), (15) has bounded inverseG⁻¹₂ .Therefore solution of linear system equatıon (14), (15) is defined by

v0 =G⁻¹₂ nh

3−τ β

t_N−³

2

(I−R^2N) + −4−τ β tN−2−^τ₂

R−R^2N−1 +

1−τ β t_N−⁷

2

+τ β t_N−³

2

R²−R^2N−2 +

N−3

P

i=2

τh β

t_i+¹

2

−β t_i−³

2

i

R^N−i−R^N+i

+τ β t³

2

R^N−1−R^N+1

×

"

2τ(I−R^2N)φ+ 4 R^N−1−R^N+1 D

N−1

P

j=1

R^N−j−R^N+j

f_jτ−4(I−R^2N)D

×

N−1

P

j=1

R^|1−j|−R^1+j

fjτ− R^N−2−R^N+2 D

N−1

P

j=1

R^N−j−R^N+j fjτ +(I−R^2N)D

N−1

P

j=1

R^|2−j|−R^2+j f_jτ

#

− R^N−1−R^N+1−R^N−2+R^N+2

× (

2τ(I−R^2N)η+

−4−τ β t_N−5

2

"

R−R^2N−1 D

N−1

P

j=1

R^N−j−R^N+j f_jτ

× −(I−R^2N)D

N−1

P

j=1

R^|N−1−j|−R^N−1+j f_jτ

# +

1−τ β t_N−7

2

+τ β t_N−3

2

×

"

R²−R^2N−2 D

N−1

P

j=1

R^N−j−R^N+j

f_jτ−(I−R^2N)D

N−1

P

j=1

R^|N−2−j|−R^N−2+j f_jτ

#

Numerical solution to elliptic ...

−

N−3

P

i=2

τ

α t_i+1− ^τ₂

−α ti−1−^τ₂

"

R^N−i−R^N+i D

N−1

P

j=1

R^N−j−R^N+j f_jτ

−(I −R^2N)D

N−1

P

j=1

R^|i−j|−R^i+j f_jτ

#

−τ α t₂−^τ₂ R^N−1−R^N+1 D

×^N

−1

P

j=1

R^N−j−R^N+j

f_jτ−(I−R^2N)D

N−1

P

j=1

R^|1−j|−R^1+j f_jτ

##)) ,

(16)

and

v_N =G⁻¹₂

−3(I−R^2N) + 4 R−R^2N−1

− R²−R^2N−2

2τ(I−R^2N)η + −4−τ β tN−2−^τ₂

"

R−R^2N−1 D

N−1

P

j=1

R^N−j−R^N+j fjτ

−(I−R^2N)D

N−1

P

j=1

R^|N−1−j|−R^N−1+j fjτ

#

+ 1−τ β tN−3−^τ₂

+τ β tN−1−^τ₂

×

"

R²−R^2N−2 D

N−1

P

j=1

R^N−j−R^N+j

fjτ−(I−R^2N)D

N−1

P

j=1

R^|N−2−j|−R^N−2+j fjτ

#

−

N−3

P

i=2

τ h

β

t_i+¹

2

−β

t_i−³

2

i

"

R^N−i−R^N+i D

N−1

P

j=1

R^N−j−R^N+j

fjτ−(I−R^2N)D

N−1

P

j=1

R^|i−j|−R^i+j fjτ

#

−τ β t³

2

"

R^N−1−R^N+1 D

N−1

P

j=1

R^N−j −R^N+j

fjτ −(I−R^2N)D

N−1

P

j=1

R^|1−j|−R^1+j fjτ

#

−h

−

4 +τ β t_N−⁵

2

R^N−1−R^N+1 +

1−τ β

t_N−⁷

2

+τ β

t_N−³

2

R^N−2−R^N+2 +τ β

t¹

2

(I−R^2N) +

N−3

P

i=2

τ h

β

t_i+¹

2

−β

t_i−³

2

i

Rⁱ−R^2N−i +τ β

t³

2

R−R^2N−1

×

"

2τ(I−R^2N)φ+ 4 R^N−1−R^N+1 D

N−1

P

j=1

R^N−j −R^N+j fjτ

−4(I−R^2N)D

N−1

P

j=1

R^|1−j|−R^1+j f_jτ

− R^N−2−R^N+2 D

N−1

P

j=1

R^N−j−R^N+j

fjτ + (I−R^2N)D

N−1

P

j=1

R^|2−j|−R^2+j fjτ

#) .

