(1)(2)ISSN 2663–5011(Online) Индексi 74618 Индекс 74618 ҚАРАҒАНДЫ УНИВЕРСИТЕТIНIҢ ХАБАРШЫСЫ ВЕСТНИК КАРАГАНДИНСКОГО УНИВЕРСИТЕТА BULLETIN OF THE KARAGANDA UNIVERSITY МАТЕМАТИКА сериясы Серия МАТЕМАТИКА MATHEMATICS Series Сәуiр–мамыр–маусым 30 маусым 2021

ISSN 2663–5011(Online)

Индексi 74618

Индекс 74618

ҚАРАҒАНДЫ

УНИВЕРСИТЕТIНIҢ

ХАБАРШЫСЫ

ВЕСТНИК

КАРАГАНДИНСКОГО УНИВЕРСИТЕТА

BULLETIN

OF THE KARAGANDA UNIVERSITY

МАТЕМАТИКА сериясы Серия МАТЕМАТИКА

MATHEMATICS Series

№ 2(102)/2021

Сәуiр–мамыр–маусым 30 маусым 2021 ж.

Апрель–май–июнь 30 июня 2021 г.

April–May–June June, 30

, 2021

1996 жылдан бастап шығады Издается с 1996 года

Founded in 1996 Жылына 4 рет шығады

Выходит 4 раза в год Published 4 times a year

Қарағанды, 2021

Караганда, 2021

Karaganda 2021

Candidate of Physics and Mathematics sciences

N.T. Orumbayeva

Responsible secretary PhD

O.I. Ulbrikht Editorial board

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

M.I. Ramazanov, E.S. Smailov, A.R. Yeshkeyev,

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

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

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

Academician of NAS RK, Professor, Dr. techn. sciences, Satbayev University, Almaty (Kazakhstan);

Academician of NAS RK, Professor, Dr. phys.-math. sciences, IMMM, Almaty (Kazakhstan);

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

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

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

Professor, Xinjiang University, Urumchi (China);

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

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

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

PhD, Buketov KU, Karaganda (Kazakhstan);

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

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

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

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

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

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

Professor, Dr. phys.-math. sciences, Buketov KU, Karaganda (Kazakhstan)

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Zh.Т. Nurmukhanova, S.S. Balkeyeva, D.V. Volkova

G.K. Kalel

Bulletin of the Karaganda University. Mathematics series.

ISSN 2518-7929 (Print). ISSN 2663–5011 (Online).

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MATHEMATICS

Ashyralyev A., Sadybekov M.Recent advances in PDE and their applications. Preface . . . . Ashyralyev A., Erdogan A.S. Parabolic time dependent source identification problem with involution and Neumann condition . . . . Ashyralyev A., Ashyralyyeva M., Batyrova O. On the boundedness of solution of the second order ordinary differential equation with damping term and involution . . . . Ashyralyev A., Ibrahim S., Hincal E.On stability of the third order partial delay differential equation with involution and Dirichlet condition . . . . Ashyralyev A., Urun M.On the Crank-Nicolson difference scheme for the time-dependent source identi- fication problem. . . . Ashyralyyev C., Akyuz G. On fourth order accuracy stable difference scheme for a multi-point overdetermined elliptic problem . . . . Cavusoglu S., Mukhtarov O.Sh. A new finite difference method for computing approximate solutions of boundary value problems including transition conditions. . . . Kanetov B.E., Baidzhuranova A.M.Paracompact-type mappings . . . . Kerimbekov A., Ermekbaeva A.T., Seidakmat kyzy E.On the solvability of the tracking problem in the optimization of the thermal process by moving point controls . . . . Mardanov M.J., Mammadov R.S., Gasimov S.Yu., Sharifov Ya.A.Existence and uniqueness results for the first-order non-linear impulsive integro-differential equations with two-point boundary conditions Muradov F.Kh.Ternary semigroups of topological transformations . . . . Mustapha U.T., Hincal E., Yusuf A., Qureshi S., Sanlidag T., Muhammad S.M., Kaymakamzade B., Gokbulut N.Transmission dynamics and control strategies of COVID-19: a modelling study. . . . Mustapha U.T., Sanlidag T., Hincal E., Kaymakamzade B., Muhammad S.M., Gokbulut N.Modelling the effect of horizontal and vertical transmissions of HIV infection with efficient control strategies . . . . Sinsoysal B., Rasulov M., Yener O. Grid method for solution of 2D Riemann type problem with two discontinuities having in initial condition . . . . Utebaev D., Utepbergenova G.Kh., Tleuov К.О.On convergence of schemes of finite element method of high accuracy for the equation of heat and moisture transfer . . . . Yildirim O., Caglak S. On the Lie symmetries of the boundary value problems for differential and difference sine-Gordon equations . . . . INFORMATION ABOUT AUTHORS . . . .

4 5 16 25 35 45 54 62 67 74 84 92 106 115 129 142 154

MSC 34K28, 35A35, 35G15, 35J15, 35J25, 35K10, 35K60, 35L05, 35L10, 35L35, 35M10, 35M12, 35N25, 35R30, 35S15, 39A14, 42A10, 42B10, 45J05, 45J99, 47A62, 47B39, 54E15, 54D20, 58J05, 58J32, 58J99, 58JXX, 60H30, 65M06, 65H10, 65H30, 65J22, 65N06, 82D75

Guest-Editors: A. Ashyralyev

, M. Sadybekov

1 Near East University, Nicosia, TRNC, Mersin 10, Turkey;

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

3 Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan (E-mail: aallaberen@gmail.com, sadybekov@math.kz)

Recent advances in PDE and their applications Preface

This issue is a collection of 15 selected papers of foreign and national scientists. All these have been accepted after peer-reviewing and contain numerous new results in the fields of construction and investigation of solutions of well-posed and ill-posed boundary value problems for partial differential equations and their related applications. The authors of the selected papers are from different countries: Turkey, Kazakhstan, USA, Russian Federation, Azerbaijan, Kirgizistan, Uzbekistan, Turkmenistan, Pakistan and Nijerya. We are especially pleased with the fact that many articles are written by co-authors who work in different universities around the world.

