Inverse Problem for a Partial Differential Equation with Gerasimov–Caputo-Type Operator and Degeneration

: In the three-dimensional open rectangular domain, the problem of the identiﬁcation of the redeﬁnition function for a partial differential equation with Gerasimov–Caputo-type fractional operator, degeneration, and integral form condition is considered in the case of the 0 < α ≤ 1 order. A positive parameter is present in the mixed derivatives. The solution of this fractional differential equation is studied in the class of regular functions. The Fourier series method is used, and a countable system of ordinary fractional differential equations with degeneration is obtained. The presentation for the redeﬁnition function is obtained using a given additional condition. Using the Cauchy–Schwarz inequality and the Bessel inequality, the absolute and uniform convergence of the obtained Fourier series is proven.


Introduction
When the boundary of a domain of a physical process is impossible to study, a nonlocal condition of integral form can be obtained as additional information sufficient for the unique solvability of the problem. Therefore, in recent years, research has intensified on the nonlocal direct and inverse boundary value problems for differential and integro-differential equations with integral conditions (see, for example, [1][2][3][4][5][6][7][8][9][10][11][12][13][14]). The many problems of gas dynamics, the theory of elasticity, and the theory of plates and shells have been described by high-order partial differential equations.
Fractional calculus plays an important role in mathematical modeling in many scientific and engineering disciplines [15][16][17][18]. In [19] where problems of continuum and statistical mechanics are considered. The construction of various models of theoretical physics using fractional calculus is described in ([20], Vol. 4,5), [21,22]. A detailed review of the application of fractional calculus to solving problems in applied sciences is provided in ( [23], Vol. [6][7][8], [24]. In [25], an inverse problem to determine the right-hand side for a mixed type integro-differential equation with fractional order Gerasimov-Caputo operators is considered. The problem of determining the source function for a degenerate parabolic equation with the Gerasimov-Caputo operator was investigated [26]. In [27], the solvability of the nonlocal boundary problem for a mixed-type differential equation with a fractional-order operator and degeneration is studied. In the applications of fractional derivatives to solving partial differential equations, interesting results have been obtained [28][29][30][31][32]. We recall some materials from the theory of fractional order integro-differentiation. Let 0; T be an interval on the set of non-negative real numbers, 0 < T < ∞. The Riemann-Liouville 0 < α-order fractional integral for the function η (t) has the form For the case n − 1 < α ≤ n, n ∈ N, the Riemann-Liouville α-order fractional derivative for the function η(t) is defined as follows: The Gerasimov-Caputo α-order fractional derivative for the function η (t) is defined by the following formula * D α These derivatives are reduced to the nth order derivatives for α = n ∈ N: In this paper, for the case of the 0 < α ≤ 1 order, we study the regular one value solvability of the inverse boundary value problem for the Gerasimov-Caputo-type fractional partial differential equation with degeneration. This partial differential equation is a fractional-order ordinary differential equation with respect to the first argument and is a higher even-order partial differential equation with respect to spatial arguments. The stability of the solution on the given functions is proved.
So, in the three-dimensional open domain Ω = {(t, x, y) | 0 < t < T, 0 < x, y < l}, a partial differential equation of the following form is considered with a nonlocal condition on the integral form where ρ, T, and l are given positive real numbers, where ω and β are non-negative parameters, ε is a positive parameter, ε > δ > 0, δ = const, 0 < α ≤ 1, k is a given positive integer, a (t) ∈ C (Ω T ), Ω T ≡ [0; T], Ω l ≡ [0; l], b (x, y) ∈ C Ω 2 l is a known function, and ϕ(x, y) is a redefinition function, Ω 2 l ≡ Ω l × Ω l . We assume that for the given functions, the following boundary conditions are true

Cauchy Problem for a Fractional Ordinary Differential Equation with Degeneration
It is well-known that the two-parametric Mittag-Leffler function is defined as (see, for example, [33]) The generalized Mittag-Lefler (Kilbas-Saigo)-type function was defined for real α, m ∈ R and complex l ∈ C by Kilbas and Saigo in the following form [33] These functions belong to the class of entire functions on the complex plane. Let us consider the Cauchy problem for an ordinary differential equation of fractional order with degeneration * D α where β, λ, u 0 ∈ R, f (t) is a given continuous function.
Proof. The uniqueness of the solution u (t) ∈ C α γ Ω T of the problem (8) was proven in [34], p. 205. In this paper, the existence of solution for the case f (t) = 0 is also proved. So, we consider the inhomogeneous problem (8) and the solution to this problem as the sum of two functions where the functions v (t) and w (t), respectively, are solutions to the following two problems: As implied by [34], p. 233, in their Theorem 4.4, the problem (13) has a unique solution We consider the problem (14). According to [34] in their Corollary 3.24, p. 202, the problem is equivalent to the one value solvability of theVolterra-type integral equation of the second kind We apply the method of successive approximations to solve the integral Equation (16). In this order, we place With the convergence of the iterative process for the integral Equation (16), i.e., the existence of a solution to Equation (16), we can similarly provide the proof of the corre-sponding part of Theorem 3.25 in [34], p. 202. Hence, it is not difficult to determine that the solution to (16) has the form where the kernel K(t, τ) is defines by the Formulas (10) and (11). According to (12), from the representations (15) and (17), we have (9). Lemma 1 is proved.

