ISSN (Print): 2077-9879 ISSN (Online): 2617-2658
Eurasian
Mathematical Journal
2022, Volume 13, Number 1
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c
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VAGIF SABIR oglu GULIYEV (to the 65th birthday)
On February 22, 2022 was the 65th birthday of Vagif Sabir oglu Guliyev, editor-in-chief of the Transactions of the Azerbaijan National Academy of Science, Issue Mathematics, Series of physical-technical and mathematics science (Scopus, Q3), deputy editor-in-chief of the Applied and Computational Mathematics (Web of Science, Q1), deputy director of the Institute of Applied Mathematics (IAM) of the Baku State University (BSU), head of the Department of Mathematical Analysis at the Institute of Mathematics and Mechanics (IMM) of the Azerbaijan National Academy of Sciences (ANAS), member of the Editorial Board of the Eurasian Mathematical Journal.
V.S. Guliyev was born in the city of Salyan in Azerbaijan. In 1978 Vagif Guliyev graduated from the Faculty of Mechanics and Mathematics of the Azerbaijan State University (now the Baku State University) with an honors degree and then completed his postgraduate studies at this university. His scientic supervisors were distinguished mathematicians A.A. Babayev and S.K. Abdullayev. In 1983 he defended his PhD thesis at the BSU. From 1983 he continued his scientic activities at the V.A. Steklov Mathematical Institute of the Academy of Sciences of the USSR. In 1987-1991 he was in internship at this institute and in 1994 defended there his DSc thesis.
From 1983 to 1995 he worked as assistant, a senior lecturer, docent and from 1995 to 2018 as a professor of Mathematical Analysis Chair of the BSU. In 1995-2008 he worked on part-time basis at the Institute of the IMM. From 2008 to 2014 he was a chief researcher of the Department of Mathematical Analysis of the IMM, from 2014 to the present day he is the head of this department.
In 2014 V.S. Guliyev was elected a corresponding member of the ANAS.
From 2015 to 2019, he worked as deputy director on science at the IMM. From 2019 to the present day, he has been working as a chief researcher at the IAM. Since May 2021, he has been working as a deputy director on science of the IAM.
Professor Vagif Guliyev has been a member of the Presidium of the Higher Attestation Commis- sion under the President of the Republic of Azerbaijan since 2014 to the present day.
V.S. Guliyev is a world recognized specialist in real and harmonic analysis, function spaces and partial dierential equations. He obtained seminal scientic results in several areas of functional analysis and the theory of partial dierential and integral equations. He was one of the rst to study local Morrey-type spaces, generalized weighted Morrey-type spaces and anisotropic Banach-valued Sobolev spaces, for which appropriate embedding theorems were established.
Some of his results and methods are named after him: the Adams-Guliyev and Spanne-Guliyev conditions for the boundedness of operators in Morrey-type spaces, Guliyev's method of local esti- mates of integral operators of harmonic analysis, the Burenkov-Guliyevs conditions for the bound- edness of operators in general local Morrey-type spaces.
On the whole, the results obtained by V.S. Guliyev have laid a groundwork for new perspec- tive scientic directions in the theory of functions spaces and its applications to partial dierential equations.
Vagif Sabir oglu Guliyev is an author of more than 250 scientic publications including 2 mono- graphs. Among his pupils there are more than 20 candidates of sciences and 5 doctors of sciences.
The results obtained by V.S. Guliyev, his pupils, collaborators and followers gained worldwide recog- nition.
The mathematical community, many his friends and colleagues and the Editorial Board of the Eurasian Mathematical Journal cordially congratulate Vagif Sabir oglu Guliyev on the occasion of his 65th birthday and wish him good health, happiness and new achievements in mathematics and mathematical education.
EURASIAN MATHEMATICAL JOURNAL ISSN 2077-9879
Volume 13, Number 1 (2022), 69 85
ON AN ALGORITHM OF FINDING AN APPROXIMATE SOLUTION OF A PERIODIC PROBLEM FOR A THIRD-ORDER DIFFERENTIAL EQUATION
N.T. Orumbayeva, A.T. Assanova, A.B. Keldibekova Communicated by V.I. Korzyuk
Key words: partial dierential equation, third order boundary value problem, algorithm, approxi- mate solution.
AMS Mathematics Subject Classication: 34A30, 35Q92.
Abstract. In this paper, we study a periodic boundary value problem for a partial dierential equation of the third order. An algorithm for nding a solution to this boundary value problem is proposed, and sucient conditions for the convergence of the proposed algorithm are obtained.
