An inverse problem for Hilfer type differential equation of higher order

UDC 517.968.73

T.K. Yuldashev

, B.J. Kadirkulov

, Kh.R. Mamedov

1National University of Uzbekistan, Tashkent, Uzbekistan;

2Tashkent State University of Oriental Studies, Tashkent, Uzbekistan;

3Igdir University, Igdir, Turkey

(E-mail: tursun.k.yuldashev@gmail.com, kadirkulovbj@gmail.com, hanlar@mersin.edu.tr)

An inverse problem for Hilfer type differential equation of higher order

In three-dimensional domain, an identification problem of the source function for Hilfer type partial differential equation of the even order with a condition in an integral form and with a small positive parameter in the mixed derivative is considered. The solution of this fractional differential equation of a higher order is studied in the class of regular functions. The case, when the order of fractional operator is

0< α <1, is studied. The Fourier series method is used and a countable system of ordinary differential

equations is obtained. The nonlocal boundary value problem is integrated as an ordinary differential equation. By the aid of given additional condition, we obtained the representation for redefinition (source) function. Using the Cauchy–Schwarz inequality and the Bessel inequality, we proved the absolute and uniform convergence of the obtained Fourier series.

Keywords:fractional order, Hilfer operator, inverse source problem, Fourier series, integral condition, unique solvability.

Introduction

The theory of the inverse boundary value problems is currently one of the most important fields of the modern theory of differential equations. Consequently, a large number of research works are devoted to study the different kind of inverse problems for differential and integro-differential equations (see, for example, [1–10]).

In cases where the boundary of the flow domain of a physical process is unavailable for measurements, nonlocal conditions in an integral form can serve as additional information sufficient for unique solvability of the problem.

Therefore, researches on the study of nonlocal boundary value problems for differential and integro-differential equations with integral conditions have been intensified (see, for example, [11–20]). In addition, we note that studies of many problems of gas dynamics, theory of elasticity, theory of plates and shells are described by higher-order partial differential equations.

Fractional calculus plays an important role for the mathematical modeling in many natural and engineering sciences [21]. In [22], it is considered problems of continuum and statistical mechanics. In [23] is studied the mathematical problems of Ebola epidemic model. In [24] and [25], it is studied the fractional model for the dynamics of tuberculosis infection and novel coronavirus (nCoV-2019), respectively. The construction of various models of theoretical physics by the aid of fractional calculus is described in [26, Vol. 4, 5], [27], [28]. A detailed review of the application of fractional calculus in solving problems of applied sciences is given in [29, Vol. 6-8], [30]. In [31], the unique solvability of boundary value problem for weak nonlinear partial differential equations of mixed type with fractional Hilfer operator is studied. In [32], the solvability of nonlocal problem for a mixed type fourth-order differential equation with Hilfer fractional operator is studied. In the direction of applications of fractional derivatives to solving partial differential equations the interesting results were also obtained in [33–42].

We recall some basic terms of fractional integro-differentiation, which have been used during the study.

Let (t0;T)⊂ R⁺ ≡ [0;∞)be an interval on the set of positive real numbers, where 0 ≤t0 < T < ∞. The Riemann–Liouville0< α-order fractional integral of a functionη(t)is defined as follows:

I_t^α

0+η(t) = 1 Γ(α)

t

Z

t₀

(t−s)^α−1η(s)ds, α >0, t∈(t₀;T),

∗Corresponding author.

E-mail: tursun.k.yuldashev@gmail.com

Buketov

university

whereΓ(α)is the Gamma function.

Letn−1< α≤n, n∈N. The Riemann–Liouvilleα-order fractional derivative of a functionη(t)is defined as follows:

D_t^α₀₊η(t) = dⁿ

dtⁿI_t^n−α₀₊ η(t), t∈ t0;T .

The Gerasimov-Caputoα-order fractional derivative of a functionη(t)is defined by

∗D_t^α

0+η(t) =I_t^n−α

0+ η⁽ⁿ⁾(t) = 1 Γ(n−α)

t

Z

t₀

η⁽ⁿ⁾(s)d s

(t−s)^α−n+1, t∈ t₀;T .

These derivatives are reduced to then-th order derivatives forα=n∈N: D_tⁿ₀₊η(t) =_∗D_tⁿ₀₊η(t) = dⁿ

dtⁿη(t), t∈ t0;T .

The Hilfer fractional derivatives ofα-order (n−1< α≤n, n∈N)andβ-type(0≤β ≤1)are defined by the following composition of three operators:

D_t^{α, β}₀₊η(t) =I_t^β(n−α)₀₊ dⁿ

dtⁿI_t^{(1−β)(n−α)}₀₊ η(t), t∈ t0;T . For β = 0, this operator is reduced to the Riemann–Liouville fractional derivative D_t^α,0

0+ =D_t^α

0+and the case β= 1corresponds to the Gerasimov–Caputo fractional derivativeD_t^α,₀₊¹=_∗D_t^α₀₊. Letγ=α+β n−α β. It is easy to see, thatα≤γ≤n. Then it is convenient to use another designation for the operatorD^{α, γ}η(t) =D_t^{α, β}

0+η(t).

The generalized Riemann–Liouville operator was introduced by R. Hilfer based on time evolutions that arise during the transition from the microscopic scale to the macroscopic time scale (see [26]).

In this paper, for 0< α < γ≤1 we study the regular solvability of an inverse boundary value problem for a Hilfer type partial differential equation of even order with positive small parameter. The source function is in the integral condition containing the Riemann–Liouville0< α <1-order fractional integral. The stability of the solution from the given functions is proved.

In the three-dimensional domain Ω ={(t, x, y)|0< t < T, 0< x, y < l}a partial differential equation of the following form is considered

D^{α, γ}_ε [U] =a(t)b(x, y) (1)

with a nonlocal condition on the integral form containing the Riemann–Liouville 0 < α <1-order fractional integral

U(T, x, y) + I₀₊^ρ U(t, x, y)

|t=T =ϕ(x, y), 0≤x , y≤l, (2)

whereρ,T andl are given positive real numbers, D_ε^{α, γ}[U] =

D^{α, γ}+εD^{α, γ} ∂^4k

∂ x^4k + ∂^4k

∂ y^4k

+ω ∂^4k

∂ x^4k + ∂^4k

∂ y^4k

U(t, x, y),

ω is a positive parameter, ε is a positive small parameter, 0 < α < γ ≤ 1, k is a given positive integer, a(t)∈C(ΩT),ΩT ≡[0;T], Ωl≡[0;l],b(x, y)∈C Ω²_l

is a known function,ϕ(x, y)is a source (redefinition) function,Ω²_l ≡Ω_l×Ω_l. We assume that for given functions are true the following boundary conditions

ϕ(0, y) =ϕ(l, y) =ϕ(x,0) =ϕ(x, l) = 0, b(0, y) =b(l, y) =b(x,0) =b(x, l) = 0.