Keywords:partial differential equations, hyperbolic-parabolic equations, integro-differential equations, boun- dary value problem, Dirichelt problem, well-posedness, regular solutions, numerical methods and solutions, difference scheme, involution, stability.

Guest-Editors:A. Ashyralyev andM. Sadybekov April 15, 2021

*Corresponding author.

E-mail: aallaberen@gmail.com

MSC 35N25, 65J22, 39A14

A. Ashyralyev

, A.S. Erdogan

1Near East University, Nicosia, Turkey;

2Peoples’ Friendship University of Russia (RUDN University), Moscow, Russia;

3Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan;

4Palm Beach State College, FL, USA

(E-mail: allaberen.ashyralyev@neu.edu.tr, aserdogan@gmail.com)

Parabolic time dependent source identification problem with involution and Neumann condition

A time dependent source identification problem for parabolic equation with involution and Neumann condi- tion is studied. The well-posedness theorem on the differential equation of the source identification parabolic problem is established. The stable difference scheme for the approximate solution of this problem and its stability estimates are presented. Numerical results are given.

Keywords:well-posedness, coercive stability, source identification, exact estimates, boundary value problem.

Introduction

The theory and applications of source identification problems (SIPs) for partial differential equations have been studied and for references we refer to articles [1–9] and the references given therein. Also, numerous source identification problems for hyperbolic-parabolic equations and their applications have been investigated too (see, e.g., [10–13] and the references given therein). In the last decade partial differential equations with involutions were investigated by several authors including Ashyralyev and Sarsenbi [14–17]. However, source identification problems for parabolic equations with involution still need more investigating.

The present paper is devoted to the study of a time SIP for parabolic equation with involution and Neumann condition. The stability theorem on the differential equation of the source identification parabolic problem is proved. The stable difference scheme (DS) for the approximate solution of this problem is constructed.

Furthermore, stability estimates for the DS of the time source identification parabolic problem are established.

Numerical results are provided.

Stability and coercive stability of the differential problem We consider the time SIP















ut(t, x)−(a(x)ux(t, x))_x−β(a(−x)ux(t,−x))_x+δu(t, x)

=p(t)q(x) +g(t, x), −l < x < l, 0< t < T, u(0, x) =ϕ(x), −l≤x≤l,

ux(t,−l) =ux(t, l) = 0, Z l

0

u(t, x)dx=γ(t), 0≤t≤T

(1)

for the one dimensional parabolic differential equation with involution and Neumann boundary condition.

Throughout this paper, we assume that the following conditions hold

a≥a(x) =a(−x)≥a >0, x∈(−`, `), a−a|β| ≥0, δ≥0, q⁰(−l) =q⁰(l) = 0,

Z l 0

q(x)dx6= 0.

Under compatibility conditions, identification problem (1) has a unique solution (u(t, x), p(t))for the smooth functionsg(t, x),(t, x)∈(0, T)×(−l, l), a(x), q(x), x∈(−l, l)andγ(t), t∈[0, T], ϕ(x), x∈[−l, l].

*Corresponding author.

E-mail: aserdogan@gmail.com

Assume that H is a Hilbert space andAis the self-adjoint positive-definite operator defined by the formula Az=− d

dx

a(x)dz(x) dx

−β d dx

a(−x)dz(−x) dx

+δz(x) (2)

with domain

D(A) ={z:z, z⁰⁰∈L₂[−l, l], z⁰(−l) =z⁰(l) = 0}.

Here and in the rest of this paper,C₀^α([0, T], H) (0< α <1)stands for Banach spaces of all abstract continuous functions ϕ(t) defined on [0, T] with values in H satisfying a H¨older condition with weight t^α for which the following norm is finite

kϕk_Cα

0([0,T],H)=kϕk_C([0,T],H)+ sup

0≤t<t+τ≤T

(t+τ)^αkϕ(t+τ)−ϕ(t)k_H

τ^α .

Here, C([0, T], H)stands for the Banach space of all abstract continuous functionsϕ(t)defined on [0, T]with values inH equipped with the norm

kϕk_C([0,T],H)= max

0≤t≤Tkϕ(t)k_H.

Moreover, let the Sobolev space W₂²[−`, `] be defined as the set of all functionsv(x)defined on[−`, `]such that bothv(x)andv⁰⁰(x)are locally integrable inL₂[−`, `], equipped with the norm

kvk_W2 2[−`,`]=





`

Z

−`

|v(x)|²dx





1/2

+





`

Z

−`

|v⁰⁰(x)|²dx





1/2

.

Theorem 1. Assume that f(t, x) and ζ(t) are continuously differentiable functions. Then the SIP (1) has a unique solution u ∈ C(L2[−l, l]) and p ∈ C[0, T], and for the solution of SIP (1) the following stability estimates hold

ku_tk_C(L

2[−l,l])+kuk_C(^W₂²[−l,l]) +kpk_C[0,T]6M(q, δ)h kϕk_W2

2[−l,l]

+kg(0,·)k_L

2[−l,l]+kg_tk_C(L

2[−l,l])+kζ_tk_C[0,T]i .