DOI: https://doi.org/10.32523/2077-9879-2022-13-1-69-85
1 Introduction
On Ω = [0, ω]×[0, T]we consider the periodic boundary value problem
∂3u
∂x2∂t =A(x, t)∂2u
∂x2 +B(x, t)∂u
∂x +C(x, t)u+f(x, t),(x, t)∈Ω, (1.1)
u(x,0) =u(x, T), x∈[0, ω], (1.2)
u(0, t) =ϕ(t), t∈[0, T], (1.3)
∂u(0, t)
∂x =ψ(t), t ∈[0, T], (1.4)
where (n×n) - matrix functions A(x, t), B(x, t), C(x, t), n-vector functions f(x, t) are continuous onΩ, n-vector functions ϕ(t), ψ(t) are continuously dierentiable on [0, T] satisfying the conditions ϕ(0) = ϕ(T), ψ(0) =ψ(T).
In particular, for A(x, t) ≡ 1, B(x, t) ≡ C(x, t) ≡ f(x, t) ≡ 0 the general solution of equation (1.1) satisfying conditions (1.3), (1.4) has the form:
u(x, t) = ϕ(t) +ψ(t)x+etV(x),
where V(x) is an arbitrary twice continuously dierentiable function. Substituting it in conditions (1.2) and taking into account the conditions ϕ(0) = ϕ(T), ψ(0) = ψ(T) we obtain that V(x) = 0.
Then the solution of problem (1.1)-(1.4) for A(x, t) ≡1, B(x, t) ≡C(x, t) ≡f(x, t)≡ 0 is u(x, t) = ϕ(t) +ψ(t)x.
Modeling of various processes of physics, mechanics, biology, and others sciences leads to the study of boundary value problems for partial dierential equations of the third order [1], [3], [4], [6], [7], [12] and construction of approximate methods for nding their solutions. Application of dier- ent approaches, ideas and methods leads to results formulated in dierent terms. In this paper we
70 N.T. Orumbayeva, A.T. Assanova, A.B. Keldibekova
investigate the existence of a solution to problem (1.1)-(1.4) and propose a method for constructing an approximate solution. With the help of additional functions [2], [8]-[10] the considered problem reduces to an equivalent problem consisting of a family of multipoint problems for an ordinary dier- ential equation of the rst order with a functional parameter and an integral relation. An algorithm for nding an approximate solution to the problem under study is proposed and its convergence is proved. Sucient conditions for the existence and uniqueness of the solution of the periodic problem for the system of partial dierential equations of the third order are established. Works [5], [11], are devoted to singular dierential equations of the third order.
To nd the solution, we introduce the functionz(x, t) = ∂u(x,t)∂x ,and we rewrite problem (1.1)-(1.4) in the form
∂2z
∂x∂t =A(x, t)∂z
∂x +B(x, t)z+C(x, t)u+f(x, t), (x, t)∈Ω, (1.5)
z(x,0) =z(x, T), x∈[0, ω], (1.6)
z(0, t) =ψ(t), t∈[0, T], (1.7)
u(x, t) =ϕ(t) +
x
Z
0
z(ξ, t)dξ. (1.8)
For a xedu(x, t)problem (1.5)-(1.7) is a periodic boundary value problem for a system of hyperbolic equations of the second order.
We next introduce the notation v(x, t) = ∂z(x,t)∂x ,and reduce problem (1.5)-(1.8) to the problem
∂v
∂t =A(x, t)v +B(x, t)z+C(x, t)u+f(x, t), (x, t)∈Ω, (1.9)
v(x,0) =v(x, T), x∈[0, ω], (1.10)
and functional relations
z(x, t) = ψ(t) +
x
Z
0
v(ξ, t)dξ, (x, t)∈Ω, (1.11)
u(x, t) =ϕ(t) +
x
Z
0
z(ξ, t)dξ, (x, t)∈Ω. (1.12)
To solve problem (1.9)-(1.12) we apply the method of a parametrization . For the step h > 0 : N h = T we partition [0, T) =
N
S
r=1
[(r−1)h, rh), N = 1,2, .... In this case, Ω is divided into N parts. By vr(x, t), zr(x, t), ur(x, t) we denote, respectively, the restrictions of the functions v(x, t), z(x, t), u(x, t) on Ωr = [0, ω] ×[(r − 1)h, rh), r = 1, N . By λr(x) we denote the value of the function vr(x, t) at t = (r −1)h, i.e. λr(x) = vr(x,(r−1)h) and denote evr(x, t) = vr(x, t) − λr(x), r = 1, N. We obtain an equivalent boundary value problem for the unknown functions λr(x):
∂evr
∂t =A(x, t)ver+A(x, t)λr(x) +B(x, t)zr(x, t) +C(x, t)ur(x, t) +f(x, t), (1.13) ver(x,(r−1)h) = 0, x∈[0, ω], r = 1, N , (1.14) λ1(x)−λN(x)− lim
t→T−0evN(x, t) = 0, x∈[0, ω], (1.15) λs(x) + lim
t→sh−0evs(x, t)−λs+1(x) = 0, x∈[0, ω], s= 1, N−1. (1.16)
On an algorithm of nding an approximate solution of a periodic problem 71
zr(x, t) =ψ(t) +
x
Z
0
evr(ξ, t)dξ+
x
Z
0
λr(ξ)dξ, (x, t)∈Ωr, r= 1, N , (1.17)
ur(x, t) =ϕ(t) +
x
Z
0
zr(ξ, t)dξ, (x, t)∈Ωr, r= 1, N , (1.18) where (1.17) is the condition of gluing functions on the internal lines of the partition.