Problem Statement. We find the pair of unknown functions {U(t, x, y);ϕ(x, y)}, first of them satisfies differential equation (1), nonlocal integral condition (2), zero boundary value conditions

U(t,0, y) =U(t, l, y) =U(t, x,0) =U(t, x, l) =

= ∂²

∂ x²U(t,0, y) = ∂²

∂ x²U(t, l, y) = ∂²

∂ x²U(t, x,0) = ∂²

∂ x²U(t, x, l) =

Buketov

university

= ∂²

∂y²U(t,0, y) = ∂²

∂y²U(t, l, y) = ∂²

∂y²U(t, x,0) = ∂²

∂y²U(t, x, l) =. . .=

= ∂^4k−2

∂x^4k−2U(t,0, y) = ∂^4k−2

∂x^4k−2U(t, l, y) = ∂^4k−2

∂x^4k−2U(t, x, 0) = ∂^4k−2

∂x^4k−2U(t, x, l) =

= ∂^4k−2

∂y^4k−2U(t,0, y) = ∂^4k−2

∂y^4k−2U(t, l, y) = ∂^4k−2

∂y^4k−2U(t, x,0) = ∂^4k−2

∂y^4k−2U(t, x, l) = 0, (3) properties of the class of functions

t^1−γU(t, x, y)∈C(Ω),

D^{α, γ}U(t, x, y)∈C_{x, y}^4k,4k(Ω)∩C_{x, y}^4k+0(Ω)∩C_{x, y}^0+4k(Ω) (4) and the additional condition

U t1, x, y

=ψ(x, y), 0< t1< T, 0≤x, y≤l, (5) ϕ(x)∈C[0;l], whereψ(x, y)are given smooth function and

ψ(0, y) =ψ(l, y) =ψ(x,0) =ψ(x, l) = 0,

C_{x, y}^4k+0(Ω) is the class of continuous functions ^∂4kU_{∂ x}^{(t, x, y)}_4k on Ω, while C_{x, y}^0+4k(Ω) is the class of continuous functions ^∂4kU_{∂ y}^{(t, x, y)}4k onΩ, _∂y^∂4k−24k−2U(t, x, l)we understand as _∂y^∂4k−24k−2U(t, x, y)

_y=l,Ω ={(t, x, y)|0≤t≤T, 0≤x, y≤l}.

1 Expansion of the solution in a Fourier series We seek nontrivial solutions of the problem in the form of Fourier series

U(t, x, y) =

∞

X

n, m=1

un , m(t)ϑn, m(x, y), (6)

where

u_{n, m}(t) =

l

Z

0 l

Z

0

U(t, x, y)ϑ_{n, m}(x, y)d x d y, (7)

ϑn, m(x, y) =2

l sin π n

l xsin π m

l y, n, m= 1,2, . . . We also suppose that the following function is expand in a Fourier series

b(x, y) =

∞

X

n, m=1

b_{n, m}ϑ_{n, m}(x, y), (8)

where

bn, m=

l

Z

0 l

Z

0

b(x, y)ϑn, m(x, y)d x d y. (9)

Substituting Fourier series (6) and (8) into given partial differential equation (1), we obtain the countable system of ordinary differential equations of a fractional0< α, γ <1-order

D^{α, γ}u_{n, m}(t) +λ^2k_{n, m}(ε)ω u_{n, m}(t) = a(t)bn, m

1 +ε µ^4k_{n, m}, (10)

where

λ^2k_{n, m}(ε) = µ^4k_{n, m}

1 +ε µ^4k_{n, m}, µ^k_{n, m}=π l

^k p

n^2k+m^2k.

Buketov

university

The general solution of the countable system of differential equations (10) has the form [30]

un, m(t) =Cn, mt^γ−1Eα, γ −λ^2k_{n, m}(ε)ω t^α

+bn, mhn, m(t), (11) where

Eα, γ(z) =

∞

X

k=0

z^k

Γ (α k+γ), z, α, γ∈C, Re(α)>0 is the Mittag–Leffler function [26, 269–295] and

h_{n, m}(t) = 1 1 +ε µ^4k_{n, m}

t

Z

0

(t−s)^α−1E_{α, α} −λ^2k_{n, m}(ε)ω(t−s)^α

a(s)d s,

C_{n, m}is an arbitrary constant.

By Fourier coefficients (7), we rewrite integral condition (2) for the countable system (10)

u_{n, m}(T) + I₀₊^ρ u_{n, m}(t)

|_t=T =

l

Z

0 l

Z

0

U(T, x, y) + I₀₊^ρ U(t, x, y)

|_t=T

ϑ_{n, m}(x, y)d x d y=

=

l

Z

0 l

Z

0

ϕ(x, y)ϑn, m(x, y)d x d y=ϕn, m. (12)

To find the unknown coefficients Cn, m in (11), we use condition (12) and from (11) we have Cn, m= 1

σ_{0n, m}[ϕn, m−bn, mσ1n, m], (13) where

σ0n, m=T^γ−1

Eα, γ −λ^2k_{n, m}(ε)ω T^α

+T^ρEα, ρ+γ −λ^2k_{n, m}(ε)ω T^α , σ_{1n, m}=h_{n, m}(T) + I₀₊^ρ h_{n, m}(t)

|t=T. Hereinafter, we use the following properties of the Mittag–Leffler function:

1) The functionE_{α, β}(−t)withα∈(0; 1], β≥αis completely monotonic fort >0, i.e.

(−1)ⁿ[Eα, β(−t)]⁽ⁿ⁾≥0, n= 0,1,2, ...

2) For allα∈(0; 2), β ∈Randargz=πthere takes place the following estimate

|E_{α, γ}(z)| ≤ M₁ 1 +|z|, where0< M1= constdoes not depend onz.

Then, from here follows that there exists numbers M2, M3>0such that 0< M2≤σ0n, m≤M3. Further, substituting the defined coefficients (13) into representation (11), we derived that

u_{n, m}(t) =ϕ_{n, m}A_{n, m}(t) +b_{n, m}B_{n, m}(t), (14) where

A_{n, m}(t) = 1 σ0n, m

t^γ−1E_{α, γ} −λ^2k_{n, m}(ε)ω t^α

, B_{n, m}(t) =h_{n, m}(t)−σ_{1n, m} σ0n, m

t^γ−1E_{α, γ} −λ^2k_{n, m}(ε)ω t^α . Substituting the representation of Fourier coefficients (14) of main unknown function into Fourier series (6), we obtain

U(t, x, y) =

∞

X

n, m=1

ϑ_{n, m}(x, y) [ϕ_{n, m}A_{n, m}(t) +b_{n, m}B_{n, m}(t)]. (15) Fourier series (15) is a formal solution of the direct problem (1)–(4).