Theorem 2.Assume thatg(t, x)andζ(t) are continuously differentiable functions andζ_t(t)is satisfying a H¨older condition with the ; weightt^α. Then the SIP (1) has a unique solutionu∈C₀^α([0, T], L₂[−l, l])and p∈C₀^α[0, T]. For the solution of SIP (1) the following coercive stability estimates hold:

kutk_Cα

0([0,T],L₂[−l,l])+kuk_Cα

0(^[0,T],W₂²[−l,l]) +kpk_Cα

0[0,T] 6M(q, δ)h kϕk_W2

2[−l,l]

+_α(1−α)¹ kgk_Cα

0([0,T],L₂[−l,l])+kζtk_Cα 0[0,T]

i. Proof. Denoted as

u(t, x) =w(t, x) +η(t)q(x), (3)

where

η(t) = Z t

0

p(s)ds, η(0) = 0 (4)

andw(t, x)is the solution of the following problem































wt(t, x)−(a(x)wx(t, x))_x+δw(t, x)

=g(t, x) +η(t) [(a(x)qx(x))_x−δq(x)], x∈(−l, l), t∈(0, T),

w(0, x) =ϕ(x), x∈[−l, l], w_x(t,−l) =w_x(t, l) = 0, t∈[0, T].

(5)

Applying the condition

Z l 0

u(t, x)dx=ζ(t) and formula (3), we can write

η(t) =q −ζ(t) + Z l

0

w(t, x)dx

!

, (6)

where

q= 1

Rl

0q(x)dx . Applying formulas (4) and (6), we get

p(t) =q −ζ⁰(t) + Z l

0

w_t(t, x)dx

!

. (7)

ApplyingRl

0q(x)dx6= 0, we get the estimate

|p(t)|6K₁(q)

|ζt(t)|+||wt(t,·)||L₂[−l,l]

(8)

for each t∈[0, T].From (7) and (8) follows it kpk_C([0,T])≤K1(q)h

kζtk_C[0,T]+kwtk_C([0,T],L

2[−l,l])

i

, (9)

kpk_Cα

0[0,T]≤K₂(q)h kζ_tk_Cα

0[0,T])+kw_tk_Cα

0([0,T],L2[−l,l])

i

. (10)

Applying (3), we get

ut(t, x) =wt(t, x) +p(t)q(x) and

kutk_C([0,T],L

2[−l,l])6kwtk_C([0,T],L

2[−l,l])+kpk_C[0,T])kqk_L

2[−l,l]), kutk_Cα

0([0,T],L₂[−l,l])6kwtk_Cα

0([0,T],L₂[−l,l])+kpk_Cα

0[0,T])kqk_L

2[−l,l]). Therefore, the following theorems will complete the proof of Theorem 1 and 2.

Theorem 3.Under assumptions of Theorem 1, inC([0, T], L₂[−l, l])the problem (5) has a unique solution and the following stability estimate is satisfied:

kwtk_C([0,T],L

2[−l,l])6K2(q, δ)h kϕk_W2

2[−l,l]+|ζ(0)|

+kg(0,·)k_L

2[−l,l]+kgtk_C([0,T],L

2[−l,l])+kζtk_C[0,T]i .

Theorem 4.Under assumptions of Theorem 2, inC₀^α([0, T], L2[−l, l])the problem (5) has a unique solution and the following coercive stability estimate is satisfied:

kwtk_Cα

0([0,T],L₂[−l,l])6K2(q, δ)h kϕk_W2

2[−l,l]

+_α(1−α)¹ kgk_Cα

0([0,T],L2[−l,l])+kζtk_Cα 0[0,T]

i . Proof. Problem (5) can be written in the following abstract form

( w⁰(t) +Aw(t) =−η(t)Aq+g(t), 0< t < T,

w(0) =ϕ (11)

in a Hilbert spaceH =L2[−`, `]with the space operatorA=A^xdefined by the formula (2). Hereg(t) =g(t, x) is given abstract function,w(t) =w(t, x)is unknown function, andq=q(x)is the unknown element ofL2[−`, `].

The proofs of Theorems 3 and 4 are based on estimates (8), (9) and (10), theorems on stability and coercive stability of the abstract problem (11) [9], the integral inequality and the self-adjointness and positive definiteness of the space operatorA^x defined by formula (2) [15].

Stability and coercive stability of DS

Letα∈(0,1)be a given number andC_τ^α(H) =C₀^α([0, T]_τ, H]), Cτ(H) =C([0, T]_τ, H)be Banach spaces of allH-valued mesh functionswτ={wk}^N_k=0 defined on

[0, T]_τ={t_k=kτ,06k6N, N τ=T} with the corresponding norms

kwτk_C

τ(H)= max

0≤k≤Nkwkk_H, kwτk_Cα

τ(H)= sup

1≤k<k+n≤N

(N−n)^−α(k)^αkwk+n−wkkH+kwτk_C

τ(H).

Moreover, let L2h = L2[−l, l]_h and W_2h² = W₂²[−l, l]_h be normed spaces of all mesh functions γ^h(x) ={γn}^M_n=−M defined on

[−l, l]_h={xn =nh,−M 6n6M, M h=l}

equipped with norms

γ^h _L

2h =



 X

x∈[−l,l]h

γ^h(x)

2h





1/2

and

γ^h _W2

2h

= γ^h

_L

2h+



 X

x∈[−l,l]h

γ^h

xx,j

2

h





1/2

, respectively. Moreover, we introduce the difference operator A^x_hdefined by the formula

A^x_hu^h(x) ={−(a(x)ux(x))_x,r−β(a(−x)ux(−x))_x,r+δu_r}^M_−M+1⁻¹ , (12) acting in the space of mesh functions u^h(x) = {un}^M_n=−M defined on [−l, l]_h satisfying the conditions u_M−u_M−1=u_−M−u_−M+1= 0.For the numerical solution

u^h_k(x) ^N_k=0 of SIP (1) we present DS of the first order of approximation



























u^k_n−u^k−1_n τ −_h¹

an+1

u^k_n+1−u^k_n h −an

u^k_n−u^k_n−1 h

−^β_h a_−n+1^u

k

−n+1−u^k_−n

h −a_−n^u

k

−n−u^k_−n−1 h

+δu^k_n =pkqn+g^k_n, g_n^k =g(tk, xn), tk∈[0, T]_τ, xn∈[−l, l]_h, k∈1, N , n∈1, M−1, u⁰_n=ϕ_n, ϕ_n=ϕ(x_n), n∈0, M ,

u^k_M−u^k_M−1=u^k_−M+1−u^k_−M = 0,PM

i=1u^k_ih=ζk, ζk=ζ(tk), k∈0, N .

