key: cord-0062812-70br24jh
authors: Yuldashev, T. K.; Kadirkulov, B. J.
title: Inverse Boundary Value Problem for a Fractional Differential Equations of Mixed Type with Integral Redefinition Conditions
date: 2021-05-06
journal: Lobachevskii J Math
DOI: 10.1134/s1995080221030227
sha: 108680068d142d33653a165d282083ed33c06ea4
doc_id: 62812
cord_uid: 70br24jh

In this paper, we consider an inverse boundary value problem for a mixed type partial differential equation with Hilfer operator of fractional integro-differentiation in a positive rectangular domain and with spectral parameter in a negative rectangular domain. The differential equation depends from another positive parameter in mixed derivatives. With respect to first variable this equation is a fractional-order nonhomogeneous differential equation in the positive part of the considering segment, and with respect to second variable is a second-order differential equation with spectral parameter in the negative part of this segment. Using the Fourier series method, the solutions of direct and inverse boundary value problems are constructed in the form of a Fourier series. Theorems on the existence and uniqueness of the problem are proved for regular values of the spectral parameter. It is proved the stability of the solution with respect to redefinition functions, and with respect to parameter given in mixed derivatives. For irregular values of the spectral parameter, an infinite number of solutions in the form of a Fourier series are constructed.

One of the most striking areas of mathematical analysis is the invention of fractional-order integrodifferential operators. Today, the theory and application of operators of fractional differentiation and integration have become a powerful industry of theoretical and applied research at the highest levels of different science and technology. At present, the operators of fractional differentiation and integration are also widely used in the study of problems associated with the study of the coronavirus COVID-19 (see, for example [1, 2] ).

In a rectangular domain Ω = {(t, x) : −a < t < b, 0 < x < l} we consider the fractional partial differential equation of mixed type

where Ω 1 = {(t, x) : 0 < t < b, 0 < x < l}, Ω 2 = {(t, x) : −a < t < 0, 0 < x < l}, ν is positive parameter, ω is positive spectral parameter, a, b are positive real numbers, is Hilfer operator and

is Riemann-Liouville integral operator, f 1 (t) ∈ C[0; b], f 2 (t) ∈ C[−a; 0], g i (x) ∈ C[0; l] are redefinition functions, i = 1, 2.

Problem IT IT IT ν,ω . It is required to find a triple of functions {U (t, x), g 1 (x), g 2 (x)}, which belongs to the class ⎡ (2) satisfies equation (1) in the domain Ω 1 ∪ Ω 2 , boundary value conditions

gluing conditions

and additional integral conditions

where ϕ(x), ψ i (x) are given sufficiently smooth functions, Θ 1 (t) ∈ C[0; b], Θ 2 (t) ∈ C[−a; 0], i = 1, 2.

The generalized integro-differentiation operator D α,γ is a continuous interpolation of the well-known fractional order differentiation operators of Riemann-Liouville and Caputo, which describe diffusion processes [2, vol. 1, 47-85] . A physical and engineering interpretation of the generalized fractional operator D α,γ is given in [2, vol. 4-8] , [3] [4] [5] [6] [7] . In [4] , in particular, were provide results on the existence and representation of solution to an initial value problem for the general ordinary linear fractional differential equation with generalized Riemann-Liouville fractional derivatives and constant coefficients by using operational calculus of Mikusinski type. In [8] , the problem of source identification was studied for the generalized diffusion equation with operator D α,γ . In the work [9] the inverse problems are investigated for a generalized fourth-order parabolic equation with the operator D α,γ .

Note that boundary value conditions of the type (3) take place in modeling problems of the flow around a profile by a subsonic velocity stream with a supersonic zone. Nonlocal boundary value problems for different type of equations were studied in the works of many authors, in particular, in [10] [11] [12] [13] [14] [15] [16] .

In our work, unlike mixed parabolic-hyperbolic equations, the problem of small denominators does not arise. In addition, in our solvability problem we impose conditions that are weaker than in the case of corresponding parabolic-hyperbolic equations. In this paper, we consider an inverse boundary value problem for a mixed type fourth-order differential equation with Hilfer operator of fractional integrodifferentiation and spectral parameter. The Fourier method of separation of variables is used taking into account the features of the fractional integro-differentiation operator. We study the solvability of the problem (1)- (7) for various values of the spectral parameter. We prove the stability of the solution with respect to redefinition functions and with respect to parameter given in mixed derivatives. This work is a further development and generalization of the results of [17] [18] [19] .

We consider the Cauchy problem for a nonhomogeneous differential equation of fractional order

where f (t) is given continuous function, u 0 = const.

. Then the solution of the problem (8) 

where

is Mittag-Leffler function [2, vol. 1, 269-295] .