Buketov

university

2 Determination of source function

Using additional condition (5) and taking into account (12), we obtain from Fourier series (15) following countable system for Fourier coefficients of the source function

ϕn, mAn, m(t1) +bn, mBn, m(t1) =ψn, m, (16) where

ψn, m=

l

Z

0 l

Z

0

ψ(x, y)ϑn, m(x, y)d x d y. (17)

From relation (16) we find the source function as

ϕ_{n, m}=ψ_{n, m}χ_{1n, m}+b_{n, m}χ_{2n, m}, (18) where

χ1n, m= 1

An, m(t1), χ2n, m=−Bn, m(t1) An, m(t1). A_{n, m}(t₁) = 1

σ0n, m

t^γ−1₁ E_{α, γ} −λ^2k_{n, m}(ε)ω t^α₁

6= 0, 0< t₁< T.

Sinceϕ_{n, m} are Fourier coefficients (see (12)), we substitute representation (18) into the Fourier series

ϕ(x, y) =

∞

X

n, m=1

ϑ_{n, m}(x, y) [ψ_{n, m}χ_{1n, m}+b_{n, m}χ_{2n, m}]. (19)

We prove absolutely and uniformly convergence of Fourier series (19) for the source function. We need to use the concepts of the following Banach spaces:

the Hilbert coordinate space`2 of number sequences{ϕn, m}_{n, m=1}^∞ with norm

kϕk_`

2= v u u t

∞

X

n, m=1

|ϕ_{n, m}|²<∞;

the spaceL₂(Ω²_l)of square-summable functions on the domainΩ²_l = Ω_l×Ω_lwith norm

kϑ(x, y)k_L

2(^Ω_l²) = v u u u t

l

Z

0 l

Z

0

|ϑ(x, y)|²d x d y <∞.

Conditions of smoothness.Let for functions

ψ(x, y), b(x, y)∈C^4k(Ω²_l)

there exist piecewise continuous 4k+ 1 order derivatives. Then by integrating in parts the functions (9) and (17)4k+ 1times over every variablex, y, we obtain the following relations

|ψn, m|= l

π ^8k+2

ψn, m^(8k+2)

n^4k+1m^4k+1, |bn, m|= l

π ^8k+2

b^(8k+2)n, m

n^4k+1m^4k+1, (20)

ψ_{n, m}^(8k+2) _`

2

≤ 2 l

∂^8k+2ψ(x, y)

∂ x^4k+1∂ y^4k+1 _L

2(^Ω_l²)

, (21)

b^(8k+2)_{n, m `}

2

≤ 2 l

∂^8k+2b(x, y)

∂ x^4k+1∂ y^4k+1 _L

2(^Ω_l²)

, (22)

Buketov

university

where

ψ_{n, m}^(8k+2)=

l

Z

0 l

Z

0

∂^8k+2ψ(x, y)

∂ x^4k+1∂ y^4k+1ϑ_{n, m}(x, y)d x d y,

b_{n, m}^(8k+2)=

l

Z

0 l

Z

0

∂^8k+2b(x, y)

∂ x^4k+1∂ y^4k+1ϑn, m(x, y)d x d y.

In obtaining estimates for the solution, we have used these formulas (20)–(22) and the above indicated properties of the Mittag–Leffler function. Then it is easy to see that

σ2= max

n, m {|χ1n, m|; |χ2n, m|}<∞, (23)

where

χ1n, m= 1

An, m(t1), χ2n, m=−Bn, m t1

An, m t1

, 0< t1< T, An, m(t) = 1

σ_{0n, m}t^γ−1Eα, γ −λ^2k_{n, m}(ε)ω t^α

, Bn, m(t) =hn, m(t)−σ1n, m

σ_{0n, m}t^γ−1Eα, γ −λ^2k_{n, m}(ε)ω t^α . Theorem 1. Suppose that the conditions of smoothness and (23) are fulfilled. Then Fourier series (19) convergence absolutely and uniformly in the domainΩ_l².

Proof. We use formulas (20)–(22) and estimate (23). Using the Cauchy–Schwartz inequality for series (19), we obtain the estimate

|ϕ(x, y)| ≤

∞

X

n, m=1

|ϑn, m(x, y)| · |ψn, mχ1n, m+bn, mχ2n, m| ≤

≤ 2 lσ2

" _∞ X

n, m=1

|ψn, m|+

∞

X

n, m=1

|bn, m|

#

≤

≤ 2 l

l π

8k+2

σ2





∞

X

n, m=1

ψn, m^(8k+2)

n^4k+1m^4k+1 +

∞

X

n, m=1

b^(8k+2)n, m

n^4k+1m^4k+1



≤

≤ 2 l

l π

^8k+2 σ2C01

ψ_{n, m}^(8k+2) _`

2

+

b_{n, m}^(8k+2) _`

2

≤

≤γ1

"

∂^8k+2ψ(x, y)

∂ x^4k+1∂ y^4k+1 _L

2(^Ωl²) +

∂^8k+2b(x, y)

∂ x^4k+1∂ y^4k+1 _L

2(^Ωl²)

#

<∞, (24)

where

γ₁=σ₂C_{0 1} 2

l ²l

π ^8k+2

, C_{0 1}= v u u t

∞

X

n, m=1

1

n^8k+2m^8k+2 <∞.

From estimate (24) the absolutely and uniformly convergence of Fourier series (19) implies. The Theorem 1 is proved.

3 Determination of main unknown function

We determined the source function as a Fourier series (19). So, the source function is known. Using representation (16), Fourier series (15), we can present the main unknown function as

U(t, x, y) =

∞

X

n, m=1

ϑn, m(x, y) [ψn, mPn, m(t) +bn, mQn, m(t)], (25)

where

Pn, m(t) =χ1n, mAn, m(t), Qn, m(t) =χ2n, mAn, m(t) +Bn, m(t).

Buketov

university

To establish the uniqueness of the function U(t, x, y)we suppose that there are two functionsU1 andU2

satisfying given conditions (1)–(5). Then their difference U = U1−U2 is a solution of differential equation (1), satisfying conditions (2)–(5) with the function ψ(x, y)≡ 0. By virtue of relations (9) and (16), we have ψ_{n, m}= 0. Hence, from formulas (7) and (25) in the domainΩwe obtain the zero identity

l

Z

0 l

Z

0

t^1−γU(t, x, y)ϑn, m(x, y)d x d y≡0.