Problem (1.13), (1.14) for xed λr(x), zr(x, t), ur(x, t) is a one-parameter family of Cauchy prob- lems for systems of ordinary dierential equations, where x ∈ [0, ω], which are equivalent to the integral equations
evr(x, t) =
t
Z
(r−1)h
A(x, τ)evr(x, τ)dτ +
t
Z
(r−1)h
A(x, τ)dτ ·λr(x) +
t
Z
(r−1)h
F(x, τ, zr, ur)dτ, (1.19)
where
t
Z
(r−1)h
F(x, τ, zr, ur)dτ =
t
Z
(r−1)h
B(x, τ)zr(x, τ)dτ +
t
Z
(r−1)h
C(x, τ)ur(x, τ)dτ +
t
Z
(r−1)h
f(x, τ)dτ.
Instead ofevr(x, τ)we substitute the corresponding right-hand side of (1.19) and repeating this process ν (ν = 1,2, ...)times we obtain
evr(x, t) = Dνr(x, t)λr(x) +Fνr(x, t, zr, ur) +Gνr(x, t,evr), r= 1, N , (1.20) where
Dνr(x, t) =
t
Z
(r−1)h
A(x, τ1)dτ1 +...+
t
Z
(r−1)h
A(x, τ1)...
τν−1
Z
(r−1)h
A(x, τν)dτν...dτ1,
Fνr(x, t, zr, ur) =
t
Z
(r−1)h
B(x, τ1)zr(x, τ1) +C(x, τ1)ur(x, τ1) +f(x, τ1) dτ1+
+
ν−1
X
j=1 t
Z
(r−1)h
A(x, τ1)...
τj−1
Z
(r−1)h
A(x, τj)
τj
Z
(r−1)h
B(x, τj+1)zr(x, τj+1)+
+C(x, τj+1)ur(x, τj+1) +f(x, τj+1)
dτj+1dτj...dτ1, Gνr(x, t,evr) =
=
t
Z
(r−1)h
A(x, τ1)...
τν−2
Z
(r−1)h
A(x, τν−1)
τν−1
Z
(r−1)h
A(x, τν)ver(x, τν)dτνdτν−1...dτ1,
τ0 =t, r = 1, N . Passing to the limit ast →rh−0 in (1.20) we have
t→rh−0lim evr(x, t) =Dνr(x, rh)λr(x) +Fνr(x, rh, zr, ur) +Gνr(x, rh,ver),
72 N.T. Orumbayeva, A.T. Assanova, A.B. Keldibekova
x∈[0, ω], r= 1, N . Substituting in (1.15), (1.16) instead of lim
t→rh−0evr(x, t), r= 1, N ,the correspond- ing to them right-hand sides, for the unknown functions λr(x), r = 1, N , we obtain the system of functional equations:
Qν(x, h)λ(x) = −Fν(x, h, z, u)−Gν(x, h,ev), (1.21)
where Qν(x, h) =
=
I 0 . . . 0 −[I +DνN(x, N h)]
I+Dν1(x, h) −I . . . 0 0
0 I+Dν2(x,2h) . . . 0 0
0 0 . . . 0 0
. . . .
0 0 . . . I+Dν,N−1(x,(N −1)h) −I
,
Fν(x, h, z, u) = (−FνN(x, N h, zN, uN), Fν1(x, h, z1, u1), ..., Fν,N−1(x,(N−1)h, zN−1, uN−1)), Gν(x, h,ev) = (−GνN(x, N h,evN), Gν1(x, h,ev1), ..., Gν,N−1(x,(N −1)h,evN−1)),
and I is the unit matrix of dimension n.
For nding a system of four functions{λr(x),evr(x, t), zr(x, t), ur(x, t)}, r= 1, N ,we have a closed system consisting of equations (1.21), (1.20), (1.18) and (1.17).
Assuming the invertibility of the matrix Qν(x, h) for all x ∈ [0, ω], from equation (1.21), where evr(x, t) = 0, zr(x, t) =ψ(t), ur(x, t) =ϕ(t),we nd λ(0)(x) =
= (λ(0)1 (x), λ(0)2 (x), ..., λ(0)N (x))0 :
λ(0)(x) =−[Qν(x, h)]−1
Fν(x, h, ψ, ϕ) +Gν(x, h,0) .