(13)

Here it is assumed thatqM−q_M−1=q_−M−q_−M+1= 0,and PM

m=1qmh6= 0.Let us give the following results on the stability of DS (13).

Theorem 5. For the solution of DS (13), the stability estimate

1

τ u^h_k−u^h_k−1 ^N

k=1

_C

τ(L2h)

+

u^h_k ^N_{k=1 C}

τ(^W2h² )+

{pk}^N_{k=1 C[0,T]}

τ

6K(q)h ϕ^h

_W2 2h

+ g₁^h

_L

2h+|ζ0| +

1

τ g^h_k−g_k−1^h N

k=2

_C

τ(L_2h)

+

1

τ(ζk−ζk−1) N

k=1

_C[0,T]



,

and coercive stability estimate

1

τ u^h_k −u^h_k−1 N

k=1

_Cα

τ(L_2h)

+

u^h_k ^N_{k=1 Cα}

τ(^W_2h² )+

{pk}^N_{k=1 Cα}

0[0,T]_τ

6K(q)h ϕ^h

_W2 2h

+ 1

α(1−α)

g_k^{h N}_{k=1 Cα}

τ(L_2h)

+

1

τ(ζ_k−ζ_k−1) ^N

k=1

_Cα

0[0,T]_τ





hold.

Proof. We will use

u^k_n =w^k_n+ηkqn, (14)

where

q_n=q(x_n), η_k=

k

X

m=1

p_mτ.

It is easy to use

w^h_k(x) ^N_k=0 as the solution of the following DS

w^k_n−w^k−1_n

τ −h

1 h

an+1

w^k_n+1−w^k_n h −an

w^k_n−w^k_n−1 h

−^β_h

a_−n+1^w

k

−n+1−w^k_−n

h −a_−n^w

k

−n−w^k_−n−1 h

+δw_n^k

=−h

−¹_h qn+1

w^k_n+1−w_n^k h −qn

w^k_n−w_n−1^k h

−^β_h q_−n+1^w

k

−n+1−w^k_−n

h −q_−n^w

k

−n−w^k_−n−1 h

+δq^k_nηk

i +g_n^k, k∈1, N , n∈1, M−1,

w⁰_n=ϕn, n∈0, M ,

w^k_M−w^k_M−1=w_−M+1^k −w^k_−M = 0, k∈k∈0, N .

(15)

Now we estimate|pk|.Using the condition PM

m=1u^k_mh=ζ_k and (14), we obtain η_k =b₁ ζ_k−

M

X

m=1

w^k_mh

! , where

b1= 1

PM m=1qmh. Then,

p_k =b1

τ ζ_k−ζ_k−1−

M

X

m=1

(w_m^k −w_m^k−1)h

!

. (16)

Applying the Cauchy-Schwartz inequality, we get

|pk|6|b1|[|ζk−ζ_k−1

τ |+

M

X

m=1

|w_m^k −w_m^k−1

τ |h]

6K(b)

"

ζk−ζk−1

τ

+

w^h_k −w_k−1^h τ

_L

2h

#

(17) for every16k6N, and

{pk}^N_{k=1 C[0,T]}

τ

6K(b)







ζ_k−ζ_k−1 τ

^N

k=1

_C[0,T]

τ

+

(w^h_k−w^h_k−1 τ

)N

k=1

_C

τ(L_2h)





. Moreover, using (16), we can write

{pk}^N_{k=1 Cα}

0[0,T]_τ

6K(b)







ζk−ζk−1

τ ^N

k=1

_Cα

0[0,T]_τ

+

(w^h_k−w^h_k−1 τ

)^N

k=1

_Cα

τ(L_2h)





. (18) Applying (14), we obtain

u^k_n−u^k−1_n

τ = w^k_n−w^k−1_n

τ +pkqn. From that it follows

1

τ u^h_k−u^h_k−1 N

k=1

_C

τ(L_2h)

6

1

τ w^h_k−w^h_k−1 ^N

k=1

_C

τ(L_2h)

+

{pk}^N_{k=1 C[0,T]}

τ

q^h _L

2h (19)

and

1

τ u^h_k −u^h_k−1 ^N

k=1

_Cα

τ(L2h)

6

1

τ w^h_k−w^h_k−1 N

k=1

_Cα

τ(L_2h)

+

{pk}^N_{k=1 Cα}

0[0,T]_τ

q^h _L

2h. Therefore, the following theorem will complete the proof of Theorem 5.

Theorem 6. For the solution of DS (15), the stability estimate

1

τ w_k^h−w_k−1^{h N}

k=1

_C

τ(L2h)

6K3(a)h ϕ^h

_W2

2h

+ g^h₁

_L

2h+|ζ0|

+

1

τ g^h_k −g^h_k−1 N

k=2

_C

τ(L2h)

+

ζ_k−ζ_k−1 τ

N k=1

_C[0,T]

τ





and coercive stability estimate

1

τ w_k^h−w_k−1^h N

k=1

_Cα

τ(L_2h)

6K3(q)



 ϕ^h

_W2

2h

+ 1

α(1−α)

g^h_k ^N_{k=1 Cα}

τ

+

ζk−ζ_k−1 τ

^N

k=1

C^α₀[0,T]τ





hold.

