A nonlocal problem for loaded partial differential equations of fourth order

MATHEMATICS

DOI 10.31489/2020M1/6-16 UDC 517.956

A.T. Assanova

, A.E. Imanchiyev

, Z.M. Kadirbayeva

1Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan;

2K. Zhubanov Aktobe Regional State University, Kazakhstan;

3International Information Technology University, Almaty, Kazakhstan (E-mail: assanova@math.kz)

A nonlocal problem for loaded

partial differential equations of fourth order

A nonlocal problem for the fourth order system of loaded partial differential equations is considered.

The questions of a existence unique solution of the considered problem and ways of its construction are investigated. The nonlocal problem for the loaded partial differential equation of fourth order is reduced to a nonlocal problem for a system of loaded hyperbolic equations of second order with integral conditions by introducing new functions. As a result of solving nonlocal problem with integral conditions is applied a method of introduction functional parameters. The algorithms of finding the approximate solution to the nonlocal problem with integral conditions for the system of loaded hyperbolic equations are proposed and their convergence is proved. The conditions of the unique solvability of the nonlocal problem for the loaded hyperbolic equations are obtained in the terms of initial data. The results also formulated relative to the original problem.

Keywords:nonlocal problem, loaded partial differential equations of fourth order, integral condition, system of loaded hyperbolic equations, algorithm, unique solvability.

Introduction

Many problems of dynamics and kinetics of gas sorption, processes of drying by air stream, movement of adsorbed mixtures and others lead to the study of nonlocal problems for the systems of hyperbolic equations with loading [1–10] and also for nonlocal problems with integral conditions for equations of hyperbolic type [11–16]. In order to solve these problems, the theoretical methods of ordinary differential equations, loaded differential equations, numerical-analytical method are applied, and new approaches and methods are developed as well. Conditions for solvability are received and ways for finding the approximate solutions are offered.

Mathematical modeling of various processes in physics, chemistry, biology, technology, ecology, economics and others are leaded to nonlocal problems for the higher order loaded differential equations with variable coefficients and parameters. Despite the presence of numerous works, general statements of nonlocal problems for the higher order loaded partial differential equations remain poorly studied up to now. Therefore, the problems of solvability of nonlocal problems for the fourth order partial differential equations with and without loading remain important for applications [17–21].

The Goal of this paper is to study boundary value problems with data on the characteristics for the fourth order system of hyperbolic equations with loading and to establish coefficient criteria for unique solubility and to construct algorithms for finding their approximate solutions. Therefore, in the present paper we study of a

Ре по зи то ри й Ка рГ У

questions the existence and uniqueness of classical solutions to nonlocal problem for the fourth order system of loaded partial differential equations and the methods of finding its approximate solutions. For these purposes, we are applied method of introduction a new functions [22, 23] for solve of this problem.

We consider on the domainΩ = [0, T]×[0, ω]a nonlocal problem for the fourth order system of the loaded partial differential equations with two independent variables

∂⁴u

∂x³∂t =

3

X

i=1

n

A_i(t, x)∂⁴⁻ⁱu

∂x⁴⁻ⁱ +B_i(t, x) ∂⁴⁻ⁱu

∂x³⁻ⁱ∂t o

+C(t, x)u+

+

3

X

i=1 m

X

k=1

n

Ki,k(t, x)∂⁴⁻ⁱu(t, x)

∂x⁴⁻ⁱ +Li,k(t, x)∂⁴⁻ⁱu(t, x)

∂x³⁻ⁱ∂t o

_t=t

k

+

m

X

k=1

Mi,k(t, x)u(tk, x) +f(t, x), (1)

P(x)∂³u(0, x)

∂x³ +S(x)∂³u(T, x)

∂x³ =ϕ(x), x∈[0, ω], (2)

u(t,0) =ψ0(t), ∂u(t, x)

∂x

_x=0=ψ1(t), ∂²u(t, x)

∂x²

_x=0=ψ2(t), t∈[0, T]. (3) Here u(t, x) =col(u1(t, x), u2(t, x), ..., un(t, x))is unknown function; the n×n matrices Ai(t, x), Bi(t, x), C(t, x), Ki,k(t, x), Li,k(t, x), Mi,k(t, x), i= 1,3, k= 1, m, and n vector functionf(t, x)are continuous on Ω;

0≤t1< t2< ... < tm≤T; then×nmatricesP(x),S(x), andnvector functionϕ(x)are continuous on[0, ω];

thenvector-functionsψ0(t), ψ1(t)andψ2(t)are continuously differentiable on[0, T].

LetC(Ω, Rⁿ)be the space of continuous vector functionsu: Ω→Rⁿ onΩwith norm

||u||0= max

(t,x)∈Ω||u(t, x)||.

A functionu(t, x)∈C(Ω, Rⁿ), having partial derivatives ∂^i+ju(t, x)

∂xⁱ∂t^j ∈C(Ω, Rⁿ), i= 1,3, j= 0,1,is called aclassical solution to problem (1)–(3) if it satisfies to system of loaded equations (1) for all(t, x)∈Ω and meets the conditions (2) and (3).