(17) Thus solution of difference problem (13) exists and it is defined by (8) with the corresponding v₀ and vN via (16) and (17). From (8), (16),(17), estimates (4), (6), it follows that for solution of difference problem (13) stability estimates

{v_k}^N−1_k=1 Cτ(H)

≤M(δ)

kφk_H +kζk_H+kηk_H +kf_τk_C

τ(H)

, (18)

{Av_k}^N_k=1⁻¹

Cα,α τ (H)

+

n_v

k+1−2v_k+vk−1

τ²

oN−1 k=1

Cα,α τ (H)

≤M(δ) 1

α(1−α)kf_τk

Cα,α

τ (H)+kAζk_H+kAφk_H +kAηk_H .

(19)

are fulfilled. (12) and estimates (18) permit us to get estimates estimates (11) (10) and (19).

C.Ashyralyyev, A. Cay

Theorem 2. Let us f_τ ∈ Cτ^α,α(H), and φ, ζ, η ∈ D(A) and inequality (1) is satisfied. Then, for solution

{u_k}^N−1_k=1 , p

of difference problem (9) the coercive stability inequality

n_u

k+1−2u_k+uk−1

τ²

oN−1 k=1

Cτ^α,α(H)

+

{Au_k}^N−1_k=1

Cα,α τ (H)

+kpk_H

≤M(δ) 1

α(1−α)kf_τk

Cα,α

τ (H) +kAζk_H +kAφk_H +kAηk_H (20)

is valid.

The proof of inequality (20) is based on formulas (8), (12), (16), (17), and (19).

Approximation of (3)

Denote by Ωe_h =

x= (h₁m₁, ..., h_nm_n);m= (m₁, ..., m_n), m_i= 0, M_i, h_iM_i = 1, i= 1, n , Ω_h=Ωe_h∩Ω, S_h=Ωe_h∩S

and byA^x_h difference operator

A^x_hu^h(x) =−

n

X

i=1

a_i(x)u^h_xi(x)

xi,ji

+σu^h(x)

acting in the space of grid functionsu^h(x),satisfying boundary conditionu^h(x) = 0 for all x∈S_h. In the beginning, by using approximation in variable x and later by approximation in variablet, one can get the following difference scheme for approximately solution of SIP (3):

−τ⁻² u^h_k+1(x)−2u^h_k(x) +u^h_k−1(x)

+Au^h_k(x) =f_k^h(x) +p^h(x), 1≤k≤N −1, x∈Ωh

−3u^h₀(x) + 4u^h₁(x)−u^h₂(x) =τ φ^h(x), u^h_l(x) +µ(u^h_l+1(x)−u^h_l(x)) =ζ^h(x) 3u^h_N(x)−4u^h_N−1(x) +u^h_N−2(x) =

N−1

P

i=1

τ α ti−^τ₂

u^h_i+1(x)−u^h_i(x)

+ 2τ η^h(x), x∈Ωe_h. .

(21)

Let L_2h =L₂(Ωe_h) and W_2h² =W₂²(Ωe_h), the Banach spaces of the grid functions u^h(x) ={u(h₁m1,· · ·, hnmn)} defined on Ωe_h,equipped with the corresponding norms

u^h

L2h= (P

x∈Ωeh|u^h(x)|²h1· · ·hn)^1/2,

u^h

W_2h² = u^h

L2h+ +(P

x∈Ωeh

Pn i=1

(u^h(x))xixi, mi

2h1· · ·hn)^1/2.

Theorem 3. Assume that (1) is valid, f_τ ∈ C_τ^α,α(L_2h), and φ^h, η^h, ζ^h ∈ D(A^x_h)∩L_2h. Then, the solution of difference problem (21) exists and for solution the stability estimates hold:

u^h_k ^N₁ ⁻¹ _C

τ(L2h)≤M(δ) [

φ^h

L2h+ η^h

L2h+ ζ^h

L2h+kf_τk_C

τ(L_2h)

,

p^h

L2h≤M(δ)

ζ^h

W_2h² + η^h

W_2h² + φ^h

W_2h² +_α(1−α)¹ kf_τk_C^α,α

τ (L2h)

.