Proof. Problem (15) can be written in the following abstract form







w^h_k−w^h_k−1

τ +A^hw_k^h=g^h_k−A^hq^hηk, t_k=kτ,16k6N, w^h₀ =ϕ^h

(20) in a Hilbert spaceH =L2h with the space operatorA^h=A^x_h defined by the formula (12). Here,g^h_k =g^h_k(x)is given abstract mesh function,w^h_k =w_k^h(x)is unknown mesh function andq^h=q^h(x)is the unknown element of L2h. The proof of Theorem 6 is based on estimates (17), (18) and (19), theorems on stability and coercive stability of the abstract problem (20) (see [9]), difference analogy of integral inequalities, and the self-adjointness and positive definiteness of the difference operatorA^x_h defined by the formula (12) [15].

Numerical experiment

In this section a numerical computation to approximate solution of a time dependent source identification problem with involution and Neumann conditions is considered to support the theoretical results. We use the first order of accuracy difference schemes. The error analysis is given.

We consider















ut(t, x)−uxx(t, x)−¹₂uxx(t,−x) +u(t, x)

=p(t) (1 + cosx) +cosx 2 −1

e^−t, x∈(−π, π), t∈(0, π), u(0, x) = 1 + cosx, x∈[−π, π],

ux(t,−π) =ux(t, π) = 0, t∈[0, π], Rπ

0 u(t, s)ds=πe^−t, t∈[0, π]

(21)

for parabolic equation with involution and Neumann condition. The integral condition is given as an overdetermi- ned condition. The exact solution of this problem is

u(t, x) = (1 + cosx)e^−t,−π≤x≤π,0≤t≤π, p(t) = e^−t, t∈[0, π].

Here we denote the set[0, π]_τ×[−π, π]_h of all grid points

[0, π]_τ×[−π, π]_h={(tk, xn) :tk=kτ,0≤k≤N, N τ =π, x_n=nh,−M ≤n≤M, M h=π}. For obtaining the solution to problem (21) we apply the substitution

u(t, x) =w(t, x) +η(t) (1 + cosx), where

η(t) = Z t

0

p(s)ds, η(0) = 0.

One can show that after the substitution, problem (21) turns to















wt(t, x)−wxx(t, x)−¹₂wxx(t,−x) +w(t, x)

=−

5 cosx 2 + 1

η(t) +cosx 2 −1

e^−t, x∈(−π, π), t∈(0, π), w(0, x) = 1 + cosx, x∈[−π, π],

wx(t,−π) =wx(t, π) = 0, t∈[0, π].

(22)

Moreover, using the overdetermined condition we can write Z π

0

u(t, s)ds= Z π

0

w(t, s)ds+η(t) Z π

0

(1 + coss)ds=πe^−t and

η(t) = πe^−t−Rπ

0 w(t, s)ds

π .

For the numerical solution of (22), we present the first order of accuracy difference scheme



























τ⁻¹ w^k_n−w^k−1_n

−h⁻² w_n+1^k −2w_n^k+w_n−1^k

−¹₂h⁻² w^k_−n+1−2w_−n^k +w^k_−n−1 +w_n^k

−¹_π

5 cosxn

2 + 1

PM−1 n=0 w^k_n h

=−(2 + 2 cosxn)e^−tk,

1≤k≤N, −M+ 1≤n≤M−1, w⁰_n= 1 + cosx_n, −M ≤n≤M,

w^k_−M+1−w^k_−M =w^k_M−w^k_M−1= 0, 0≤k≤N.

(23)

For obtaining the solution of difference scheme (23), we rewrite it in the matrix form

A W^k+B W^k−1=Rϕ^k, 1≤k≤N, W⁰=ϕ , (24)

where

A=

























1 −1 0 0 . . . 0 0 0 0

a b a 0 . . d−M+1 . . d−M+1 d−M+1+c d−M+1−a c

0 a b a . . d−M+2 . . d−M+2+c d−M+2−a d−M+2+c 0

. . . .

0 0 0 0 . a+c b−a+d0 a+c+d0 . d0 d0 d0 0

. . . .

0 c −a c . . dM−2 . . a+dM−2 b+dM−2 a+dM−2 0

c −a c 0 . . dM−1 . . dM−1 dM−1 dM−1 a

0 0 0 0 . . . 0 0 1 −1























 ,

B =





















0 0 0 0 . 0 0 0 0 e 0 0 . 0 0 0 0 0 e 0 . 0 0 0 . . . . 0 0 0 0 . e 0 0 0 0 0 0 . 0 e 0 0 0 0 0 . 0 0 0





















and

a=−1

h², b= 1 τ + 2

h² + 1, c=− 1

2h², e= 1 τ, d_i =−h

π

5 cosxi

2 + 1

, i=−M + 1,−M + 2, ...M−1,

W^s=





 W_−M^s ... W_M^s





for s=k, k−1,

R=









1 0 . 0 0 1 . 0 . . . . 0 0 . 1









, ϕ^k =













 0 ϕ^k_−M+1 ... ϕ^k_M−1 0













 .

So, we have a first order difference equation with respect to kwith matrix coefficients. From (24) it follows that

W^k =−A⁻¹BW^k−1+A⁻¹Rϕ^k k= 1,· · ·, N.

In the second step, using the formulas

u^k_n=w^k_n+ηk(1 + cosxn), 0≤k≤N, −M ≤n≤M,

ηk = πe^−tk−PM−1 n=0 w^k_n h

π , 1≤k≤N, η0= 0, pk = ηk−η_k−1

τ , 1≤k≤N, we can find the approximate solutions foru(t, x)andp(t).