In [9] , the Laplace method was applied to prove this Lemma 1. In [4] a solution was found using operational calculus for a more general problem than (8) in a specially constructed functional space. We note that there is a more rational way to solve this problem (8) , which allows us to obtain an explicit solution (9) (see [20] ).

The solution of the differential equation (1) in the domain Ω is sought in the form of a Fourier series

where

We suppose also

where g in = l 0 g i (x)ϑ n (x)dx. Substituting series (10) and (12) into equation (1), we obtain a countable systems of differential equations

where λ 2 n (ν) = μ 2 n 1+νμ 2 n , μ n = nπ l , n ∈ N. Taking (11) into account from the conditions (5) we obtain

Analogously we find from (4) that

where

Applying Lemma 1 to (13) and variation method of arbitrary constants to (14) , we obtain the general forms of solutions

where A in are arbitrary constants, i = 1, 3, n = 1, 2, ...,

Taking into account that h 1n (0) = h 2n (0) = 0 and satisfying functions (18) and (19) to conditions (15) and (16), we obtain the following systems of algebraic equations

Applying the condition (17) and representation (20) to (18) and (19), we derive

if there holds the condition

Substituting (21) into (20) , for (18) and (19) we obtain the representations

where

Substituting representations (23) and (24) into the Fourier series (10), we obtain

To establish the uniqueness of the solution U (t, x, ν, ω) of the direct problem (1)-(5) we suppose that there are two solutions U 1 and U 2 of this direct problem. Then their difference U = U 1 − U 2 is a solution of equation (1), satisfying conditions (2)-(5) with functions ϕ(x) ≡ 0. Then for ϕ n = 0 it follows from representations (25) and (26) in the domain Ω that

Hence, by virtue of the completeness of the system of eigenfunctions {ϑ n (x)} in L 2 (0; l), we deduce that

Since

Therefore, the solution of direct problem is unique in the domain Ω. Thus, we have proved that the following theorem holds: Theorem 1. Suppose that there exists a solution of the direct problem (1)- (5) . Then this solution is unique, if condition (22) is fulfilled for all n ∈ N.

Now we consider the case, when condition (22) is violated. Let Δ 0m (ω, ν) = 0 be for all ω, γ ∈ (0; 1). Then the direct problem (1)

where

From the condition Δ 0n (ω, ν) = 0 we come to the trigonometric equation

where ρ n = arcsin

. Hence we obtain, that the quantity Δ 0n (ω) vanishes at

The set of positive solutions of trigonometric equation (28) with respect to spectral parameter ω is called a set of irregular values of the spectral parameter ω. The set of the remaining all values of the spectral parameter ℵ = (0; ∞)\ is called a set of regular values of the spectral parameter ω. For all regular values of the spectral parameter ω the quantity Δ 0n (ω, ν) is nonzero. So, for large n the values of Δ 0n (ω, ν) can not become quite small and there the problem of "small denominators" does not arise. Therefore, for regular values of the spectral parameter ω the quantity Δ 0n (ω, ν) is separated from zero.

Indeed, from the relations λ 2 n (ν) = μ 2 n 1+νμ 2 n , μ n = nπ l we see that λ 2 n (ν) → 1 ν as n → ∞. So, for regular values of the spectral parameter ω we have 

Proof. From (28) for all n and a, b > 0 we derive

We use the following properties of the Mittag-Leffler function [2, vol. 1, 269-295]:

is completely monotonous and there holds

2) For all α ∈ (0; 2), γ ∈ R and arg z = π there takes place the following estimate

where 0 < M 1 = const does not depend from z. Then, from (30) and (31) implies that there exists a number M 0 such that

Consequently, for regular values of the spectral parameter ω there takes place (29): |Δ n (ω)| ≥ M 0 > 0. Lemma 2 is proved. P

We apply the conditions (6) and (7) to Fourier series (25) and (26), respectively:

We suppose here that

where ψ in = l 0 ψ i (x)ϑ n (x)dx. Substituting Fourier series (34) into (32) and (33), we obtain

This system of equations we rewrite as

where

From the system (35) we determine the Fourier coefficients of redefinition functions

if there fulfilled condition

where

Substituting presentations (36) and (37) into Fourier series (12), we found the redefinition functions Then, by integrating in parts desired number of times over the variable x of the integrals

we obtain

Bessel inequalities are true here

Proof. We prove the absolutely and uniformly convergence of series (39). The absolutely and uniformly convergence of series (40) is proved similarly. Taking Lemma 2 into account by the aid of (41)-(44) we obtain

where

Theorem 2 is proved. P

Substituting representations (36) and (37) into the Fourier series (25) and (26), respectively, we obtain 

and for t ∈ [−a; 0]

where C k , k = 1, 5 are positive constants. Now formally differentiating term-by-term the series (45) and (46) the required number of times, we obtain the series