Theorem 4.Assume that (1) is true, fτ ∈Cτ^α,α(W_2h² ), andφ^h, η^h, ζ^h ∈D(A^x_h)∩W_2h² . Then, for the solution of difference problem (21) the coercive stability estimate obeys

u^h_k+1−2u^h_k+u^k_k−1

τ² )

N−1 1

Cτ(L2h)

+

u^h_k ^N−1₁

Cτ(W_2h² )+ p^h

L2h

≤M(δ) [

ζ^h

W_2h² + η^h

W_2h² + φ^h

W_2h² +_α(1−α)¹ kf_τk_C^α,α

τ (W_2h² )

.

Numerical solution to elliptic ...

The proofs of Theorems 3 and 4 are based on the symmetry property of the operatorA^x_h in the Hilbert spaceL_2h and the corresponding theorem in [20] on the coercivity stability inequality for the solution of the elliptic difference problem inL2h with first kind boundary condition.

Test examples

In the present section, we illustrate computed results for twodimensional and threedimentional examples of inverse elliptic problem with Neumann-type overdetermination and integral condition. All computed results are carried out by using MATLAB.

2D example Notice that pair functions (p(x), u(t, x)) = ( π²+ 1

sin(πx), e^−t +t+ 1

sin(πx)) is exact solution of the following 2D overdetermined elliptic problem with integral boundary condition:











−u_tt(t, x)−u_xx (t, x) +u(t, x) =f(t, x) +p(x), t, x∈(0,1), ut(0, x) = 0, u(0.3, x) =ζ(x), ut(1, x) =

R1 0

e^−λu_λ(λ)dλ +η(x), x∈[0,1], u(t,0) = 0, u(t,1) = 0, t∈[0,1],

(22)

where

f(t, x) =

−e^−t+ π²+ 1

e^−t+t sin(πx), ζ(x) = e^−0.3 + 1.3

sin(πx) η(x) =₁

2−¹₂e⁻²

sin(πx).

The notation [0,1]_τ×[0,1]_h means the set of grid points

[0,1]τ×[0,1]h ={(t_i, xn) :ti=iτ, i= 0, N , xn=nh, n= 0, M}, which depends on the small parameters τ and h such thatN τ = 1, M h= 1.Let us

l0 =

0.3τ⁻¹

, µ0 = 0.3τ⁻¹−l0;φn= 0, ηn=η(xn), ζn=ζ(xn), n= 0, M; f_n^k=f(t_k, x_n), k= 0, N , n= 0, M .

To approximately soving (22), we use algorithm which contains three stages. Firstly, we find approximately solution of auxiliary NBVP















τ⁻² v_n^k+1−2v_n^k+v_n^k−1

+h⁻² v^k_n+1−2v^k_n+v_n−1^k

−v_n^k=−f(t_k, x_n), k= 1, N −1, n= 1, M −1,

v^k₀ =v^k_M = 0, k= 0, N , −3v⁰_n+ 4v_n¹−v_n² = 0, 3v_n^N −4v_n^N−1+v_n^N−2=

N−1

P

j=1 τ

2e⁻(^tj−^τ

2)

v^j+1n −vn^j−1+vn^j −v^j−2n

+ 2τ ηn, n= 0, M .

(23)

Secondly, we find p_n.It is caried out by

pn=−_h¹₂[(ζn+1−(µ0v_n+1^l0+1−(µ0−1)v^l_n+1⁰ ))−2(ζn−(µ0v^l_n⁰⁺¹−(µ0−1)v^l_n⁰))

+ (ζn−1−(µ0v_n−1^l0+1−(µ0−1)v^l_n−1⁰ ))] +ζn−(µ0v^l_n⁰⁺¹−(µ0−1)v_n^l0), , n= 1, M −1.