We will investigate the existence of a unique solution to the nonlocal problem for the fourth order loaded partial differential equation (1)–(3). We use method of introduction a new functions for solve of the problem (1)–(3) and construct of its approximate solutions. The nonlocal problem for the fourth order system of loaded partial differential equations is reduced to a nonlocal problem for a system of loaded hyperbolic equations of second order with integral conditions by introducing new functions. An algorithms of finding the approximate solution to the equivalent nonlocal problem with integral conditions are constructed and their convergence is proved. The conditions of the unique solvability of the nonlocal problem for the system of loaded hyperbolic equations with integral conditions are established in the terms of initial data. The results also formulated relative to the original of the nonlocal problem for the fourth order system of loaded partial differential equations.

1 Scheme of the method

Introduce a new unknown functions w(t, x) =∂²u(t, x)

∂x² , v(t, x) = ∂u(t, x)

∂x . Taking into account of first and second conditions in (3), we have

v(t, x) =ψ1(t) + Z x

0

w(t, ξ)dξ, u(t, x) =ψ0(t) +ψ1(t)x+ Z x

0

Z ξ

0

w(t, ξ1)dξ1dξ.

Then problem (1)–(3) is reduced to a following problem

∂²w

∂x∂t =A₁(t, x)∂w

∂x +B₁(t, x)∂w

∂t +A₂(t, x)w+

+

m

X

k=1

n

K1,k(t, x)∂w(tk, x)

∂x +L1,k(t, x)∂w(t, x)

∂t t=tk

+K2,k(t, x)w(tk, x)o

+f(t, x) +g(t, x, v, u), (4)

P(x)∂w(0, x)

∂x +S(x)∂w(T, x)

∂x =ϕ(x), (5)

Ре по зи то ри й Ка рГ У

w(t,0) =ψ2(t), t∈[0, T], (6) v(t, x) =ψ₁(t) +

Z x

0

w(t, ξ)dξ, u(t, x) =ψ₀(t) +ψ₁(t)x+ Z x

0

Z ξ

0

w(t, ξ₁)dξ₁dξ, (7) where

g(t, x, v, u) =A₃(t, x)v(t, x) +B₂(t, x)∂v

∂t +B₃(t, x)∂u

∂t +C(t, x)u+

+

m

X

k=1

n

K3,k(t, x)v(tk, x) +L2,k(t, x)∂v(t, x)

∂t _t=t

k

+L3,k(t, x)∂u(t, x)

∂t _t=t

k

+Mi,k(t, x)u(tk, x)o . From (7) it follows

∂v(t, x)

∂t = ˙ψ1(t) + Z x

0

∂w(t, ξ)

∂t dξ, ∂u(t, x)

∂t = ˙ψ0(t) + ˙ψ1(t)x+ Z x

0

Z ξ

0

∂w(t, ξ1)

∂t dξ1dξ. (8) A triple functions(w(t, x), v(t, x), u(t, x)), wherew(t, x)∈C(Ω,Rⁿ), ∂w(t, x)

∂x ∈C(Ω,Rⁿ),∂w(t, x)

∂t ∈C(Ω,Rⁿ),

∂²w(t, x)

∂x∂t ∈C(Ω,Rⁿ), andv(t, x)∈C(Ω,Rⁿ), ∂v(t, x)

∂t ∈C(Ω,Rⁿ), u(t, x)∈C(Ω,Rⁿ),∂u(t, x)

∂t ∈C(Ω,Rⁿ), is called a solution to problem (4)–(7), if it satisfies the system of loaded hyperbolic equations second order (4) for all(t, x)∈Ω, the boundary conditions (5), (6), and integral relations (7).

The problem (4)–(6) at fixedv(t, x),u(t, x), is a nonlocal problem for system of loaded hyperbolic equations of second order with respect to w(t, x) on Ω. The integral relations (7) allow us to determine the unknown functionsv(t, x)andu(t, x).

From (8) we define the partial derivatives ∂v(t, x)

∂t and ∂u(t, x)

∂t for all(t, x)∈Ω.

The problem (4)–(6) can be interpreted:

• as a nonlocal problem for the system of loaded hyperbolic equations of second order with distributed parametersv(t, x)andu(t, x);

• as an inverse problem for the system of loaded hyperbolic equations of second order, where the unknown functionsv(t, x), u(t, x)determine from integral relations (7);

• as a control problem for the system of loaded hyperbolic equations of second order, where the control functionsv(t, x), u(t, x)satisfy integral constrains (7).

Since the function w(t, x) and the functions v(t, x), u(t, x) are unknown together to find a solution to problem (4)–(7) we use an iterative method.

2 Algorithm for finding of solution to problem (4)–(7)

A triple functions (w^∗(t, x), v^∗(t, x), u^∗(t, x)) we determine as a limit of sequences of triple functions (w^(p)(t, x), v^(p)(t, x), u^(p)(t, x))andp= 0,1,2, ..., by the following algorithm:

Step - 0.1) Let v(t, x) =ψ1(t), u(t, x) =ψ0(t) +ψ1(t)x, ∂v(t, x)

∂t = ˙ψ1(t), ∂u(t, x)

∂t = ˙ψ0(t) + ˙ψ1(t)xin right-hand side of system (4). Then from nonlocal problem for the system of loaded hyperbolic equations (4)–(6) we findw⁽⁰⁾(t, x)for all(t, x)∈Ω. Also we find its partial derivatives ∂w⁽⁰⁾(t, x)

∂x , ∂w⁽⁰⁾(t, x)

∂t and ∂²w⁽⁰⁾(t, x) for all(t, x)∈Ω; ∂x∂t

2) From integral relations (7) we determinev⁽⁰⁾(t, x)andu⁽⁰⁾(t, x):

v⁽⁰⁾(t, x) =ψ₁(t) + Z x

0

w⁽⁰⁾(t, ξ)dξ, u⁽⁰⁾(t, x) =ψ₀(t) +ψ₁(t)x+ Z x

0

Z ξ

0

w⁽⁰⁾(t, ξ₁)dξ₁dξ, (t, x)∈Ω.