We compute the error between the exact solution and numerical solution by











kEuk∞= max

0≤k≤N,−M≤n≤M

u(t_k, x_n)−u^k_n , kE_ηk_∞= max

1<k<N|η(t_k)−η_k|, kEpk_∞= max

1<k<N|p(tk)−pk|,

whereu(t, x), p(t), η(t)represent the exact solutions,u^k_n represents the numerical solution at (tk, xn), and pk andηk represent the numerical solutions attk.The numerical results are given in Table 1.

T a b l e 1

Errors kEuk∞ kEηk∞ kEpk∞

N =M = 30 6.6666·10⁻² 7.3894·10⁻² 1.1917·10⁻¹ N =M = 60 3.3333·10⁻² 3.6655·10⁻² 6.6532·10⁻² N=M= 120 1.6667·10⁻² 1.8262·10⁻² 3.5167·10⁻² N=M= 240 8.3333·10⁻³ 9.1165·10⁻³ 1.8081·10⁻²

Conclusion

In this paper we considered a time dependent source of identification problem for parabolic equation with involution and Neumann condition. The theoretical considerations that prove well-posedness theorem on the differential equation of the source identification parabolic problem and stability estimates for the difference scheme of the source identification parabolic problem were given. To support the theoretical results by a numeri- cal experiment we constructed a stable difference scheme for the approximate solution of the problem. Obtained results given in Table 1 support the theoretical results.

Acknowledgements

We would like to thank the following institutions for their support. The publication has been prepared with the support of the "RUDN University Program 5-100". This research has been funded by the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant No. AP08855352).

References

1 Choulli M. Generic well-posedness of a linear inverse parabolic problem with respect to diffusion parameters / M. Choulli, M. Yamamoto // Journal of Inverse and III-Posed Problems. — 1999. — 7. — No.3. — P. 241–

254.

2 Ashyralyev A. On source identification problem for a delay parabolic equation / A. Ashyralyev, D. Agi- rseven // Nonlinear Analysis: Modelling and Control. — 2014. — 19. — No.3. — P. 335–349.

3 Ashyralyev A. On the problem of determining the parameter of an elliptic equation in a Banach space/

A. Ashyralyev, C. Ashyralyyev // Nonlinear Analysis: Modelling and Control. — 2014. — 19. — No. 3. — P. 350–366.

4 Erdogan A.S. On the second order implicit difference schemes for a right hand side identification problem / A.S. Erdogan, A. Ashyralyev // Appl. Math. Comput. — 2014. — 226. — P. 212–228.

5 Ashyralyev A. On the determination of the right-hand side in a parabolic equation / A. Ashyralyev, A.S. Erdogan, O. Demirdag // Applied Numerical Mathematics. — 2012. — 62. — No. 11. — P. 1672–

1683.

6 Ashyralyyev C. High order approximation of the inverse elliptic problem with Dirichlet-Neumann condi- tions / C. Ashyralyyev // Filomat. — 2014. — 28. — No. 5. — P. 947–962.

7 Blasio G. Di. Identification problems for parabolic delay differential equations with measurement on the boundary / G. Di. Blasio, A. Lorenzi // Journal of Inverse and Ill-Posed Problems. — 2007. — 15. — No. 7. — P. 709–734.

8 Jator S. Block unification scheme for elliptic, telegraph, and Sine-Gordon partial differential equations / S. Jator // American Journal of Computational Mathematics. — 2015. — 5. — No. 2. — P. 175–185.

9 Ashyralyev A. New Difference Schemes for Partial Differential Equations / A. Ashyralyev, P.E. Sobolevskii // Birkh¨auser Verlag, Basel, Boston, Berlin. — 2004.

10 Ashyralyev A. On source identification problem for a hyperbolic-parabolic equation / A. Ashyralyev, M.A. Ashyralyyeva // Contemporary Analysis and Applied Mathematics. — 2015. — 3. — No. 1. — P. 88–103.

11 Ashyralyyeva M.A. Stable difference scheme for the solution of the source identification problem for hyperbolic-parabolic equations / M.A. Ashyralyyeva, A. Ashyralyyev // AIP Conference Proceedings. — 2015. — 1676. Article Number: 020024.

12 Ashyralyyeva M.A. On the numerical solution of identification hyperbolic-parabolic problems with the Neumann boundary condition / M.A. Ashyralyyeva, M. Ashyraliyev // Bulletin of the Karaganda University-Mathematics. — 2018. — 91. — No.3. — P. 69–74.

13 Ashyralyyeva M.A. Numerical solutions of source identification problem for hyperbolic-parabolic equations / M.A. Ashyralyyeva, M. Ashyraliyev // AIP Conference Proceedings. — 2018. — 1997. Article Number:

020048.

14 Ashyralyev A. Well-posedness of an elliptic equation with involution / A. Ashyralyev, A. Sarsenbi //

Electron. J. Differential Equations. — 2015. — 2015. — 284. — P. 1–8.

15 Ashyralyev A. Well-Posedness of a parabolic equation with involution / A. Ashyralyev, A. Sarsenbi //

Numerical Functional Analysis and Optimization. — 2017. — 38. — No.10. — P. 1295–1304.

16 Ashyralyev A. Stability of a hyperbolic equation with the involution, in: Functional Analysis in Interdi- sciplinary Applications / A. Ashyralyev, A. Sarsenbi // Vol. 216 of Springer Proceedings in Mathematics

& Statistics Book Series. — 2016. — P. 204–212.