By virtue of the completeness of the systems of eigenfunctionsnq

2

lsin^{π n}_l xo , nq

2

lsin^{π m}_l yo

inL2 Ω²_l we deduce thatU(t, x, y)≡0 for allx∈Ω²_l ≡[0;l]² andt∈ΩT ≡[0;T].

Sincet^1−γU(t, x, y)∈C Ω

thent^1−γU (t, x, y)≡0in the domainΩ. Therefore, the solution to problem (1)–(5) is unique in the domainΩ.

Theorem 2. Let the conditions of the Theorem 1 be fulfilled. Then the series (25) converges in the domain Ω. At the same time, in this domain their term-by-term differentiation is possible.

Proof. By virtue of conditions of the theorem 1 and properties of Mittag–Leffler function, as in the case of (23), the functions t^1−γP_{n, m}(t), t^1−γQ_{n, m}(t) are uniformly bounded on the segment [0;T]. So, for any positive integersn, mthere exists a finite constantσ₃, that there takes place the following estimate

maxn, m

0≤t≤Tmax

t^1−γPn, m(t) ; max

0≤t≤T

t^1−γQn, m(t)

≤σ3. (26)

Using estimates (20)–(22) and (26), analogously to estimate (24), for series (25) we obtain

t^1−γU(t, x, y) ≤

∞

X

n, m=1

|ϑ_{n, m}(x, y)| ·

ψ_{n, m}t^1−γP_{n, m}(t) +b_{n, m}t^1−γQ_{n, m}(t) ≤

≤γ2

"

∂^8k+2ψ(x, y)

∂ x^4k+1∂ y^4k+1 _L

2(^Ω_l²) +

∂^8k+2b(x, y)

∂ x^4k+1∂ y^4k+1 _L

2(^Ω_l²)

#

<∞, (27)

whereγ2=C01σ3 2 l

2 l π

^8k+2 .

From estimate (27) the absolutely and uniformly convergence of Fourier series (25) implies. We differentiate the required number of times function (25)

∂^4k

∂ x^4kt^1−γU(t, x, y) =

∞

X

n, m=1

π n l

4k

ϑn, m(x, y)

ψn, mt^1−γPn, m(t) +bn, mt^1−γQn, m(t)

, (28)

∂^4k

∂ y^4kt^1−γU(t, x, y) =

∞

X

n, m=1

π m l

4k

ϑ_{n, m}(x, y)

ψ_{n, m}t^1−γP_{n, m}(t) +b_{n, m}t^1−γQ_{n, m}(t)

. (29)

The expansions of the following functions in a Fourier series are defined in a similar way t^1−γD^{α, γ}U(t, x, y), ∂^4k

∂ x^4kt^1−γD^{α, γ}U(t, x, y), ∂^4k

∂ y^4kt^1−γD^{α, γ}U(t, x, y).

We show the convergence of series (28) and (29). As in the case of estimate (27), applying the Cauchy–

Schwarz inequality, we obtain:

∂^4k

∂ x^4kt^1−γU(t, x, y)

≤

∞

X

n, m=1

π n l

4k

t^1−γu_{n, m}(t)

· |ϑ_{n, m}(x, y)| ≤

≤2 l

π l

^4k σ3

" _∞ X

n, m=1

n^4k|ψn, m|+

∞

X

n, m=1

n^4k|bn, m|

#

≤

Buketov

university

≤ 2 l

l π

^4k+2 σ3





∞

X

n, m=1

ψ^(8k+2)n, m

n m^4k+1 +

∞

X

n, m=1

b^(8k+2)n, m

n m^4k+1



≤

≤ 2 l

l π

4k+2

σ3C02

ψ^(8k+2)_{n, m `}

2

+ b^(8k+2)_{n, m}

_`

2

≤

≤γ₃

"

∂^8k+2ψ(x, y)

∂ x^4k+1∂ y^4k+1 _L

2(^Ω_l²) +

∂^8k+2b(x, y)

∂ x^4k+1∂ y^4k+1 _L

2(^Ω_l²)

#

<∞, (30)

whereγ3= ²_l² l π

^4k+2

σ3C02, C0 2= s ∞

P

n, m=1 1

n m^8k+2 <∞;

∂^4k

∂ y^4kt^1−γU(t, x, y)

≤

∞

X

n, m=1

π m l

^4k

t^1−γun, m(t)

· |ϑn, m(x, y)| ≤

≤2 l

π l

4k

σ3

" _∞ X

n, m=1

m^4k|ψn, m|+

∞

X

n, m=1

m^4k|bn, m|

#

≤

≤ 2 l

l π

^4k+2 σ₃





∞

X

n, m=1

ψ^(8k+2)n, m

n^4k+1m +

∞

X

n, m=1

b^(8k+2)n, m

n^4k+1m



≤

≤ 2 l

l π

^4k+2 σ3C03

ψ^(8k+2)_{n, m `}

2

+

b^(8k+2)_{n, m `}

2

≤

≤γ4

"

∂^8k+2ψ(x, y)

∂ x^4k+1∂ y^4k+1 _L

2(^Ωl²) +

∂^8k+2b(x, y)

∂ x^4k+1∂ y^4k+1 _L

2(^Ωl²)

#

<∞, (31)

where

γ₄= 2

l ²l

π ^4k+2

σ₃C₀₃, C_{0 3}= v u u t

∞

X

n, m=1

1

n^8k+2m <∞.

It is easy to prove the convergence of Fourier series for functions t^1−γD^{α, γ}U(t, x, y), ∂^4k

∂ x^4kt^1−γD^{α, γ}U(t, x, y), ∂^4k

∂ y^4kt^1−γD^{α, γ}U(t, x, y),

since the necessary estimates can be obtained by a similar way as for the cases of estimates (29), (30) and (31).

Therefore, the function U(t, x, y)belongs to the class of functions (4). Theorem 2 is proved.

4 Stability of the solution U(t, x, y) with respect to the given functions and the source function

Theorem 3. Suppose that all the conditions of Theorem 2 are fulfilled. Then, the functionU(t, x, y)as a solution to problem (1)–(5) is stable with respect to a given functionψ(x, y).

Proof. We show that the solution U(t, x, y) of differential equation (1) is stable with respect to a given functionψ(x, y). LetU₁(t, x, y)andU₂(t, x, y)be two different solutions of the inverse boundary value problem (1)–(5), corresponding to two different values of the functionψ1(x, y)andψ2(x, y), respectively.

We put that |ψ1n, m−ψ2n, m| < δn, m, where 0 < δn, m is a sufficiently small positive quantity and the series

∞

P

n, m=1

|δn, m|is convergent. Then, considering this fact by virtue of the conditions of the theorem, from Fourier series (25), it is easy to obtain that

t^1−γ[U1(t, x, y)−U2(t, x, y)]

_C(Ω)≤ 2 lσ3

∞

X

n, m=1

|ψ1n, m−ψ2n, m|< 2 lσ3

∞

X

n, m=1

|δn, m|<∞.