Difference problem (23) can be rewritten in the matrix form

Avn+1+Bvn+Cvn−1=Ig⁽ⁿ⁾, n= 1, M −1, v₀=−→

0, v_M =−→

0. (24)

C.Ashyralyyev, A. Cay

Here,A, B, C, Iare(N+1)×(N+1)square matrices, andI is identity matrix,v_s, s=n−1, n, n+1, g⁽ⁿ⁾ are column matrices with(N+ 1) rows,v_s=

v_s⁰ . . . v_s^N t

.Denote by a= 1

h², c= 1

h², q=− 2 h² − 2

τ² −1, r= 1 τ². Then,

A_n=diag(0, a, a, . . . , a,0), C_n=A_n, g_k⁽ⁿ⁾=−f(t_k, x_n), k= 1, N −1, n= 1, M −1,

b_i,i =q, bi−1,i=r, bi,i−1 =r, i= 2, N , b_1,1 =−3, b_1,2 = 4, b_1,3 =−1, b_N+1,N+1= 2τ

e

tN−3 2

4 +e^−tN−32

−3, b_N+1,N = 2τ

e^−tN−5

4 +^e

−tN−3 2

4 −e^−tN−12

+ 4, bN+1,N−1 = ^{τ e}

−tN−7 2

2 +^{τ e}

−tN−5 2

2 −^{τ e}

−tN−3 2

2 −1, b_N+1,1 = 2τ

−^e

−t3 2

4 −e^−t12

, b_N+1,2 = 2τ

−^e

−t3 2

4 − ^e

−t5 2

4 +e^−t12

,

bN+1,3 = ^{τ e}

−t3 2

2 − ^{τ e}

−t5 2

2 − ^{τ e}

−t7 2

2 , b_N+1,j = ^τ₂

e^−tj−32 +e^−tj−12 −e^{−tj+ 12} −e^{−tj+ 32}

, j= 4, ..., N −2;

bi,j = 0, for otheriand j;g⁰_n= 2τ φn, g^N_n = 2τ ηn, n= 1, M−1.

To solve (24), we use modified Gauss elimination method.

Thirdly, we define

u^k_n by u^k_n=v_n^k+ζ_n− µ₀v^l_n⁰⁺¹−(µ₀−1)v^l_n⁰ .

Errors are presented in Tables 1-3 for second order ADS in case N=M=10,20, 40, 80, 160 and 320.

It can be seen from Tables 1-3 when N, M are increased two times that errors are decreased with approximately ratio ¹₄.

T a b l e 1 Test example (22) - errorv

DS\(N, M) (10,10) (20,20) (40,40) (80,80) (160,160) (320,320) 2nd order of ADS 6.29×10⁻³ 1.57×10⁻³ 3.93×10⁻⁴ 9.84×10⁻⁵ 2.46×10⁻⁵ 6.15×10⁻⁶

T a b l e 2 Test example (22) - erroru

DS \ (N, M) (10,10) (20,20) (40,40) (80,80) (160,160) (320,320) 2nd order of ADS 3.13×10⁻⁴ 7.95×10⁻⁵ 2.02×10⁻⁵ 5.10×10⁻⁶ 1.28×10⁻⁶ 3.22×10⁻⁷

T a b l e 3 Test example (22) - errorp

Appr. \ (N, M) (10,10) (20,20) (40,40) (80,80) (160,160) (320,320) 2nd order 5.03×10⁻³ 1.28×10⁻³ 3.21×10⁻⁴ 8.06×10⁻⁵ 2.02×10⁻⁵ 5.05×10⁻⁶

Numerical solution to elliptic ...

3D example

Now, consider the three dimensional inverse elliptic problem with integral condition















−u_tt(t, x, y)−u_xx(t, x, y)−u_yy(t, x, y) +u(t, x, y) =f(t, x, y) +p(x, y), x, y, t∈(0,1), u(t,0, y) =u(t,1, y) = 0, y, t∈[0,1], u(t, x,0) =u(t, x,1) = 0, x, t∈[0,1],

u_t(0, x, y) =φ(x, y), u(0.6, x, y) =ζ(x, y), u_t(1, x, y)−

1

R

0

e^−λu_λ(λ, x, y)dλ =η(x, y), x, y∈[0,1],

(25)

where

f(t, x, y) = 2π²e^−tq(x, y), φ(x, y) =−q(x, y), η(x, y) =

−e⁻¹+¹₃ e^−0.6+e^−1.2

q(x, y), ζ(x, y) =

e⁻³⁵ + 1

q(x, y),q(x, y) = sin(πx) sin(πy) It is clear that pair funcions p(x, y) = 2π²+ 1

q(x, y) and u(t, x, y) = e^−t+ 1

q(x, y) is exact solution of (25).