17 Cabada A. Differential Equations with Involutions / A. Cabada, F. Tojo. Atlantis Press.

A. Ашыралыев

, А.С. Ердоган

1Таяу Шығыс университетi, Никосия, Түркия;

2Ресей халықтар достығы университетi, Мәскеу, Ресей;

3Математика және математикалық модельдеу институты, Алматы, Қазақстан;

4Палм-Бич мемлекеттiк колледжi, Флорида, АҚШ

Инволюциямен және Дирихле шартымен сәйкестендiрудiң параболалық мәселесi туралы ескерту

Инволюция және Дирихле шарты бар параболалық теңдеу үшiн дереккөздi анықтаудың кеңiстiк- тiк есептерi зерттелдi. Параболалық дифференциалдық теңдеу үшiн дереккөздi анықтау есебiнiң дұрыстығы теоремасы анықталды. Бұл есептiң жуық шешiмiн табу үшiн тұрақты айырымдық схема берiлген. Сонымен қатар, дереккөздi сәйкестендiрудiң параболалық есебiнiң айырымдық схемасының тұрақтылығының бағалаулары ұсынылған. Сандық нәтижелер келтiрiлген.

Кiлт сөздер:корректiлiк, эллипстiк теңдеулер, коэрцитивтi тұрақтылық, дереккөздi идентификация- лау, дәл бағалаулар, шеттiк есеп.

A. Ашыралыев

, А.С. Ердоган

1Ближневосточный университет, Никосия, Турция;

2Российский университет дружбы народов, Москва, Россия;

3Институт математики и математического моделирования, Алматы, Казахстан;

4Государственный колледж Палм-Бич, Флорида, США

Замечание о параболической проблеме идентификации с инволюцией и условием Дирихле

Исследованы пространственные задачи идентификации источника для параболического уравнения с инволюцией и условием Дирихле. Установлена теорема корректности задачи идентификации источни- ка для параболического дифференциального уравнения. Представлена устойчивая разностная схема для приближенного решения этой задачи. Кроме того, даны оценки устойчивости разностной схемы параболической задачи идентификации источника. Приведены численные результаты.

Ключевые слова:корректность, эллиптические уравнения, положительность, коэрцитивная устойчи- вость, идентификация источника, точные оценки, краевая задача.

References

1 Choulli, M., & Yamamoto, M. (1999). Generic well-posedness of a linear inverse parabolic problem with respect to diffusion parameters.Journal of Inverse and III-Posed Problems, 7(3), 241–254.

2 Ashyralyev, A., & Agirseven, D. (2014). On source identification problem for a delay parabolic equation.

Nonlinear Analysis: Modelling and Control, 19(3), 335–349.

3 Ashyralyev, A., & Ashyralyyev, C. (2014). On the problem of determining the parameter of an elliptic equation in a Banach space.Nonlinear Analysis: Modelling and Control, 19(3), 350–366.

4 Erdogan, A.S., & Ashyralyev, A. (2014). On the second order implicit difference schemes for a right hand side identification problem.Appl. Math. Comput., 226, 212–228.

5 Ashyralyev, A., Erdogan, A.S., & Demirdag, O. (2012). On the determination of the right-hand side in a parabolic equation.Applied Numerical Mathematics, 62(11), 1672–1683.

6 Ashyralyyev, C. (2014). High order approximation of the inverse elliptic problem with Dirichlet-Neumann conditions.Filomat, 28(5), 947–962.

7 Blasio, G. Di., & Lorenzi, A. (2007). Identification problems for parabolic delay differential equations with measurement on the boundary.Journal of Inverse and Ill-Posed Problems, 15(7), 709–734.

8 Jator, S. (2015). Block unification scheme for elliptic, telegraph, and Sine-Gordon partial differential equations.American Journal of Computational Mathematics, 5(2), 175–185.

9 Ashyralyev, A., & Sobolevskii, P.E.New Difference Schemes for Partial Differential Equations. Birkh¨auser Verlag, Basel, Boston, Berlin, 2004.

10 Ashyralyev, A., & Ashyralyyeva, M.A. (2015). On source identification problem for a hyperbolic-parabolic equation.Contemporary Analysis and Applied Mathematics, 3(1), 88–103.

11 Ashyralyyeva, M.A., & Ashyralyyev, A. (2015). Stable difference scheme for the solution of the source identification problem for hyperbolic-parabolic equations. AIP Conference Proceedings, 1676, Article Number: 020024.

12 Ashyralyyeva, M.A., & Ashyraliyev, M. (2018). On the numerical solution of identification hyperbolic- parabolic problems with the Neumann boundary condition.Bulletin of the Karaganda University-Mathe- matics, 91(3), 69–74.

13 Ashyralyyeva, M.A., & Ashyraliyev, M. (2018). Numerical solutions of source identification problem for hyperbolic-parabolic equations.AIP Conference Proceedings, 1997. Article Number: 020048.

14 Ashyralyev, A., & Sarsenbi, A. (2015). Well-posedness of an elliptic equation with involution. Electron.

J. Differential Equations, 2015, 284, 1–8.

15 Ashyralyev, A., & Sarsenbi, A. (2017). Well-Posedness of a parabolic equation with involution.Numerical Functional Analysis and Optimization, 38(10), 1295–1304.

16 Ashyralyev, A., & Sarsenbi, A. (2016). Stability of a hyperbolic equation with the involution, in: Functional Analysis in Interdisciplinary Applications.Vol. 216 of Springer Proceedings in Mathematics & Statistics Book Series, 204–212.

17 Cabada, A., & Tojo, F.Differential Equations with Involutions. Atlantis Press, 2015.

MSC 35J25, 47E05, 34B27

A. Ashyralyev

, M. Ashyralyyeva

, O. Batyrova

1Near East University, Nicosia, Turkey;

2Peoples’ Friendship University of Russia (RUDN University), Moscow, Russia;

3Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan;

4Magtymguly Turkmen State University, Ashgabat, Turkmenistan;

5Oguz Han Engineering and Technology University of Turkmenistan, Ashgabat, Turkmenistan (E-mail: aallaberen@gmail.com, allaberen.ashyralyev@neu.edu.tr, ashyrmaral2010@mail.ru,

ogulbabek93@gmail.com)

On the boundedness of solution of the second order ordinary differential equation with damping term and involution

In the present paper the initial value problem for the second order ordinary differential equation with damping term and involution is investigated. We obtain equivalent initial value problem for the fourth order ordinary differential equations to the initial value problem for second order linear differential equations with damping term and involution. Theorem on stability estimates for the solution of the initial value problem for the second order ordinary linear differential equation with damping term and involution is proved. Theorem on existence and uniqueness of bounded solution of initial value problem for second order ordinary nonlinear differential equation with damping term and involution is established.