Buketov

university

We putε=²_lσ₃

∞

P

n, m=1

|δ_{n, m}|<∞. Then, from last estimate we finally obtain assertions about the stability of the solution of the differential equation (1) with respect to a given functionψ(x, y)in (5). The Theorem 3 is proved.

By a similar way we have have proved that there hold the following two theorems.

Theorem 4.Suppose that all conditions of Theorem 2 are fulfilled. Then, the functionU(t, x, y)as a solution to problem (1)–(5) is stable with respect to the given functionb(x, y)in the right-hand side of equation (1).

Theorem 5.Suppose that all conditions of Theorem 2 are fulfilled. Then, the functionU(t, x, y)as a solution to problem (1)–(5) is stable with respect to the source functionϕ(x, y).

Remark.It is easy to study the stability of functionU(t, x, y)with respect to asmall parameterε(see [43]).

Conclusions

In three-dimensional domain, an inverse problem of identification of a source function for Hilfer type partial differential equation (1) of the higher even order with integral form condition (2) and a small positive parameter in mixed derivative is considered. Suppose that the conditions of smoothness are fulfilled. Then the solution to this fractional differential equation of the higher order for0 < α < γ ≤1 is studied in the class of regular functions. The Fourier series method have been used and a countable system of ordinary differential equations has been obtained (10). The nonlocal inverse boundary value problem is integrated as an ordinary differential equation. By the aid of given additional condition, we obtained the representation for the source function. Using the Cauchy–Schwarz inequality and the Bessel inequality, we proved the absolute and uniform convergence of the obtained Fourier series (19) for the source function ϕ(x, y)and (25) for the unknown function U(t, x, y) and its derivatives. It is proved that solution of problem (1)–(5) U(t, x, y)is stable with respect to the given functionsψ(x, y), b(x, y)and the source functionϕ(x, y).

References

1 Ашуров Р.Р. Обратная задача по определению плотности тепловых источников для уравнения суб- диффузии / Р.Р. Ашуров, А.Т. Мухиддинова // Дифф. уравн. — 2020. — 56. — № 12. — С. 1596–1609.

2 Ashyralyyev C. Numerical solution to elliptic inverse problem with Neumann-type integral condition and overdetermination / C. Ashyralyyev, A. Cay // Bulletin of the Karaganda University. Mathematics Series.

— 2020. — № 3(99). — P. 5–17.

3 Денисов А.М. Итерационный метод численного решения обратной коэффициентной задачи для системы уравнений в частных производных / А.М. Денисов, A.A. Ефимов // Дифф. уравн. — 2020.

— 56. — № 7. — С. — 927–935.

4 Kabanikhin S.I. An algorithm for source reconstruction in nonlinear shallow-water equations / S.I. Kabanikhin, O.I. Krivorotko // Computational Mathematics and Mathematical Physics. — 2018.

— 58. — № 8. — P. 1334–1343.

5 Kostin A.B. The inverse problem of recovering the source in a parabolic equation under a condition of nonlocal observation / A.B. Kostin // Collection Mathematics. — 2013. — 204. — № 10. — P. 1391–1434.

6 Mamedov Kh.R. Uniqueness of the solution to the inverse problem of scattering theory for the Sturm–Liouville operator with a spectral parameter in the boundary condition / Kh.R. Mamedov //

Mathematical Notes. — 2003. — 74. — № 1. — P. 136–140.

7 Прилепко А.И. Обратные задачи нахождения источника и коэффициентов для эллиптических и параболических уравнений в пространствах Гёльдера и Соболева / А.И. Прилепко, А.Б. Костин, В.В. Соловьёв // Сиб. журн. чист. и прикл. мат. — 2017. — 17. — 3. — С. 67–85.

8 Romanov V.G. Inverse phaseless problem for the electrodynamic equations in an anisotropic medium / V.G. Romanov // Doklady Mathematics. — 2019. — 100. — № 2. — P. 495–500.

9 Romanov V.G. Phaseless inverse problems with interference waves / V.G. Romanov, M. Yamamoto //

Journal of Inverse and Ill-Posed Problems. — 2018. — 26. — № 5. — P. 681–688.

10 Yuldashev T.K. Nonlocal inverse problem for a pseudohyperbolic-pseudoelliptic type integro-differential equations / T.K. Yuldashev // Axioms. — 2020. — 9. — № 2. — ID 45. — 21 p.

Buketov

university

11 Амангалиева М.М. Об одной однородной задаче для уравнения теплопроводности в бесконечной угловой области / М. М. Амангалиева, М.Т. Дженалиев, М.Т. Космакова, М.И. Рамазанов // Сиб.

мат. журн. — 2015. — 56. — № 6. — С. 1234–1248.

12 Assanova A.T. An integral-boundary value problem for a partial differential equation of second order / A.T. Assanova // Turkish Journal of Mathematics. — 2019. — 43. — № 4. — P. 1967–1978.

13 Assanova A.T. A nonlocal problem for loaded partial differential equations of fourth order / A.T. Assanova, A.E. Imanchiyev, Zh.M. Kadirbayeva // Bulletin of the Karaganda University.

Mathematics. — 2020. — 97. — № 1. — P. 6–16.

14 Ильин В.А. Единственность обобщенных решений смешанных задач для волнового уравнения с нелокальными граничными условиями / В.А. Ильин // Дифф. уравн. — 2008. — 44. — № 5. — С.

672–680.

15 Дженалиев М.Т. О граничной задаче для спектрально-нагруженного оператора теплопроводности / М.Т. Дженалиев, М.И. Рамазанов // Сиб. мат. журн. — 2006. — 47. — № 3. — С. 527–547.

16 Jenaliyev M.T. On the solution to a two-dimensional heat conduction problem in a degenerate domain / M.T. Jenaliyev, M.I. Ramazanov, M.T. Kosmakova, Zh.M. Tuleutaeva // Eurasian Mathematical Journal.

— 2020. — 11. — № 3. — P. 89–94.

17 Yuldashev T.K. Nonlocal mixed-value problem for a Boussinesq-type integro-differential equation with degenerate kernel / T.K. Yuldashev // Ukrainian Mathematical Journal. — 2016. — 68. — № 8. — P. 1278–

1296.

18 Юлдашев Т.К. Смешанная задача для псевдопараболического интегро-дифференциального уравне- ния с вырожденным ядром / Т.К. Юлдашев // Дифф. уравн. — 2017. — 53. — № 1. — С. 101–110.