Then from (8) we find ∂v⁽⁰⁾(t, x)

∂t and ∂u⁽⁰⁾(t, x)

∂t :

∂v⁽⁰⁾(t, x)

∂t = ˙ψ1(t) + Z x

0

∂w⁽⁰⁾(t, ξ)

∂t dξ, ∂u⁽⁰⁾(t, x)

∂t = ˙ψ0(t) + ˙ψ1(t) + Z x

0

Z ξ

0

∂w⁽⁰⁾(t, ξ1)

∂t dξ1dξ, And so on.

Ре по зи то ри й Ка рГ У

Step - p.1) Suppose that v(t, x) =v^(p−1)(t, x), u(t, x) =u^(p−1)(t, x), ∂v(t, x)

∂t = ∂v^(p−1)(t, x)

∂t and

∂u(t, x)

∂t = = ∂u^(p−1)(t, x)

∂t in right-hand side of system (4). Then from nonlocal problem for the system of hyperbolic equations (4)–(6) we findw^(p)(t, x)for all(t, x)∈Ω. Also we find its partial derivatives ∂w^(p)(t, x)

∂x ,

∂w^(p)(t, x)

∂t and ∂²w^(p)(t, x)

∂x∂t for all(t, x)∈Ω.

2) From integral relations (7) we determinev^(p)(t, x)andu^(p)(t, x):

v^(p)(t, x) =ψ1(t) + Z x

0

w^(p)(t, ξ)dξ, u^(p)(t, x) =ψ0(t) +ψ1(t)x+ Z x

0

Z ξ

0

w^(p)(t, ξ1)dξ1dξ, (t, x)∈Ω.

Then from (8) we find ∂v^(p)(t, x)

∂t and ∂u^(p)(t, x)

∂t :

∂v^(p)(t, x)

∂t = ˙ψ1(t) + Z x

0

∂w^(p)(t, ξ)

∂t dξ, ∂u^(p)(t, x)

∂t = ˙ψ0(t) + ˙ψ1(t) + Z x

0

Z ξ

0

∂w^(p)(t, ξ1)

∂t dξ1dξ, p= 1,2, ....

3 Nonlocal problem for system of loaded hyperbolic equations

We also consider an auxiliary nonlocal problem for system of loaded hyperbolic equations second order

∂²w

∂x∂t =A1(t, x)∂w

∂x +B1(t, x)∂w

∂t +A2(t, x)w+

+

m

X

k=1

nK_1,k(t, x)∂w(t_k, x)

∂x +L_1,k(t, x)∂w(t, x)

∂t _t=t

k

+K_2,k(t, x)w(t_k, x)o

+F(t, x), (9)

P(x)∂w(0, x)

∂x +S(x)∂w(T, x)

∂x =ϕ(x), x∈[0, ω], (10)

w(t,0) =ψ2(t), t∈[0, T]. (11)

Here the functionsF(t, x)∈C(Ω,Rⁿ).

Lett₀= 0,t_m+1=T.

By lines of loading t = tk, k = 1, m, we divide of domain Ω =

m+1

S

r=1

Ωr, where Ωr = [t_r−1, tr]×[0, ω], r= 1, m+ 1. Byw_r(t, x)denote the restriction of functionw(t, x)to the subdomain Ω_rsuch thatw_r: Ω_r→Rⁿ andw_r(t, x) =w(t, x)for all(t, x)∈Ω_rand r= 1, m+ 1.

Further, by λ_r(x) denote the value of w_r(t, x) under t = t_r−1, r = 1, m+ 1. We replace w_r(t, x) by wer(t, x) +λr(x)in each domainΩr, r= 1, m+ 1. This implieswer(t_r−1, x) = 0, and ∂wer(tr−1, x)

∂x = 0, for all x∈[0, ω]andr= 1, m+ 1.

Then the problem (9)–(11) is equivalent to the problem with unknown functions λr(x):

∂²we_r

∂x∂t =A1(t, x)∂we_r

∂x +A1(t, x) ˙λr(x) +B1(t, x)∂we_r

∂t +A2(t, x)wer+A2(t, x)λr(x)+

+

m

X

k=1

n

K1,k(t, x) ˙λk+1(x) +K2,k(t, x)λk+1(x)o +

m

X

k=1

L1,k(t, x)∂wek+1(t, x)

∂t t=tk

+F(t, x), (12)

we_r(t_r−1, x) = 0, x∈[0, ω], r= 1, m+ 1, (13) we_r(t,0) =ψ₂(t)−ψ₂(t_r−1), t∈[t_r−1, t_r], r= 1, m+ 1, (14)

Ре по зи то ри й Ка рГ У

P(x) ˙λ₁(x) +S(x) ˙λ_m+1(x) +S(x)∂we_m+1(t_m+1, x)

∂x =ϕ(x), x∈[0, ω], (15)

∂wes(ts, x)

∂x + ˙λ_s(x) = ˙λ_s+1(x), x∈[0, ω], s= 1, m. (16) Here relations (16) are conditions of continuity at interior linest=t_s,s= 1, mof desired functionw(t, x).