Keywords: differential equation with damping term and involution, stability, boundedness, existence and uniqueness.

Introduction

Differential equations with involution appear in mathematical models of ecology, biology, and population dynamics (see, e.g, [1–6] and the reference given therein).

Our goal in this paper is to investigate the boundedness of the solution of the initial value problem for the second order ordinary differential equation with damping term and involution

y⁰⁰(t) =f(t, y(t), y⁰(t), y(u(t)), t∈I= (−∞,∞), y(t0) =y0, y⁰(t0) =y₀⁰. (1) Here and in future u(t) is involution function, that is u(u(t)) =t, and t0 is a fixed point of u. Problem (1) does not seem to yield directly to any techniques that can be used for ordinary differential equations without involution term [1, 2]. Therefore, we consider the second order linear differential equations with damping term and involution. We obtain equivalent initial value problem for the fourth order ordinary differential equations to the initial value problem for second order linear differential equations with damping term and involution.

Theorem on stability estimates for the solution of the initial value problem for the second order ordinary linear differential equation with damping term and involution is proved. Finally, theorem on existence and uniqueness of bounded solution of initial value problem for the second order nonlinear ordinary differential equation with damping term and involution is established. Note that some of the results of this work was presented, without proof, in [7].

Linear ordinary differential equation with damping term and involution LetC^∞[I]be the set of all differentiable functions for all degrees.

Theorem 1.Leta(t), b(t), α(t)be functions of classC^∞onI, such thatb(t)does not vanish on the interval I, then the problem

y⁰⁰(t) +α(t)y⁰(t) =a(t)y(t) +b(t)y(−t) +f(t), t∈I, y(0) =ϕ, y⁰(0) =ψ

*Corresponding author.

E-mail: ogulbabek93@gmail.com

is equivalent to the following problem for the fourth order ordinary differential equation



















y⁽⁴⁾(t) =p(t)y(t) +q(t)y⁰(t) +r(t)y⁰⁰(t) +s(t)y⁰⁰⁰(t) +F(t), t∈I, y(0) =ϕ, y⁰(0) =ψ,

y⁰⁰(0) =a(0)ϕ+b(0)ϕ−α(0)ψ+f(0), y⁰⁰⁰(0) =h

−α(0) [a(0) +b(0)] +a⁰(0) +b⁰(0)i ϕ +h

−α⁰(0) +α²(0) +a(0)−b(0)i

ψ+f⁰(0)−α(0)f(0), where

p(t) =a⁰⁰(t) +b(−t)b(t)−h

2b⁰(t) +b(t)α(−t)i 1 b(t)a⁰(t)

−

b⁰⁰(t) +b(t)a(−t)−h

2b⁰(t) +b(t)α(−t)i 1 b(t)b⁰(t)

1 b(t)a(t), q(t) =−α⁰⁰(t) + 2a⁰(t) +h

2b⁰(t) +b(t)α(−t)i 1 b(t)

hα⁰(t)−a(t)i

−

b⁰⁰(t) +b(t)a(−t)−h

2b⁰(t) +b(t)α(−t)i 1 b(t)b⁰(t)

1 b(t)α(t), r(t) =−2α⁰(t) +a(t) +h

2b⁰(t) +b(t)α(−t)i 1 b(t)α(t) +

b⁰⁰(t) +b(t)a(−t)−h

2b⁰(t) +b(t)α(−t)i 1 b(t)b⁰(t)

1 b(t), s(t) =−α(t) +h

2b⁰(t) +b(t)α(−t)i 1 b(t), and

F(t) =−

b⁰⁰(t) +b(t)a(−t)−h

2b⁰(t) +b(t)α(−t)i 1 b(t)b⁰(t)

1 b(t)f(t)

−h

2b⁰(t) +b(t)α(−t)i 1

b(t)f⁰(t) +b(t)f(−t) +f⁰⁰(t).

The proof of Theorem 1 is based on approaches of proof of Theorem 1 of paper [1] on the first order linear differential equation with involution.

Now, we consider the initial value problem

y⁰⁰(t) +αy⁰(t) =by(−t) +ay(t) +f(t), t∈ I, y(0) =ϕ, y⁰(0) =ψ (2) for the second order involutory ordinary differential equation with damping term. We are interested in studying the stability of problem (2) onI.In general cases ofα, aandbthe solution of (2) is not bounded onI.Applying Theorem 1, we get the equivalent initial value problem











y⁽⁴⁾(t) + (a²−b²)y(t)− 2a+α²

y⁰⁰(t) =F(t), F(t) =−af(t) +bf(−t)−αf⁰(t) +f⁰⁰(t), t∈I, y(0) =ϕ, y⁰(0) =ψ, y⁰⁰(0) = (b+a)ϕ−αψ+f(0), y⁰⁰⁰(0) =−α(b+a)ϕ+ −b+a+α²

ψ+f⁰(0)−αf(0)

(3)

for the fourth order ordinary differential equation. We will obtain the solution of problem (3). Assume that

|b|<|a|, a∈

−

α² 4 +_α^b22

,−^α₂²

.Then, it is easy to see that d⁴y(t)

dt⁴ − 2a+α²d²y(t)

dt² + a²−b² y(t)

= d²

dt²− a+α² 2 +

r

aα²+α⁴ 4 +b²

!! d²

dt² − a+α² 2 −

r

aα²+α⁴ 4 +b²

!!

y(t).