19 Юлдашев Т.К. О разрешимости одной краевой задачи для дифференциального уравнения типа Буссинеска / Т.К. Юлдашев // Дифф. уравн. — 2018. — 54. — № 10. — С. 1411–1419.

20 Yuldashev T.K. On a boundary-value problem for Boussinesq type nonlinear integro-differential equation with reflecting argument / T.K. Yuldashev // Lobachevskii Journal of Mathematics. — 2020. — 41. —

№ 1. — P. 111–123.

21 Samko S.G. Fractional integrals and derivatives / S.G. Samko, A.A. Kilbas, O.I. Marichev // Theory and Applications. Gordon and Breach, Yverdon, 1993.

22 Mainardi F. Fractional calculus: some basic problems in continuum and statistical mechanics / F. Mainardi // In: Carpinteri, A., Mainardi, F. (Eds.) Fractals and Fractional Calculus in Continuum Mechanics.

Springer, Wien, 1997.

23 Area I. On a fractional order Ebola epidemic model / I. Area, H. Batarfi, J. Losada, J.J. Nieto, W. Shammakh, A. Torres // Advances in Difference Equations. — 2015. — 2015. — № 1. — ID 278.

— P. 1–12.

24 Hussain A. Existence of solution and stability for the fractional order novel coronavirus (nCoV-2019) model / A. Hussain, D. Baleanu, M. Adeel // Advances in Difference Equations. — 2020. — 384.

25 Ullah S. A fractional model for the dynamics of tuberculosis infection using Caputo–Fabrizio derivative / S. Ullah, M.A. Khan, M. Farooq, Z. Hammouch, D. Baleanu // Discrete and Continuous Dynamical Systems. Ser. S. — 2020. — 13. — 3. — P. 975–993.

26 Handbook of fractional calculus with applications. Vols. 1-8. Tenreiro Machado J.A. (ed.). Berlin, Boston:

Walter de Gruyter GmbH, 2019.

27 Kumar D. Editorial: fractional calculus and its applications in physics / D. Kumar, D. Baleanu // Frontiers Physics. — 2019. — 7. — № 6.

28 Sun H. A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications / H. Sun, A. Chang, Y. Zhang, W. Chen // Fractional Calculus and Applied Analysis. — 2019. — 22. — № 1. — P. 27–59.

29 Patnaik S. Applications of variable-order fractional operators: a review / S. Patnaik, J.P. Hollkamp, F. Semperlotti // Proceedins of the Royal Society. A. — 2020. — 476. — № 2234. — P. 1–32.

30 Yuldashev T.K. Boundary value problem for weak nonlinear partial differential equations of mixed type with fractional Hilfer operator / T.K. Yuldashev, B.J. Kadirkulov // Axioms. — 2020. — 9. — № 2. — ID 68. — P. 1–19.

Buketov

university

31 Yuldashev T.K. Nonlocal problem for a mixed type fourth-order differential equation with Hilfer fractional operator / T.K. Yuldashev, B.J. Kadirkulov // Ural Mathematical Journal. — 2020. — 6. — № 1. — P. 153–

167.

32 Yuldashev T.K. Inverse problem for a mixed type integro-differential equation with fractional order Caputo operators and spectral parameters / T.K. Yuldashev, E.T. Karimov // Axioms. — 2020. — 9. — 4. — ID 121. — P. 1–19.

33 Abdullaev O.Kh. Non-local problems with integral gluing condition for loaded mixed type equations involving the Caputo fractional derivative / O.Kh. Abdullaev, K.B. Sadarangani // Electronic Journal of Differential Equations. — 2016. — 2016. — № 164. — P. 1–10.

34 Berdyshev A.S. On a nonlocal problem for a fourth-order parabolic equation with the fractional Dzhrbashyan–Nersesyan operator / A.S. Berdyshev, J.B. Kadirkulov // Differential Equations. — 2016.

— 52. — № 1. — P. 122–127.

35 Islomov B.I. On a boundary value problem for a parabolic-hyperbolic equation with fractional order Caputo operator in rectangular domain / B.I. Islomov, U.Sh. Ubaydullayev // Lobachevskii Journal of Mathematics. — 2020. — 41. — № 9. — P. 1801–1810.

36 Karimov E.T. Frankl-type problem for a mixed type equation with the Caputo fractional derivative / E.T. Karimov // Lobachevskii Journal of Mathematics. — 2020. — 41. — № 9. — P. 1829–1836.

37 Karimov E. Nonlocal initial problem for second order time-fractional and space-singular equation / E. Karimov, M. Mamchuev, M. Ruzhansky // Hokkaido Mathematical Journal. — 2020. — 49. — № 2. — P. 349–361.

38 Malik S.A. An inverse source problem for a two parameter anomalous diffusion equation with nonlocal boundary conditions / S.A. Malik, S. Aziz // Computers and Mathematics with Applications. — 2017.

— 73. — № 12. — P. 2548–2560.

39 Serikbaev D. A source inverse problem for the pseudo-parabolic equation with the fractional Sturm–

Liouville operator / D. Serikbaev, N. Tokmagambetov // Bulletin of the Karaganda University.

Mathematics. — 2020. — № 4 (99). — P. 143–151.

40 Sadarangani K.B. A non-local problem with discontinuous matching condition for loaded mixed type equation involving the Caputo fractional derivative / K.B. Sadarangani, O.Kh. Abdullaev // Advances in Difference Equations. — 2016. — № 241. — P. 1–10.

41 Sadarangani K.B. About a problem for loaded parabolic-hyperbolic type equation with fractional derivatives / K.B. Sadarangani, O.Kh. Abdullaev // International Journal of Differerential Equations. — 2016. — 2016. — ID 9815796. — P. 1–6.

42 Yuldashev T.K. Inverse boundary value problem for a fractional differential equations of mixed type with integral redefinition conditions / T.K. Yuldashev, B.J. Kadirkulov // Lobachevskii Journal of Mathematics. — 2021. — 42. — № 3. — P. 649–662.

43 Юлдашев Т.К. Смешанная задача для нелинейного псевдопараболического уравнения высокого по- рядка / Т.К. Юлдашев, К.Х. Шабадиков // Итоги науки и техники. Сер. Современная математика и ее приложения. Тематические обзоры. — 2018. — 156. — С. 73–83.