The problems (9)–(11) and (12)–(16) are equivalent in the following sense. If the function w(t, x) is a classical solution to (9)–(11), then system of pairs (λr(x),wer(t, x)), where λr(x) = w(t_r−1, x) and wer(t, x) =w(t, x)−w(t_r−1, x), and(t, x)∈Ωr, andr= 1, m+ 1is a solution to problem (12)–(16). Conversely, if the system of pairs(λ^∗_r(x),we_r^∗(t, x)),(t, x)∈Ωr, andr= 1, m+ 1, is a solution to (12)–(16), then the function w^∗(t, x)defined by the equalities

w^∗(t, x) =λ^∗_r(x) +we_r^∗(t, x) for all (t, x)∈Ωr, and r= 1, m+ 1, is a classical solution to problem (9)–(11).

From compatibility condition at (0,0)we obtain:

λr(0) =ψ2(t_r−1), r= 1, m+ 1. (17)

At fixedλrproblem (12)–(14) is Goursat problem for system of loaded hyperbolic equations of second order on Ω_rwith respect to we_r(t, x), r= 1, m+ 1.

LetVer(t, x) =^∂wer_∂x^(t,x),Wfr(t, x) = ^∂wer_∂t^(t,x).

Goursat problem (12)–(14) is equivalent to the system of three integral equations onΩr at fixedλr(x)

Ver(t, x) = Z t

tr−1

A1(τ, x)Ver(τ, x) +B1(τ, x)fWr(τ, x) +A2(τ, x)wer(τ, x) +

m

X

k=1

L1,k(τ, x)fWk+1(tk, x)+

+F(τ, x) +A₁(τ, x) ˙λ_r(x) +A₂(τ, x)λ_r(x) +

m

X

k=1

n

K_1,k(τ, x) ˙λ_k+1(x) +K_2,k(τ, x)λ_k+1(x)o

dτ, (18)

Wfr(t, x) = ˙ψ2(t) + Z x

0

A1(t, ξ)eVr(t, ξ) +B1(t, ξ)fWr(t, ξ) +A2(t, ξ)wer(t, ξ) +

m

X

k=1

L1,k(t, ξ)fWk+1(tk, ξ)+

+F(t, ξ) +A1(t, ξ) ˙λr(ξ) +A2(t, ξ)λr(ξ) +

m

X

k=1

n

K1,k(t, ξ) ˙λk+1(ξ) +K2,k(t, ξ)λk+1(ξ)o

dξ, (19)

wer(t, x) =ψ2(t)−ψ2(tr−1) + Z t

t_r−1

Wfr(τ, x)dτ. (20)

SubstitutingVe_r(τ, x) = ^∂wer_∂x^(τ,x) in the right-hand side (18) and repeating the processν times, andν ∈N, we obtain

Ver(t, x) =Dν,r(t, x) ˙λr(x) +

m

X

k=1

Deν,r,k(t, x) ˙λk+1(x) +Eν,r(t, x)λr(x) +

m

X

k=1

Eeν,r,k(t, x)λk+1(x)+

+G_ν,r(t, x,Ve_r) +H_ν,r(t, x,fW_r,we_r) +F_ν,r(t, x), (21) where

Dν,r(t, x) = Z t

tr−1

A1(τ, x)dτ+ Z t

tr−1

A1(τ1, x) Z τ₁

tr−1

A1(τ2, x)dτ2dτ1+...+

+ Z t

tr−1

A₁(τ₁, x) Z τ1

tr−1

A₁(τ₂, x)...

Z τν−1

tr−1

A₁(τ_ν, x)dτ_νdτ_ν−1...dτ₂dτ₁,

Deν,r,k(t, x) = Z t

tr−1

K1,k(τ, x)dτ + Z t

tr−1

A1(τ, x) Z τ

tr−1

K1,k(τ1, x)dτ1dτ+

+ Z t

tr−1

A₁(τ, x) Z τ

tr−1

A₁(τ₁, x) Z τ₁

tr−1

K_1,k(τ₂, x)dτ₂dτ₁dτ+...+

Ре по зи то ри й Ка рГ У

+ Z t

tr−1

A₁(τ₁, x)...

Z τν−1

tr−1

A₁(τ_ν, x) Z τν

tr−1

K_1,k(τ_ν+1, x)dτ_ν+1dτ_ν...dτ₁,

Eν,r(t, x) = Z t

tr−1

A2(τ, x)dτ+ Z τ

tr−1

A1(τ, x) Z τ₁

tr−1

A2(τ, x)dτ1dτ+

+ Z t

tr−1

A1(τ, x) Z τ

tr−1

A1(τ1, x) Z τ₁

tr−1

A2(τ2, x)dτ2dτ1dτ+...+

+ Z t

tr−1

A1(τ1, x)...

Z τν−1

tr−1

A1(τν, x) Z τ_ν

tr−1

A2(τν+1, x)dτν+1dτν...dτ1,

Ee_ν,r,k(t, x) = Z t

tr−1

K_2,k(τ, x)dτ+ Z t

tr−1

A₁(τ, x) Z τ

tr−1

K_2,k(τ₁, x)dτ₁dτ+

+ Z t

tr−1

A₁(τ, x) Z τ

tr−1

A₁(τ₁, x) Z τ1

tr−1

K_2,k(τ₂, x)dτ₂dτ₁dτ+...+

+ Z t

tr−1

A1(τ1, x)...