Using equation (1.20), at λr(x) =λ(0)r (x) we nd the functions {evr(0)(x, t)}, r= 1, N , i.e.
evr(0)(x, t) =Dνr(x, t)λ(0)r (x) +Fνr(x, t, ψ, ϕ) +Gνr(x, t,0).
The functions zr(0)(x, t), u(0)r (x, t), r= 1, N ,are dened from the relations
zr(0)(x, t) =ψ(t) +
x
Z
0
evr(0)(ξ, t)dξ+
x
Z
0
λ(0)r (ξ)dξ, (x, t)∈Ωr, r= 1, N ,
u(0)r (x, t) =ϕ(t) +
x
Z
0
zr(0)(ξ, t)dξ, (x, t)∈Ωr, r = 1, N .
2 Main results
For the initial approximation of problem (1.13)-(1.18) we take the system(λ(0)r (x),ev(0)r (x, t), zr(0)(x, t), u(0)r (x, t)), r= 1, N and construct successive approximations by using the following algorithm.
Step 1. A) Assuming that
zr(x, t) =zr(0)(x, t), ur(x, t) = u(0)r (x, t), r= 1, N ,
we nd the rst approximations ofλr(x),evr(x, t), r= 1, N ,by solving problem (1.13)-(1.16). Taking λ(1,0)r (x) = λ(0)r (x), evr(1,0)(x, t) =ev(0)r (x, t),
On an algorithm of nding an approximate solution of a periodic problem 73 we nd the system of couples {λ(1)r (x),ver(1)(x, t)}, r = 1, N , as the limit of the sequence λ(1,m)r (x),ver(1,m)(x, t), dened in the following way.
Step 1.1. Assuming the invertibility of the matrix Qν(x, h), x ∈ [0, ω], from equation (1.21), where
evr(x, t) = evr(1,0)(x, t), we nd λ(1,1)(x) = (λ(1,1)1 (x), λ(1,1)2 (x), ..., λ(1,1)N (x))0 :
λ(1,1)(x) =−[Qν(x, h)]−1n
Fν(x, h, z(0), u(0)) +Gν(x, h,ev(1,0))o . Substituting the found λ(1,1)r (x), r= 1, N , in (1.20) we nd
evr(1,1)(x, t) =Dνr(x, t)λ(1,1)r (x) +Fνr(x, t, z(0), u(0)) +Gνr(x, t,ev(1,0)).
Step 1.2. From equation (1.21), where ver(x, t) =ev(1,1)r (x, t), we dene λ(1,2)(x) = −[Qν(x, h)]−1n
Fν(x, h, z(0), u(0)) +Gν(x, h,ev(1,1))o .
Using expression (20) again, we nd the functions{ver(1,2)(x, t)}, r= 1, N , ev(1,2)r (x, t) = Dνr(x, t)λ(1,2)r (x) +Fνr(x, t, z(0), u(0)) +Gνr(x, t,ev(1,1)).
On step (1, m)we obtain the system of couples
{λ(1,m)r (x),ev(1,m)r (x, t)}, r= 1, N .
Suppose that the solution of problem (1.13)-(1.16) is a sequence of systems of couples {λ(1,m)r (x), evr(1,m)(x, t)} which are dened for x ∈ [0, ω], respectively, and converge as m → ∞ to continuous functions λ(1)r (x), ev(1)r (x, t), r= 1, N .
B) The functions zr(1)(x, t), u(1)r (x, t), r= 1, N , are dened from the relations
zr(1)(x, t) =ψ(t) +
x
Z
0
evr(1)(ξ, t)dξ+
x
Z
0
λ(1)r (ξ)dξ, (x, t)∈Ωr, r= 1, N ,
u(1)r (x, t) =ϕ(t) +
x
Z
0
zr(1)(ξ, t)dξ, (x, t)∈Ωr, r = 1, N . Step 2. A) Assuming that
zr(x, t) =zr(1)(x, t), ur(x, t) = u(1)r (x, t), r= 1, N ,
we nd the second approximations of λr(x),evr(x, t), r = 1, N , by solving problem (1.13)-(1.16).
Taking
λ(2,0)r (x) = λ(1)r (x), evr(2,0)(x, t) =ev(1)r (x, t),
we nd the system of couples {λ(2)r (x),ver(2)(x, t)}, r = 1, N , as the limit of the sequence λ(2,m)r (x),ver(2,m)(x, t), dened in the following way:
Step 2.1. Assuming the invertibility of the matrixQν(x, h), x∈[0, ω], from equation (1.21), where
evr(x, t) = evr(2,0)(x, t), we nd λ(2,1)(x) = (λ(2,1)1 (x), λ(2,1)2 (x), ..., λ(2,1)N (x))0 :
74 N.T. Orumbayeva, A.T. Assanova, A.B. Keldibekova
λ(2,1)(x) =−[Qν(x, h)]−1n
Fν(x, h, z(1), u(1)) +Gν(x, h,ev(2,0))o . Substituting the found λ(2,1)r (x), r= 1, N , in (1.20) we nd
evr(2,1)(x, t) =Dνr(x, t)λ(2,1)r (x) +Fνr(x, t, z(1), u(1)) +Gνr(x, t,ev(2,0)).