Expansion of the Solution into Fourier Series
Nontrivial solutions of the inverse problem are sought as a Fourier series where ϑ n, m (x, y) = 2 l sin π n l x sin π m l y, n, m = 1, 2 , . . .
We also suppose that the following function is expand to Fourier series b(x, y) = ∞ ∑ n, m=1 b n, m ϑ n, m (x, y), Substituting Fourier series (25) and (27) into partial differential Equation (1), we obtain the countable system of ordinary fractional differential equations of 0 < α < 1-order with degeneration * D α 0+ u n, m (t) + ω λ 2k n, m t β u n, m (t) = where λ 2k n, m = According to Lemma 1, the general solution of the countable system of differential Equation (29) has the form u n, m (t) = C n, m E n, m t α+β + b n, m h n, m (t), where h n, m (t) = 1 1 + ε µ 4k n, m t 0 K(t, τ) a (τ) d τ, C n, m is arbitrary constant, function K (t, τ) is defined by the formula (11).

Determination of the Redefinition Function
Using the additional condition (5) and considering presentation (33), we obtain from the Fourier series (34) the following countable system for the Fourier coefficients of the redefinition function ϕ n, m A n, m (t 1 ) + b n, m B n, m (t 1 ) = ψ n, m , where From the relation (35), we find the redefinition function as ϕ n, m = ψ n, m χ 1 n, m + b n, m χ 2 n, m , where Since ϕ n, m are Fourier coefficients defined by (31), we substitute presentation (35) into the Fourier series We prove the absolute and uniform convergence of the Fourier series (38) of the redefinition function. We need this to use the concepts of the following well-known Banach spaces and the Hilbert coordinate space 2 of number sequences { ϕ n, m } ∞ n, m=1 with norm The space L 2 (Ω 2 l ) of square-summable functions on the domain Ω 2 l = Ω l × Ω l has norm

Smoothness conditions. For functions
ψ(x, y), b (x, y) ∈ C 4k (Ω 2 l ), let there exist piecewise continuous 4k + 1-order derivatives. Then, by integrating the functions (28) and (36) 4k + 1 times over every variable x, y in parts, we obtain the following relations where By obtaining estimates for the solution, we use the above properties of the Kilbas-Saigo function and Lemma 2. Then, it is easy to see that where Theorem 1. Suppose that the conditions of smoothness are fulfilled. Then, Fourier series (38) is absolute and uniform convergent.

Determination of the Main Unknown Function
So, the redefinition function is determined as a Fourier series (38). Now, the redefinition function is known. Substituting representation (37) into the Fourier series (34), the main unknown function can be presented as where P n, m (t) = χ 1 n, m A n, m (t), Q n, m (t) = χ 2 n, m A n, m (t) + B n, m (t).
Hence, by virtue of the completeness of the systems of eigenfunctions 2 l sin π n l x , 2 l sin π m l y in L 2 Ω 2 l , we deduce that U (t, x, y) ≡ 0 for all x ∈ Ω 2 l ≡ [0, l] 2 and t ∈ Ω T ≡ [0; T].
Since U (t, x, y) ∈ C Ω , then U (t, x, y) ≡ 0 in the domain Ω. Therefore, the solution of the problem (1)-(5) is unique in the domain Ω.
The expansions of the following functions into Fourier series are defined similarly * D α 0+ U (t, x, y), We show the convergence of series (47) and (48). As in the case of estimate (43), applying the Cauchy-Schwarz inequality, we obtain Similarly, we proved that the following two theorems hold.

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

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

Conclusions
In three-dimensional domain, an inverse boundary value problem (1)-(5) of the identification of the redefinition function ϕ (x, y) for a partial differential equation with degeneration and integral form condition was studied in this paper. The case of a 0 < α ≤ 1-order Gerasimov-Caputo-type fractional operator was considered. The solution of the partial differential equation was studied in the class of regular functions. Equation (1) depends on three independent arguments: t, x, y. The first argument is the time argument, and with respect to this argument, Equation (1) is a fractional Gerasimov-Caputo-type ordinary differential equation with degeneration. The other two variables x, y are spatial, and Equation (1) with respect to them is a partial differential equation of higher even order. The Fourier series method was used and a countable system of ordinary differential equations was obtained. Using the given additional condition (5), the presentation for the redefinition function was obtained. When conditions of smoothness are fulfilled, then using the Cauchy-Schwarz inequality and the Bessel inequality, the absolute and uniform convergence of the obtained Fourier series was proved. The stability of main unknown function u (t, x, y) of the problem (1)-(5) with respect to given functions b (x, y), ψ (x, y) and redefinition function ϕ (x, y) was studied.
This work is theoretical in nature and develops the theory of differential equations with fractional operators. We studied the unique solvability in the classical sense of a nonlocal inverse problem for a partial differential equation. In the future, we intend to continue our research in the direction of superposition of several fractional-order operators.