Z τ_ν−1

tr−1

A1(τν, x) Z τ_ν

tr−1

K2,k(τν+1, x)dτν+1dτν...dτ1,

Gν,r(t, x,Ver) = Z t

tr−1

A1(τ1, x)...

Z τν−1

tr−1

A1(τν, x)eVr(τν, x)dτν...dτ2dτ1,

H_ν,r(t, x,Wf_r,we_r) = Z t

tr−1

h

B₁(τ, x)fW_r(τ, x) +A₂(τ, x)we_r(τ, x) +

m

X

k=1

L_1,k(τ, x)fW_k+1(t_k, x)i dτ+

+ Z t

t_r−1

A1(τ1, x) Z τ₁

t_r−1

h

B1(τ2, x)fWr(τ2, x) +A2(τ2, x)wer(τ2, x) +

m

X

k=1

L1,k(τ2, x)fWk+1(tk, x)i

dτ2dτ1+

+...+ Z t

tr−1

A1(τ1, x)...

Z τν−2

tr−1

A1(τν−1, x) Z τν−1

tr−1

h

B1(τν, x)fWr(τν, x) +A2(τν, x)wer(τν, x)+

+

m

X

k=1

L1,k(τν, x)fWk+1(tk, x)i

dτνdτ_ν−1...dτ1,

Fν,r(t, x) = Z t

tr−1

F(τ, x)dτ + Z t

tr−1

A1(τ1, x) Z τ₁

tr−1

F(τ2, x)dτ2dτ1+...+

+ Z t

t_r−1

A1(τ1, x)...

Z τν−2

t_r−1

A1(τν−1, x) Z τν−1

t_r−1

F(τν, x)dτνdτν−1...dτ1, (t, x)∈Ωr, r= 1, m+ 1, ν ∈N, k= 1, m.

From (21) we findVer(tr, x) =^∂wer_∂x^(tr,x) for allx∈[0, ω], andr= 1, m+ 1. Then, substituting their into (15) and (16), and multiplying both sides (15) by hm=tm+1−tm, we obtain the system of differential equations with respect to functionsλr(x), andr= 1, m+ 1:

hmP(x) ˙λ1(x) +hmS(x)

m

X

k=1

Deν,m+1,k(t, x) ˙λk+1(x)+

+hmS(x)[I+Dν,m+1(tm+1, x)] ˙λm+1(x) =

=−hmS(x)h

Eν,m+1(tm+1, x)λm+1(x) +

m

X

k=1

Eeν,m+1,k(tm+1, x)λk+1(x)i

−

−hmS(x)Gν,m+1(tm+1, x,Vem+1)−hmS(x)Hν,m+1(tm+1, x,fWm+1,wem+1)−

−hmS(x)Fν,m+1(tm+1, x) +hmϕ(x), x∈[0, ω], (22)

Ре по зи то ри й Ка рГ У

[I+Dν,s(ts, x)] ˙λs(x) +

m

X

k=1

Deν,s,k(ts, x) ˙λk+1(x)−λ˙s+1(x) =−Eν,s(ts, x)λs(x)−

m

X

k=1

Eeν,s,k(ts, x)λk+1(x)−

−Gν,s(ts, x,Ves)−Hν,s(ts, x,Wfs,wes)−Fν,s(ts, x), s= 1, m, x∈[0, ω]. (23) We denote byQ_ν(x)andE_ν(x)then(m+ 1)×n(m+ 1)matrices composed of the coefficientsλ˙_r(x)andλ_r(x) in (22), (23), respectively,r= 1, m+ 1.

So, we can rewrite the equations (12) and (13) in the compact form

Qν(x) ˙λ(x) =−Eν(x)λ(x)−Fν(x)−Gν(x,Ve)−Hν(x,fW ,w),e (24) whereFν(x) = hmS(x)Fν,m+1(tm+1, x)−hmϕ(x), Fν,1(t1, x), ..., Fν,m(tm, x)⁰

, Gν(x,Ve) = hmS(x)h

Gν,m+1(tm+1, x,Vem+1), Gν,1(t1, x,Ve1), ..., Gν,m(tm, x,Vem)⁰ ,

Hν(x,W ,f w) =e hmS(x)Hν,m+1(tm+1, x,fWm+1,wem+1), Hν,1(t1, x,Wf1,we1), ..., Hν,m(tm, x,Wfm,wem)0

.

System (24) with conditions (17) given us Cauchy problem for ordinary differential equations with respect toλr(x),r= 1, m+ 1.

If we know wer(t, x) and its partial derivatives Ver(t, x), Wfr(t, x), then from Cauchy problem (24), (17) we find λ˙r(x) and λr(x) for all x ∈ [0, ω], where r = 1, m+ 1. Conversely, if we know λr(x) and its derivative λ˙_r(x), then from Goursat problem (12)–(14) we can find we_r(t, x) and its partial derivativesVe_r(t, x), Wf_r(t, x) for all(t, x)∈Ω_r,r= 1, m+ 1. For solve Goursat problem (12)–(14) we use equivalent system of three integral equations (18)–(20).