Step 2.2. From equation (1.21), where
evr(x, t) = evr(2,1)(x, t), we dene
λ(2,2)(x) =−[Qν(x, h)]−1n
Fν(x, h, z(1), u(1)) +Gν(x, h,ev(2,1))o . Using expression (1.20), we nd the functions{ev(2,2)r (x, t)}, r= 1, N :
evr(2,2)(x, t) =Dνr(x, t)λ(2,2)r (x) +Fνr(x, t, z(1), u(1)) +Gνr(x, t,ev(2,1)).
On step (2, m)we obtain the system of couples
{λ(2,m)r (x),ev(2,m)r (x, t)}, r= 1, N .
Suppose that the solution of problem (1.13)-(1.16) is a sequence of systems of couples {λ(2,m)r (x), evr(2,m)(x, t)}which as m → ∞converges to {λ(2)r (x), evr(2)(x, t)}, r= 1, N .
B) The functions zr(2)(x, t), u(2)r (x, t), r= 1, N , are dened from the relations zr(2)(x, t) =ψ(t) +
x
Z
0
evr(2)(ξ, t)dξ+
x
Z
0
λ(2)r (ξ)dξ, (x, t)∈Ωr, r= 1, N ,
u(2)r (x, t) =ϕ(t) +
x
Z
0
zr(2)(ξ, t)dξ, (x, t)∈Ωr, r = 1, N .
Continuing the process, at thek-th step we obtain the system{λ(k)r (x),ev(k)r (x, t), zr(k)(x, t), u(k)r (x, t)}, r= 1, N .
The conditions of the following statement ensure the feasibility and convergence of the proposed algorithm, as well as the unique solvability of problem (1.13)-(1.18).
Theorem 2.1. Let for some 0< µ <1, h >0 :N h=T, N = 1,2, ..., and ν, ν∈N, (nN ×nN) the matrix Qν(x, h) be invertible at all x∈[0, ω] and let the following inequalities be satised
1) k[Qν(x, h)]−1k ≤γν(x, h);
2) qν(x, h)(α(x)h)ν! ν ≤µ < 1, where
qν(x, h) = 1 +γν(x, h)
ν
X
j=1
(α(x)h)j
j! , α(x) = max
t∈[0,T]kA(x, t)k,
kA(x, t)k= max
i=1,n n
X
j=1
|aij(x, t)|.
Then there exists a unique solution (λ∗r(x),ev∗(x, t), z∗(x, t), u∗(x, t))to problem (1.13)-(1.18) and the following estimates are valid
a) max
r=1,N
kλ∗r(x)−λ(k)r (x)k+ max
r=1,N
sup
t∈[(r−1)h,rh)
kver∗(x, t)−ver(k)(x, t)k
On an algorithm of nding an approximate solution of a periodic problem 75
≤ ρν(x, h) (k−1)!
Zx
0
ρν(ξ, h)dξ k−1
e
x
R
0
ρν(ξ,h)dξ x
Z
0
dν(ξ, h)dξmax
t∈[0,T]max kϕ(t)k, max
t∈[0,T]kψ(t)k,kfk0
, b) max
r=1,N
sup
t∈[(r−1)h,rh)
kzr∗(x, t)−zr(k)(x, t)k
≤
x
Z
0
max
r=1,N
kλ∗r(ξ)−λ(k)r (ξ)kdξ+
x
Z
0
max
r=1,N
sup
t∈[(r−1)h,rh)
kevr∗(ξ, t)−ver(k)(ξ, t)kdξ,
c) max
r=1,N
sup
t∈[(r−1)h,rh)
ku∗r(x, t)−u(k)r (x, t)k
≤
x
Z
0
max
r=1,N
sup
t∈[(r−1)h,rh)
kzr∗(ξ, t)−zr(k)(ξ, t)kdξ, k = 1,2, ..., where β(x) = max
t∈[0,T]kB(x, t)k, σ(x) = max
t∈[0,T]kC(x, t)k, kfk0 = max
(x,t)∈Ωkf(x, t)k, ku(x, t)k= max
i=1,n
|ui(x, t)|, ρν(x, h) = θν(x, h)[β(x) +xσ(x)]
1−qν(x, h)(α(x)h)ν! ν , θν(x, h) = [γν(x, h) +qν(x, h)]h
ν−1
X
j=0
(α(x)h)j j! ,
dν(x, h) = θν(x, h)β(x) 1−qν(x, h)(α(x)h)ν! ν
x
Z
0
[β(ξ) +σ(ξ) + 1]θν(ξ, h)dξ
+ θν(x, h)σ(x) 1−qν(x, h)(α(x)h)ν! ν
x
Z
0 ξ
Z
0
[β(ξ1) +σ(ξ1) + 1]θν(ξ1, h)dξ1dξ
+(α(x)h)ν
ν! qν(x, h)[β(x) +σ(x) + 1]θν(x, h).