Denote by [0,1]_τ×[0,1]_h ×[0,1]_h set of grid points depending on the small parameters τ andh [0,1]_τ×[0,1]²_h ={(t_i, x_n, y_m) :t_i =iτ, i= 0, N , x_n=nh, n= 0, M ,

ym=mh, m= 0, M , τ N = 1, hM = 1}.

Let us l0 =

0.3τ⁻¹

, µ0= 0.3τ⁻¹−l0 , φm,n =φ(xn, ym), ηm,n =η(xn, ym), ζm,n =ξ(xn, ym), n= 0, M , m= 0, M;f_m,nⁱ =f(t_i, x_n, y_m), i= 0, N , n= 0, M, m= 0, M .

Firstly, difference scheme for approximate solution of NBVP can be written in the following form:



























−τ⁻² v^k+1_m,n −2v_m,n^k +v^k−1_m,n

−h⁻² v_m,n+1^k −2v^k_m,n+v^k_m,n−1

−h⁻² v_m+1,n^k −2v^k_m,n+v_m−1,n^k

+v^k_m,n =f_m,n^k , k= 1, N −1, n= 1, M −1, m= 1, M −1,

v_0,n^k =v_M,n^k =v^k_m,n=v_m,M^k = 0, k = 0,· · · , N, n= 1, M−1, m= 1, M −1,

−3v⁰_m,n+ 4v_m,n¹ −v_m,n² = 2τ φ_m,n, 3v^N_m,n−4v_m,n^N−1+v_m,n^N−2

=

N−1

P

j=1 τ

2e⁻(^tj−^τ

2)

v^j+1m,n−vm,n^j−1+vm,n^j −vm,n^j−2

+ 2τ ηm,n, n= 1, M−1, n= 1, M −1.

(26)

Secondly, calculation ofpn n= 1, M −1, m= 1, M −1

is caried out by pm,n =−_h¹2

nh

ζm,n+1−

µ0v_m,n+1^l0+1 −(µ0−1)v_m,n+1^l0 i

−2

ζm,n− µ0v_m,n^l0+1−(µ0−1)v_m,n^l0 +h

ζm,n−1−

µ₀v_m,n−1^l0+1 −(µ₀−1)v_m,n−1^l0 io

−_h¹₂ nh

ζ_m+1,n−

µ₀v_m+1,n^l0+1 −(µ₀−1)v_m+1,n^l0 i

−2

ζ_m,n− µ₀v_m,n^l0+1−(µ₀−1)v_m,n^l0 +h

ζm−1,n−

µ₀v^l_m−1,n⁰⁺¹ −(µ₀−1)v^l_m−1,n⁰ io . Thirdly, we calculate

u^k_n by

u^k_m,n=v^k_m,n+ζ_m,n−

µ₀v^l_m,n⁰⁺¹−(µ₀−1)v_m,n^l0 .

Difference problem (26) can be rewritten in the matrix form (24). In this case, gn is a column matrix with(N + 1)(M+ 1) elements,A, B, C, I are square matrices with(N+ 1)(M+ 1)rows and columns, and I is the identity matrix, v_s is column matrix with (N + 1)(M+ 1)elements such that

v_s =

v_0,s⁰ · · · v_0,s^N v⁰_1,s · · · v^N_1,s · · · v⁰_M,s · · · v_M,s^N t

, s=n−1, n, n+ 1.

C.Ashyralyyev, A. Cay

Denote by

a= 1

h², q= 1 + 2 τ² + 4

h², r= 1 τ². Then,

A=C=















O O · · · O O O E · · · O O

· · · . .. · · · · O O · · · E O O O · · · O O















, B =















Q O · · · O O O D · · · O O

· · · . .. · · · · O O · · · D O O O · · · O Q













 ,

E=diag(0, a, a, . . . , a,0), Q=I_(N+1)×(N+1), O=O_(N+1)×(N+1), g^k_m,n=−f(t_k, xn, ym), k= 1, N −1, n= 1, M−1, m= 1, M −1, d_i,i=q, di−1,i=r, di,i−1 =r, i= 2, N , d_1,1 =−3, d<