Т.К. Юлдашев

, Б.Ж. Кадиркулов

, Х.Р. Мамедов

1Өзбекстан ұлттық университетi, Ташкент, Өзбекстан;

2Ташкент мемлекеттiк шығыстану университетi, Ташкент, Өзбекстан;

3Ыгдыр университетi, Ыгдыр, Түркия

Жоғары реттi Хильфер типiнiң жартылай туындылы дифференциалдық теңдеудiң керi есебi

Үшөлшемдi облыста интегралдық формадағы және аралас туындылы кiшi оң параметрi бар жұп реттi Хильфер типiнiң жартылай туындылы теңдеу үшiн функция көзiн анықтау есебi қарастырыл- ған. Бұл жоғары реттi бөлшектi дифференциалдық теңдеудiң шешiмi тұрақты функциялар класында

Buketov

university

зерттелген. Бөлшектi оператордың ретi0< α <1болатын жағдай қарастырылды. Фурье қатарлары әдiсi қолданылды және қарапайым дифференциалдық теңдеулердiң есептеу жүйесi алынды. Локал- ды емес шеттiк есебi қарапайым дифференциалдық теңдеу ретiнде интегралданады. Қосымша шарт арқылы қайта анықтау функциясы туралы түсiнiк берiлген. Коши-Шварц теңсiздiгi мен Бессель теңсiздiгiн қолдана отырып, алынған Фурье қатарларының абсолюттi және бiрқалыпты жинақтылы- ғы дәлелдендi.

Кiлт сөздер: бөлшектi рет, Хилфер операторы, функция көзi туралы керi есеп, Фурье қатарлары, интегралдық шарт, бiрмәндi шешiлуi.

Т.К. Юлдашев

, Б.Ж. Кадиркулов

, Х.Р. Мамедов

1Национальный университет Узбекистана им. Мирзо Улугбека, Ташкент, Узбекистан;

2Ташкентский государственный университет востоковедения, Ташкент, Узбекистан;

3Игдирский университет, Игдир, Турция

Обратная задача для дифференциального уравнения в частных производных типа Хильфера высшего порядка

В трехмерной области рассмотрена задача идентификации функции источника для уравнения в част- ных производных типа Хильфера четного порядка с условием в интегральной форме и малым по- ложительным параметром в смешанной производной. Решение этого дробного дифференциального уравнения высшего порядка получено в классе регулярных функций. Авторами изучен случай для порядка дробного оператора 0 < α <1. Применен метод рядов Фурье, и получена счетная систе- ма обыкновенных дифференциальных уравнений. Нелокальная краевая задача интегрирована как обыкновенное дифференциальное уравнение. С помощью дополнительного условия получено пред- ставление для функции переопределения. С помощью неравенств Коши–Шварца и Бесселя доказана абсолютная и равномерная сходимость полученных рядов Фурье.

Ключевые слова:дробный порядок, оператор Хильфера, обратная задача об источнике, ряды Фурье, интегральное условие, однозначная разрешимость.

References

1 Ashurov, R.R., & Mukhiddinova, A.T. (2020). Obratnaia zadacha po opredeleniiu plotnosti teplovykh istochnikov dlia uravneniia subdiffuzii [Inverse problem of determining the heat source density for the subdiffusion equation. Differentsialnye uravneniia — Differential Equations, 56, 12, 1550–1563 [in Russian].

2 Ashyralyyev, C., & Cay, A. (2020). Numerical solution to elliptic inverse problem with Neumann-type integral condition and overdetermination.Bulletin of the Karaganda University. Mathematics Series, 99, 3,5–17.

3 Denisov, A.M., & Efimov, A.A. (2020). Iteratsionnyi metod chislennogo resheniia obratnoi koeffitsientnoi zadachi dlia sistemy uravnenii v chastnykh proizvodnykh [Iterative method for the numerical solution of an inverse coefficient problem for a system of partial differential equations].Differentsialnye uravneniia

— Differential Equations, 56, 7,900–909 [in Russian].

4 Kabanikhin, S.I., & Krivorotko, O.I. (2018). An algorithm for source reconstruction in nonlinear shallow- water equations.Computational Mathematics and Mathematical Physics, 58, 8,1334–1343.

5 Kostin, A.B. (2013). The inverse problem of recovering the source in a parabolic equation under a condition of nonlocal observation.Collection Mathematics, 204, 10,1391–1434.

6 Mamedov, Kh.R. (2003). Uniqueness of the solution to the inverse problem of scattering theory for the Sturm-Liouville operator with a spectral parameter in the boundary condition.Mathematical Notes, 74, 1, 136–140.

Buketov

university

7 Prilepko, A.I., Kostin, A.B. & Solovev, V.V. (2017). Obratnye zadachi nakhozhdeniia istochnika i koeffitsientov dlia ellipticheskikh i parabolicheskikh uravnenii v prostranstvakh Geldera i Soboleva [Inverse problems of finding the source and coefficients for elliptic and parabolic equations in H¨older and Sobolev spaces].Sibirskii zhurnal chistoi i prikladnoi matematiki – Siberian Journal of Pure and Applied Mathematics, 17, 3,67–85 [in Russian].

8 Romanov, V.G. (2019). Inverse phaseless problem for the electrodynamic equations in an anisotropic medium.Doklady Mathematics, 100, 2, 495–500.

9 Romanov, V.G., & Yamamoto, M. (2018). Phaseless inverse problems with interference waves.Journal of Inverse and Ill-Posed Problems, 26, 5,681–688.

10 Yuldashev, T.K. (2020). Nonlocal inverse problem for a pseudohyperbolic-pseudoelliptic type integro- differential equations.Axioms, 9, 2, ID 45,1–21.

11 Amangalieva, M.M., Dzhenaliev, M.T., Kosmakova, M.T., & Ramazanov, M.I. (2015). Ob odnoi odnorodnoi zadache dlia uravneniia teploprovodnosti v beskonechnoi uglovoi oblasti [On one homogeneous problem for the heat equation in an infinite angular domain].Sibirskii matematicheskii zhurnal — Siberian Mathematical Journal, 56, 6,982–995 [in Russian].

12 Assanova, A.T. (2019). An integral-boundary value problem for a partial differential equation of second order.Turkish Journal of Mathematics, 43, 4,1967–1978.

13 Assanova, A.T., Imanchiyev, A.E., & Kadirbayeva, Zh.M. (2020). A nonlocal problem for loaded partial differential equations of fourth order.Bulletin of the Karaganda University. Mathematics Series, 97, 1, 6–16.

14 Ilin, V.A. (2008). Edinstvennost obobshchennykh reshenii smeshannykh zadach dlia volnovogo uravneniia s nelokalnymi granichnymi usloviiami [Uniqueness of generalized solutions of mixed problems for the wave equation with nonlocal boundary conditions].Differentsialnye uravneniia — Differential Equations, 44, 5,692–700 [in Russian].

15 Dzhenaliev, M.T., & Ramazanov, M.I. (2006). O granichnoi zadache dlia spektralno-nagruzhennogo operatora teploprovodnosti [On the boundary value problem for the spectrally loaded heat conduction operator].Sibirskii matematicheskii zhurnal — Siberian Mathematical Journal, 47, 3,433–451 [in Russian].