Proof. The following inequalities take place kFν(x, h, z, u)k ≤h
ν−1
X
j=0
(α(x)h)j
j! max
r=1,N
sup
t∈[(r−1)h,rh)
β(x)kzr(x, t)k+σ(x)kur(x, t)k+kf(x, t)k ,
kGν(x, h,ev)k ≤ (α(x)h)ν
ν! max
r=1,N
sup
t∈[(r−1)h,rh)
kevr(x, t)k,
max
r=1,N
sup
t∈[(r−1)h,rh)
kDνr(x, t)k ≤
ν
X
j=1
(α(x)h)j j! . From the zero step of the algorithm, the following estimates follow:
max
r=1,N
kλr(0)(x)k
≤[β(x) +σ(x) + 1]γν(x, h)h
ν−1
X
j=0
(α(x)h)j
j! max
t∈[0,Tmax]kϕ(t)k, max
t∈[0,T]kψ(t)k,kfk0
,
76 N.T. Orumbayeva, A.T. Assanova, A.B. Keldibekova
max
r=1,N
sup
t∈[(r−1)h,rh)
kev(0)r (x, t)k
≤[β(x) +σ(x) + 1]qν(x, h)h
ν−1
X
j=0
(α(x)h)j
j! max
max
t∈[0,T]kϕ(t)k, max
t∈[0,T]kψ(t)k,kfk0
,
max
r=1,N
sup
t∈[(r−1)h,rh)
kzr(0)(x, t)−ψ(t)k
≤
x
Z
0
[β(ξ) +σ(ξ) + 1]θν(ξ, h)dξmax
max
t∈[0,T]kϕ(t)k, max
t∈[0,T]kψ(t)k,kfk0
, max
r=1,N
sup
t∈[(r−1)h,rh)
ku(0)r (x, t)−ϕ(t)k
≤
x
Z
0 ξ
Z
0
[β(ξ1) +σ(ξ1) + 1]θν(ξ1, h)dξ1dξmax
t∈[0,Tmax]kϕ(t)k, max
t∈[0,T]kψ(t)k,kfk0
. The following estimates are valid:
max
r=1,N
kλr(1,1)(x)−λr(1,0)(x)k
≤γν(x, h)h
ν−1
X
j=0
(α(x)h)j
j! β(x) max
r=1,N
sup
t∈[(r−1)h,rh)
kzr(0)(x, t)−ψ(t)k
+γν(x, h)h
ν−1
X
j=0
(α(x)h)j
j! σ(x) max
r=1,N
sup
t∈[(r−1)h,rh)
ku(0)r (x, t)−ϕ(t)k
+γν(x, h)(α(x)h)ν
ν! max
r=1,N
sup
t∈[(r−1)h,rh)
kevr(0)(x, t)k, max
r=1,N
sup
t∈[(r−1)h,rh)
kevr(1,1)(x, t)−ev(1,0)r (x, t)k
≤qν(x, h)h
ν−1
X
j=0
(α(x)h)j
j! β(x) max
r=1,N
sup
t∈[(r−1)h,rh)
kzr(0)(x, t)−ψ(t)k
+qν(x, h)h
ν−1
X
j=0
(α(x)h)j
j! σ(x) max
r=1,N
sup
t∈[(r−1)h,rh)
ku(0)r (x, t)−ϕ(t)k
+qν(x, h)(α(x)h)ν
ν! max
r=1,N
sup
t∈[(r−1)h,rh)
kevr(0)(x, t)k.