Since thewer(t, x)andλr(x)are unknown to find a solution to problem (12)–(16) we use the iterative method:

1) At fixed wer(t, x) from the Cauchy problem (24), (17) we find the introducing parameters λr(x) and their derivativeλ˙r(x)for allx∈[0, ω], r= 1, m+ 1; 2) At fixedλr(x)from the Goursat problem (12)–(14) we find the unknown functionswer(t, x)and their partial derivativesVer(t, x),fWr(t, x)for all(t, x)∈Ωr,r= 1, m+ 1.

Let h= max

i=1,m+1

(ti−t_i−1), α(x) = max

t∈[0,T]||A1(t, x)||, βk(x) = max

t∈[0,T]||K1,k(t, x)||,k= 1, m.

The following assertion given us a sufficient conditions of unique solvability to problem (12)–(16) and a convergence this iterative process.

Theorem 1. Let for some ν, ν ∈N the (n(m+ 1)×n(m+ 1)) matrixQν(x) is invertible for all x∈[0, ω]

and the following conditions are valid:

1) ||[Qν(x)]⁻¹|| ≤γν(x), whereγν(x)is positive and continuous on[0, ω]function;

2) q_ν(x) =γ_ν(x)·

e^α(x)h−

ν

P

j=0 [α(x)h]^j

j! +h

e^α(x)h−

ν−1

P

j=0 [α(x)h]^j

j!

ih

m

P

k=1

β_k(x)

≤χ <1, whereχ - const.

Then problem with parameters (12)–(16) has unique solution.

Theorem 2. Let for some ν, ν ∈N the (n(m+ 1)×n(m+ 1)) matrixQν(x) is invertible for all x∈[0, ω]

and conditions 1)-2) of Theorem 1 are fulfilled.

Then nonlocal problem for system of loaded hyperbolic equations of the second order (9)–(11) has unique classical solution.

The proofs of Theorem 1 and 2 are similar of proof Theorem 1 in [22].

Therefore, for problem (4)–(7) we have the following statement.

Theorem 3. Let

i) the n×nmatrices A_i(t, x), B_i(t, x),K_i,k(t, x), L_i,k(t, x),M_i,k(t, x),i= 1,3, k= 1, m,C(t, x), and n vector functionf(t, x)are continuous onΩ;

ii) the n×nmatricesP(x),S(x), andn vector functionϕ(x)are continuous on[0, ω];

iii) the n vector-functionsψ₀(t),ψ₁(t)andψ₂(t)are continuously differentiable on[0, T];

iv) the nonlocal problem for system of loaded hyperbolic equations of the second order (9)–(11) is uniquely solvable for any F(t, x)∈C(Ω,Rⁿ),ϕ(x)∈C([0, ω],Rⁿ)andψ2(t)∈C¹([0, T],Rⁿ).

Then problem with integral conditions (4)–(7) has a unique solution.

This Theorem is proved on the basis of the above algorithm and is similar of proof Theorem 2 [23].

From equivalence of problem (1)–(3) and (4)–(7) it follows Theorem 4. Let

1) the conditions i)–iii) of Theorem 3 are fulfilled;

2) for some ν, ν ∈ N the (n(m+ 1)×n(m+ 1)) matrix Qν(x) is invertible for all x ∈ [0, ω] and

||[Qν(x)]⁻¹|| ≤γν(x), whereγν(x)is positive and continuous on [0, ω] function;

Ре по зи то ри й Ка рГ У

3) the following inequality holds:

qν(x) =γν(x)·

e^α(x)h−

ν

X

j=0

[α(x)h]^j j! +h

e^α(x)h−

ν−1

X

j=0

[α(x)h]^j j!

i h

m

X

k=1

βk(x)

≤χ <1,

whereχ - const.

Then problem (1)–(3) has a unique classical solution.

So, the nonlocal problem for system of loaded partial differential equations of the fourth order (1)–(3) is reduced to an equivalent nonlocal problem with integral conditions for system of loaded hyperbolic equations of the second order. For solve of the nonlocal problem with integral conditions for system of loaded hyperbolic equations of the second order results of articles [22–23] are used. Algorithms of finding solutions to the nonlocal problem with integral conditions for system of loaded hyperbolic equations of the second order are constructed and their convergence is proved. The conditions of the unique solvability to the nonlocal problem for system of loaded partial differential equations of the fourth order are established.

Acknowledgement

Results of this paper are announced at International Scientific Conference «Theoretical and applied questions of mathematics, mechanics and computer science», Buketov Karagandy State University, June 12–14, 2019, Karagandy; and are supported by the grant of Project No. AP 05131220, titled: «Methods for solving the initial-boundary value problems for higher order partial differential equations and their applications» of the Ministry of Education and Science of the Republic of Kazakhstan, 2018–2020 years.

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2 Дженалиев М.Т. Нагруженные уравнения как возмущения дифференциальных уравнений / М.Т. Дженалиев, М.И. Рамазанов. — Алматы: Гылым, 2010. — 334 с.

3 Нахушев А.М. Нагруженные уравнения и их приложения / А.М. Нахушев. — М.: Наука, 2012. — 428 с.

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6 Asanova A.T. On the unique solvability of a nonlocal boundary-value problem for systems of loaded hyperbolic equations with impulsive actions / A.T. Asanova, Zh.M. Kadirbaeva, E.A. Bakirova // Ukrainian Mathematical Journal. — 2018. — Vol. 69, No.8. — P. 1175–1195.