As the case of proof of Theorem 2, it is easy to check the convergence of series (45), (46) and (51)-(54), using the properties (41)-(44) and (47)-(50). Therefore, for regular values of the spectral parameter ω the function U (t, x, ν, ω), represented by series (45) and (46), possesses properties (2) and satisfies conditions (3)-(7). This is finished the proof of the theorem 3. P

For regular values of the spectral parameter ω we consider the question of the stability of the solution of the inverse boundary value problem IT ν,ω with respect to the redefinition functions g i (x), i = 1, 2 and with respect to parameter ν. To this end, we introduce the norm in the space of continuous functions as follows

x, ν, ω)| . Proof. We show that the solution of the mixed differential equation (1) U (t, x, ν, ω) is stable with respect to redefinition functions g i (x), i = 1, 2. Let U 1 (t, x, ν, ω) and U 2 (t, x, ν, ω) be two different solutions of the inverse boundary value problem IT ν,ω , corresponding to two different values of the functions g 11 (x), g 12 (x) and g 21 (x), g 22 (x).

We put that |g 11n − g 12n | < δ 1n , |g 21n − g 22n | < δ 2n , where 0 < δ in are sufficiently small quantities and the series ∞ n=1 |δ in | are convergent, i = 1, 2. Then, taking this fact into account, by virtue of the conditions of the theorem, from (25) and (26) we see that where C 8 = C 6 + C 7 , C 6 = max{C 6.1 ; C 6.2 ; C 6.3 }, C 7 = max{C 7.1 ; C 7.2 ; C 7.3 },

|ξ in (t, ν, ω)| , i = 1, 2, 3.

By virtue of these estimates we finally obtain assertions on the stability of the solution of differential

The Theorem 4 is proved. P

Now we show that the solution of the mixed differential equation (1) U (t, x, ν, ω) is stable with respect to a given parameter ν.

Theorem 5. Suppose that all the conditions of theorem 2 are fulfilled. Then, the solution of the problem IT ν,ω for regular values of the spectral parameter ω is stable with respect to given parameter ν.

Proof. Let U 1 (t, x, ν, ω) and U 2 (t, x, ν, ω) be two different solutions of the inverse boundary value problem IT ν,ω , corresponding to two different values of the parameter ν 1 and ν 1 , respectively. We put that |ν 1 − ν 2 | < δ, where 0 < δ is sufficiently small real number. By virtue of λ 2 n (ν) = μ 2 n 1+νμ 2 n , we derive the following estimates

where C 9 , C 10 , ..., C 13 are constants. Similarly, we can show that the following estimate is also valid ||θ ijn (t, ν 1 , ω) − θ ijn (t, ν 2 , ω)|| ≤ C 14 |ν 1 − ν 2 | , C 14 = const, i = 1, 2, j = 1, 2, 3.

Therefore, for series (45) and (46) 

where C 15 is constant. If we put

then we obtain

The Theorem 5 is proved. P

We note that Δ m (ω, ν) = 0 for irregular values of the spectral parameter ω and m = k 1 , ..., k s (γ = 1). Then, for the solvability of systems (18) and (19), it is necessary and sufficient that the orthogonality conditions are satisfied

In this case, by virtue of (27), the solutions of direct problem are represented as the sum of the series

where m = k 1 , ..., k s , C m are arbitrary constants.

For irregular values of the spectral parameter ω and for some m = k 1 , ..., k s direct problem has an infinite number of solutions in the form of series (56) and (57). The solvability condition is formula (55). Substituting Fourier series (56) and (57) into (6) and (7), respectively, we obtain

where m = k 1 , ..., k s . By virtue of (34), from (58) and (59) we derive coefficients of redefinition functions

Substituting Fourier coefficients (60) and (61) into series (12), (56) and (57) we obtain Fourier series for redefinition functions g 1 (x), g 2 (x) and for main unknown function U (t, x):

(63)

where m = k 1 , ..., k s and C m are arbitrary constants.

The absolutely and uniformly convergence of obtained series (63)-(65) is clear. Because C m are arbitrary numbers. Them we can select so that these series (63)-(65) converge. The coefficients ψ 1m and ψ 2m in (60) and (61) are satisfy the properties (42) and (44).

We studied the inverse boundary value problem IT ν,ω with following assumption For regular values of the spectral parameter ω we studied also the questions of the stability of the solution of the inverse boundary value problem IT ν,ω with respect to the redefinition functions g 1 (x), g 2 (x) and with respect to parameter ν.

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