Next we establish the inequality
∆(1,1)(x) = max
r=1,N
sup
t∈[(r−1)h,rh)
kver(1,1)(x, t)−ev(1,0)r (x, t)k+ max
r=1,N
kλr(1,1)(x)−λr(1,0)(x)k
≤θν(x, h)β(x)
x
Z
0
[β(ξ) +σ(ξ) + 1]θν(ξ, h)dξmax
max
t∈[0,T]kϕ(t)k, max
t∈[0,T]
kψ(t)k,kfk0
On an algorithm of nding an approximate solution of a periodic problem 77
+θν(x, h)σ(x)
x
Z
0 ξ
Z
0
[β(ξ1) +σ(ξ1) + 1]θν(ξ1, h)dξ1dξmax
t∈[0,T]max kϕ(t)k, max
t∈[0,T]kψ(t)k,kfk0
+θν(x, h)(α(x)h)ν
ν! qν(x, h)[β(x) +σ(x) + 1] max
max
t∈[0,T]kϕ(t)k, max
t∈[0,T]kψ(t)k,kfk0
. Thus,
max
r=1,N
kλr(1,m+1)(x)−λr(1,m)(x)k
≤γν(x, h)(α(x)h)ν
ν! max
r=1,N
sup
t∈[(r−1)h,rh)
kev(1,m)r (x, t)−evr(1,m−1)(x, t)k, (2.1) max
r=1,N
sup
t∈[(r−1)h,rh)
kevr(1,m+1)(x, t)−ev(1,m)r (x, t)k
≤qν(x, h)(α(x)h)ν
ν! max
r=1,N
sup
t∈[(r−1)h,rh)
kevr(1,m)(x, t)−ev(1,m−1)r (x, t)k. (2.2) Owing to the inequality qν(x, h)(α(x)h)ν! ν < 1 follows the uniform convergence of the sequence vr(1,m+1)(x, t), at (x, t) ∈ Ωr, to vr(1)(x, t) and the convergence of the sequence of systems of func- tions λ(1,m+1)r (x) to continuous onx∈[0, ω] functions λ(1)r (x)for all r= 1, N :
max
r=1,N
sup
t∈[(r−1)h,rh)
kevr(1,m+1)(x, t)−evr(1,0)(x, t)k
≤
m
X
j=0
qν(x, h)(α(x)h)ν ν!
j
max
r=1,N
sup
t∈[(r−1)h,rh)
kevr(1,1)(x, t)−ver(1,0)(x, t)k, max
r=1,N
kλr(1,m+1)(x)−λr(1,0)(x)k
≤
m
X
j=0
qν(x, h)(α(x)h)ν ν!
j
h
1 +γν(x, h)(α(x)h)ν ν!
i max
r=1,N
sup
t∈[(r−1)h,rh)
kev(1,1)r (x, t)−evr(1,0)(x, t)k + max
r=1,N
kλr(1,1)(x)−λr(1,0)(x)k.
Passing to the limit atm → ∞,we obtain the estimates:
∆(1)(x) = max
r=1,N
sup
t∈[(r−1)h,rh)
kev(1)r (x, t)−ev(0)r (x, t)k+ max
r=1,N
kλr(1)(x)−λr(0)(x)k
≤dν(x, h) max
t∈[0,Tmax]kϕ(t)k, max
t∈[0,T]kψ(t)k,kfk0
,
max
r=1,N
sup
t∈[(r−1)h,rh)
kzr(1)(x, t)−zr(0)(x, t)k ≤
x
Z
0
∆(1)(ξ)dξ,
max
r=1,N
sup
t∈[(r−1)h,rh)
ku(1)r (x, t)−u(0)r (x, t)k ≤
x
Z
0 ξ
Z
0
∆(1)(ξ1)dξ1dξ.
78 N.T. Orumbayeva, A.T. Assanova, A.B. Keldibekova
For the systems of the of dierencesλr(k+1)(x)−λr(k)(x), evr(k+1)(x, t)−evr(k)(x, t),
zr(k+1)(x, t)−zr(k)(x, t), u(k+1)r (x, t)−u(k)r (x, t), r= 1, N , k = 1,2, ...the following estimates are valid:
max
r=1,N
kλr(k+1,1)(x)−λr(k+1,0)(x)k
≤γν(x, h)h
ν−1
X
j=0
(α(x)h)j
j! β(x) max
r=1,N
sup
t∈[(r−1)h,rh)
kzr(k)(x, t)−zr(k−1)(x, t)k
+γν(x, h)h
ν−1
X
j=0
(α(x)h)j
j! σ(x) max
r=1,N
sup
t∈[(r−1)h,rh)
ku(k)r (x, t)−u(k−1)r (x, t)k, max
r=1,N
sup
t∈[(r−1)h,rh)
kevr(k+1,1)(x, t)−ev(k+1,0)r (x, t)k
≤qν(x, h)h
ν−1
X
j=0
(α(x)h)j
j! β(x) max
r=1,N
sup
t∈[(r−1)h,rh)
kzr(k)(x, t)−zr(k−1)(x, t)k
+qν(x, h)h
ν−1
X
j=0
(α(x)h)j
j! σ(x) max
r=1,N
sup
t∈[(r−1)h,rh)
ku(k)r (x, t)−u(k−1)r (x, t)k, max
r=1,N
kλr(k+1,m+1)(x)−λr(k+1,m)(x)k
≤γν(x, h)(α(x)h)ν
ν! max
r=1,N
sup
t∈[(r−1)h,rh)
kevr(k+1,m)(x, t)−evr(k+1,m−1)(x, t)k, max
r=1,N
sup
t∈[(r−1)h,rh)
kevr(k+1,m+1)(x, t)−ev(k+1,m)r (x, t)k
≤qν(x, h)(α(x)h)ν
ν! max
r=1,N
sup
t∈[(r−1)h,rh)
kevr(k+1,m)(x, t)−ev(k+1,m−1)r (x, t)k, max
r=1,N
sup
t∈[(r−1)h,rh)
kev(k+1,m+1)r (x, t)−evr(k+1,0)(x, t)k
≤
m
X
j=0
h
qν(x, h)(α(x)h)ν ν!