7 Assanova A.T. Periodic problem for an impulsive system of the loaded hyperbolic equations / A.T. Assa- nova, Zh.M. Kadirbayeva // Electronic Journal of Differential Equations. — 2018. — Vol. 2018, No.72. — P. 1–8.

8 Асанова А.Т. О разрешимости многоточечной задачи для нагруженного дифференциального урав- нения в частных производных третьего порядка / А.Т. Асанова, А.Е. Иманчиев, Ж.М. Кадирбаева // Математический журн. — 2018. — Т. 18, №1. — С. 27–35.

9 Assanova A.T., Imanchiyev A.E., Kadirbayeva, Zh.M. On the boundary value problem with data on the characteristics for partial loaded differential equations of hyperbolic type // Нелокальные краевые задачи и родственные проблемы математической биологии, информатики и физики (B&Nak 2018):

материалы V Междунар. науч. конф. (4–7 декабря 2018 г.). — Нальчик, 2018. — C. 233.

10 Assanova A.T. Well-posedness of a periodic boundary value problem for the system of hyperbolic equations with delayed argument / A.T. Assanova, N.B. Iskakova, N.T. Orumbayeva // Bulletin of the Karaganda university. Series Mathematics. — 2018. — Vol. 89, No. 1. — P. 8–14.

Ре по зи то ри й Ка рГ У

11 Asanova A.T. Well-posedness of nonlocal boundary value problems with integral condition for the system of hyperbolic equations / A.T. Asanova, D.S. Dzhumabaev // J. Math. Analysis and Applications. — 2013. — Vol. 402, No. 1. — P. 167–178.

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13 Assanova A.T. Nonlocal problem with integral conditions for systems of hyperbolic equations in characte- ristic rectangle / A.T. Assanova // Russian Mathematics (Iz.VUZ). — 2017. — Vol. 61, No. 5. — P. 7–20.

14 Assanova A.T. Solvability of a nonlocal problem for a hyperbolic equation with integral conditions / A.T. Assanova // Electronic J. Differ. Equ. — 2017. — Vol. 2017, No. 170. — P. 1–12.

15 Assanova A.T. On a nonlocal problem with integral conditions for the system of hyperbolic equations / A.T. Assanova // Differ. Equ. — 2018. — Vol. 54, No. 2. — P. 201–214.

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DOI: https:// doi.org/10.1515/ gmj-2019-2011.

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20 Kiguradze T. On solvability and well-posedness of boundary value problems for nonlinear hyperbolic equations of the fourth order / T. Kiguradze // Georgian Math. J. — 2008. — Vol. 15, No. 3. — P. 555–

569.

21 Ferraioli D.C. Fourth order evolution equations which describe pseudospherical surfaces / D.C. Ferraioli, K. Tenenblat // J. Differential Equations. — 2014. — Vol. 257. — P. 3165–3199.

22 Assanova A.T. Solution of initial-boundary value problem for a system of partial differential equations of the third order / A.T. Assanova // Russian Mathematics. — 2019. — Vol. 63, No. 4. — P. 12–22.

23 Assanova A.T. On solvability of nonlocal problem for system of Sobolev-type differential equations with multipoint conditions / A.T. Assanova, A.E. Imanchiyev, Zh.M. Kadirbayeva // Russian Mathematics (Iz.VUZ). — 2019. — Vol. 63, No. 12. — P. 13–25.

А.Т. Асанова, А.Е. Иманчиев, Ж.М. Кадирбаева

Төртiншi реттi дербес туындылы жүктелген дифференциалдық теңдеу үшiн бейлокал есеп

Төртiншi реттi дербес туындылы жүктелген дифференциалдық теңдеулер жүйесi үшiн бейлокал есеп қарастырылған. Қарастырылып отырған есептiң жалғыз шешiмiнiң бар болуы мәселелерi мен оны табу жолдары зерттелдi. Жаңа функциялар енгiзу әдiсi арқылы төртiншi реттi дербес туындылы жүктелген дифференциалдық теңдеулер жүйесi үшiн бейлокал есеп екiншi реттi жүктелген гипербо- лалық теңдеулер жүйесi үшiн интегралдық шарттары бар бейлокал есепке келтiрiледi. Нәтижесiнде алынған интегралдық шарттары бар бейлокал есептi шешу үшiн функционалдық параметрлер енгiзу әдiсi қолданылды. Жүктелген гиперболалық теңдеулер жүйесi үшiн интегралдық шарттары бар бей- локал есептiң жуық шешiмдерiн табу алгоритмдерi ұсынылған және оның жинақтылығы дәлелден- ген. Жүктелген гиперболалық теңдеулер жүйесi үшiн интегралдық шарттары бар бейлокал есептiң бiрмәндi шешiлiмдiлiгiнiң шарттары бастапқы берiлiмдер терминiнде алынған. Нәтижелер сәйкесiн- ше бастапқы төртiншi реттi дербес туындылы жүктелген дифференциалдық теңдеулер жүйесi үшiн бейлокал есепке қатысты тұжырымдалған.

Кiлт сөздер: бейлокал есеп, төртiншi реттi дербес туындылы жүктелген дифференциалдық теңдеу- лер, интегралдық шарт, жүктелген гиперболалық теңдеулер жүйесi, алгоритм, бiрмәндi шешiлiмдiлiк.