16 Jenaliyev, M.T., Ramazanov, M.I., Kosmakova, M.T., & Tuleutaeva, Zh.M. (2020). On the solution to a two-dimensional heat conduction problem in a degenerate domain. Eurasian Mathematical Journal, 11, 3,89–94.

17 Yuldashev, T.K. (2016). Nonlocal mixed-value problem for a Boussinesq-type integro-differential equation with degenerate kernel.Ukrainian Mathematical Journal, 68, 8,1278–1296.

18 Yuldashev, T.K. (2017). Smeshannaia zadacha dlia psevdoparabolicheskogo integro-differentsialnogo uravneniia s vyrozhdennym iadrom [Mixed problem for pseudoparabolic integro-differential equation with degenerate kernel].Differentsialnoe uravneniia — Differential Equations, 53, 1,99–108 [in Russian].

19 Yuldashev, T.K. (2018). O razreshimosti odnoi kraevoi zadachi dlia differentsialnogo uravneniia tipa Bussineska [Solvability of a boundary value problem for a differential equation of the Boussinesq type.

Differentsialnoe uravneniia — Differential Equations, 54, 10,1384–1393 [in Russian].

20 Yuldashev, T.K. (2020). On a boundary-value problem for Boussinesq type nonlinear integro-differential equation with reflecting argument.Lobachevskii Journal of Mathematics, 41, 1,111–123.

21 Samko, S.G., Kilbas, A.A., & Marichev, O.I. (1993). Fractional integrals and derivatives. Theory and Applications. Gordon and Breach, Yverdon, 1993.

22 Mainardi, F. (1997).Fractional calculus: some basic problems in continuum and statistical mechanics.In:

Carpinteri, A., Mainardi, F. (Eds.) Fractals and Fractional Calculus in Continuum Mechanics. Springer, Wien, 1997.

23 Area, I., Batarfi, H., Losada, J., Nieto, J.J., Shammakh, W., & Torres A. (2015). On a fractional order Ebola epidemic model.Advances in Difference Equations, 2015, 1, ID 278,1–12.

24 Hussain, A., Baleanu, D., & Adeel, M. (2020). Existence of solution and stability for the fractional order novel coronavirus (nCoV-2019) model.Advances in Difference Equations, 2020, 384.

25 Ullah, S., Khan, M.A., Farooq, M., Hammouch, Z., & Baleanu, D. (2020). A fractional model for the dynamics of tuberculosis infection using Caputo–Fabrizio derivative.Discrete and Continuous Dynamical Systems, Ser. S, 13, 3,975–993.

Buketov

university

26 Tenreiro Machado, J.A. (Ed.) (2019).Handbook of fractional calculus with applications.Vols. 1-8. Berlin, Boston: Walter de Gruyter GmbH, 2019.

27 Kumar, D., & Baleanu, D. (2019). Editorial: fractional calculus and its applications in physics.Frontiers Physics, 7, 6.

28 Sun, H., Chang, A., Zhang, Y., & Chen, W. (2019). A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications. Fractional Calculus and Applied Analysis, 22, 1,27–59.

29 Patnaik, S., Hollkamp, J.P., & Semperlotti F. (2020). Applications of variable-order fractional operators:

a review.Proceedings of the Royal Society, A., 476, 2234, 1–32.

30 Yuldashev, T.K., & Kadirkulov, B.J.(2020). Boundary value problem for weak nonlinear partial differential equations of mixed type with fractional Hilfer operator.Axioms, 9, 2, ID 68,1–19.

31 Yuldashev, T.K., & Kadirkulov, B.J. (2020). Nonlocal problem for a mixed type fourth-order differential equation with Hilfer fractional operator. Ural Mathematical Journal, 6, 1,153–167.

32 Yuldashev, T.K., & Karimov, E.T. (2020). Inverse problem for a mixed type integro-differential equation with fractional order Caputo operators and spectral parameters. Axioms, 9, 4, ID 121,1–24.

33 Abdullaev, O.Kh., & Sadarangani, K.B. (2016). Nonlocal problems with integral gluing condition for loaded mixed type equations involving the Caputo fractional derivative. Electronic Journal of Differential Equations, 2016, 164,1–10.

34 Berdyshev, A.S., & Kadirkulov, J.B. (2016). On a nonlocal problem for a fourth-order parabolic equation with the fractional Dzhrbashyan–Nersesyan operator. Differential Equations, 52, 1,122–127.

35 Islomov, B.I., & Ubaydullayev, U.Sh. (2020). On a boundary value problem for a parabolic-hyperbolic equation with fractional order Caputo operator in rectangular domain. Lobachevskii Journal of Mathematics, 41, 9,1801–1810.

36 Karimov, E.T. (2020). Frankl-type problem for a mixed type equation with the Caputo fractional derivative. Lobachevskii Journal of Mathematics, 41, 9,1829–1836.

37 Karimov, E., Mamchuev, M., & Ruzhansky M. (2020). Nonlocal initial problem for second order time- fractional and space-singular equation. Hokkaido Mathematical Journal, 49, 2,349–361.

38 Malik, S.A., & Aziz, S. (2017). An inverse source problem for a two parameter anomalous diffusion equation with nonlocal boundary conditions. Computers and Mathematics with Applications, 73, 12, 2548–2560.

39 Serikbaev, D., & Tokmagambetov, N. (2020). A source inverse problem for the pseudo-parabolic equation with the fractional Sturm-Liouville operator. Bulletin of the Karaganda University. Mathematic Series, 4 (99),143–151.

40 Sadarangani, K.B., & Abdullaev, O.Kh. (2016). A nonlocal problem with discontinuous matching condition for loaded mixed type equation involving the Caputo fractional derivative. Advances in Difference Equations, 2016, 241.1–12.

41 Sadarangani, K.B., & Abdullaev, O.Kh. (2016). About a Problem for Loaded Parabolic-Hyperbolic Type Equation with Fractional Derivatives. International Journal of Differential Equations, 2016, ID 9815796, 1–6.

42 Yuldashev, T.K., & Kadirkulov, B.J. (2021). Inverse boundary value problem for a fractional differential equations of mixed type with integral redefinition conditions. Lobachevskii Journal of Mathematics, 42, 3,649–662.

43 Yuldashev, T.K., & Shabadikov, K.H. (2018). Smeshannaia zadacha dlia nelineinogo psevdoparaboliche- skogo uravneniia vysokogo poriadka [Mixed problem for a nonlinear pseudoparabolic equation of higher order]. Itogi nauki i tekhniki. Seriia Sovremennaia matematika i ee prilozheniia. Tematicheskie obzory – Results of Science and Technology. Series: Contemporary mathematics and its applications. Thematic reviews., 156,

Buketov

university