ij
max
r=1,N
sup
t∈[(r−1)h,rh)
kevr(k+1,1)(x, t)−ver(k+1,0)(x, t)k, max
r=1,N
kλr(k+1,m+1)(x)−λr(k+1,0)(x)k
≤
m−1
X
j=0
h
qν(x, h)(α(x)h)ν ν!
ij
γν(x, h)(α(x)h)ν
ν! max
r=1,N
sup
t∈[(r−1)h,rh)
kevr(k+1,1)(x, t)−evr(k+1,0)(x, t)k + max
r=1,N
kλr(k+1,1)(x)−λr(k+1,0)(x)k.
Passing to the limit as m→ ∞, we obtain the following estimates:
max
r=1,N
sup
t∈[(r−1)h,rh)
kevr(k+1)(x, t)−ev(k)r (x, t)k
≤
qν(x, h)h
ν−1
P
j=0
(α(x)h)j j! β(x) 1−qν(x, h)(α(x)h)ν! ν max
r=1,N
sup
t∈[(r−1)h,rh)
kzr(k)(x, t)−zr(k−1)(x, t)k
On an algorithm of nding an approximate solution of a periodic problem 79
+
qν(x, h)h
ν−1
P
j=0
(α(x)h)j j! σ(x) 1−qν(x, h)(α(x)h)ν! ν max
r=1,N
sup
t∈[(r−1)h,rh)
ku(k)r (x, t)−u(k−1)r (x, t)k, (2.3) max
r=1,N
kλr(k+1)(x)−λr(k)(x)k
≤
γν(x, h)h
ν−1
P
j=0
(α(x)h)j j! β(x) 1−qν(x, h)(α(x)h)ν! ν max
r=1,N
sup
t∈[(r−1)h,rh)
kzr(k)(x, t)−zr(k−1)(x, t)k
+
γν(x, h)h
ν−1
P
j=0
(α(x)h)j j! σ(x) 1−qν(x, h)(α(x)h)ν! ν max
r=1,N
sup
t∈[(r−1)h,rh)
ku(k)r (x, t)−u(k−1)r (x, t)k, (2.4) max
r=1,N
sup
t∈[(r−1)h,rh)
kzr(k+1)(x, t)−zr(k)(x, t)k
≤
x
Z
0
max
r=1,N
kλr(k+1)(ξ)−λr(k)(ξ)kdξ+
x
Z
0
max
r=1,N
sup
t∈[(r−1)h,rh)
kevr(k+1)(ξ, t)−evr(k)(ξ, t)kdξ,
max
r=1,N
sup
t∈[(r−1)h,rh)
ku(k+1)r (x, t)−u(k)r (x, t)k
≤
x
Z
0
max
r=1,N
sup
t∈[(r−1)h,rh)
kz(k+1)r (x, t)−zr(k)(x, t)kdξ.
Summing, respectively, the left and right sides of inequalities (2.3), (2.4) we have
∆(k+1)(x) = max
r=1,N
sup
t∈[(r−1)h,rh)
kev(k+1)r (x, t)−evr(k)(x, t)k+ max
r=1,N
kλr(k+1)(x)−λr(k)(x)k
≤ θν(x, h)β(x)
1−qν(x, h)(α(x)h)ν! ν max
r=1,N
sup
t∈[(r−1)h,rh)
kzr(k)(x, t)−zr(k−1)(x, t)k
+ θν(x, h)σ(x)
1−qν(x, h)(α(x)h)ν! ν max
r=1,N
sup
t∈[(r−1)h,rh)
ku(k)r (x, t)−u(k−1)r (x, t)k, (2.5)
max
r=1,N
sup
t∈[(r−1)h,rh)
kzr(k+1)(x, t)−zr(k)(x, t)k ≤
x
Z
0
∆(k+1)(ξ)dξ,
max
r=1,N
sup
t∈[(r−1)h,rh)
ku(k+1)r (x, t)−u(k)r (x, t)k ≤
x
Z
0 ξ
Z
0
∆(k+1)(ξ1)dξ1dξ.
For the function∆(k+1)(x)on the basis of (2.5) we establish the inequality
∆(k+1)(x)≤ρν(x, h)
x
Z
0
∆(k)(ξ)dξ, (2.6)