Ре по зи то ри й Ка рГ У

А.Т. Асанова, А.Е. Иманчиев, Ж.М. Кадирбаева

Нелокальная задача для нагруженного дифференциального уравнения в частных производных четвертого порядка

Рассмотрена нелокальная задача для системы нагруженных дифференциальных уравнений в част- ных производных четвертого порядка. Исследованы вопросы существования единственного решения рассматриваемой задачи и способы его построения. Методом введения новых функций нелокальная задача для системы нагруженных дифференциальных уравнений в частных производных четвертого порядка сведена к нелокальной задаче с интегральными условиями для системы нагруженных гипер- болических уравнений второго порядка. Для решения полученной нелокальной задачи с интеграль- ными условиями применен метод введения функциональных параметров. Предложены алгоритмы нахождения приближенного решения нелокальной задачи с интегральными условиями для систе- мы нагруженных гиперболических уравнений второго порядка и доказана их сходимость. Получены условия однозначной разрешимости нелокальной задачи с интегральными условиями для системы на- груженных гиперболических уравнений в терминах исходных данных. Результаты сформулированы относительно исходной нелокальной задачи для системы нагруженных дифференциальных уравне- ний в частных производных четвертого порядка.

Ключевые слова:нелокальная задача, нагруженные дифференциальные уравнения в частных произ- водных четвертого порядка, интегральное условие, система нагруженных гиперболических уравне- ний, алгоритм, однозначная разрешимость.

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5 Assanova, A.T., & Kadirbayeva, Z.M. (2018). On the numerical algorithms of parametrization method for solving a two-point boundary-value problem for impulsive systems of loaded differential equations.

Computational and Applied Mathematics. Vol. 37,No. 4, 4966–4976.

6 Asanova, A.T., Kadirbaeva, Zh.M., & Bakirova, E.A. (2018). On the unique solvability of a nonlocal boundary-value problem for systems of loaded hyperbolic equations with impulsive actions. Ukrainian Mathematical Journal. Vol. 69,No. 8, 1175–1195.

7 Assanova, A.T., & Kadirbayeva, Z.M. (2018). Periodic problem for an impulsive system of the loaded hyperbolic equations.Electronic Journal of Differential Equations. Vol. 2018,No. 72, 1–8.

8 Assanova, A.T., Imanchiyev, A.E., & Kadirbayeva, Z.M. (2018). O razreshimosti mnohotochechnoi zadachi dlia nahruzhennoho differentsialnoho uravnenia v chastnykh proizvodnykh treteho poriadka [On the solvability of multipoint problem for loaded partial differential equation of third order].Matematicheskii zhurnal — Mathematical journal, Vol. 18,No. 1, 27–35 [in Russian].

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11 Asanova, A.T., & Dzhumabaev, D.S. (2013). Well-posedness of nonlocal boundary value problems with integral condition for the system of hyperbolic equations.J. Math. Analysis and Applications, Vol. 402, No. 1, 167–178.

12 Asanova, A.T. (2015). On solvability of nonlinear boundary value problems with integral condition for the system of hyperbolic equations.Electronic J. Qualitative Theory of Differ. Equ., Vol. 63, 1–13.

13 Assanova, A.T. (2017). Nonlocal problem with integral conditions for systems of hyperbolic equations in characteristic rectangle.Russian Mathematics (Iz.VUZ), Vol. 61,No. 5, 7–20.

14 Assanova, A.T. (2017). Solvability of a nonlocal problem for a hyperbolic equation with integral conditions.

Electronic J. Differ. Equ., Vol. 2017,No. 170, 1–12.

15 Assanova, A.T. (2018). On a nonlocal problem with integral conditions for the system of hyperbolic equations.Differ. Equ., Vol. 54,No. 2, 201–214.

16 Assanova, A.T. On the solvability of a nonlocal problem for the system of Sobolev-type differential equations with integral condition.Georgian Math. J.Published Online: 02/19/2019; DOI: https:// doi.org /10.1515/ gmj-2019-2011.

17 Kiguradze, T., & Lakshmikantham, V. (2002). On the Dirichlet problem for fourth order linear hyperbolic equations.Nonlinear Analysis, Vol. 49,No. 2, 197–219.

18 Midodashvili, B. (2002). A nonlocal problem for fourth order hyperbolic equations with multiple charac- teristics.Electr. J. of Differential Equations, Vol. 2002,No. 85, 1–7.

19 Midodashvili, B. (2005). Generalized Goursat problem for a spatial fourth order hyperbolic equation with dominated low terms.Proc. of A. Razmadze Math. Institute, Vol. 138,43–54.

20 Kiguradze, T. (2008). On solvability and well-posedness of boundary value problems for nonlinear hyper- bolic equations of the fourth order.Georgian Math. J., Vol. 15,No. 3, 555–569.

21 Ferraioli, D.C., & Tenenblat, K. (2014). Fourth order evolution equations which describe pseudospherical surfaces.J. Differential Equations, Vol. 157,3165–3199.

22 Assanova, A.T. (2019). Solution of initial-boundary value problem for a system of partial differential equations of the third order.Russian Mathematics (Iz.VUZ), Vol. 63,No. 4, 12–22.

23 Assanova, A.T., Imanchiyev, A.E., & Kadirbayeva, Zh.M. (2019). On solvability of nonlocal problem for system of Sobolev-type differential equations with multipoint conditions.Russian Mathematics (Iz.VUZ), Vol. 63,No. 12, 13–25.

Ре по зи то ри й Ка рГ У