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A boundary-value problem for a mixed type equation involving hyper-Bessel fractional differential operator and Hilfer's bi-ordinal fractional derivative

In a rectangular domain, a boundary-value problem is considered for a mixed-type equation with a regularized Caputo-like counterpart of hyper-Bessel differential operator and the bi-ordinal Hilfer's fractional derivative. Using the method of separation of variables, Laplace transform, a unique solvability of the considered problem has been established. Moreover, we have found the explicit solution of initial problems for a differential equation with the bi-ordinal Hilfer's derivative and regularized Caputo-like counterpart of the hyper-Bessel differential operator with the non-zero starting point.


Introduction
The study of fractional order differential equations has been attracting many scientists because of its adequate and interesting applications in modeling of real-life problems related to several fields of science [1]- [5]. Initial-value problems (IVPs) and boundary-value problems (BVPs) involving the Riemann-Liouville and Caputo derivatives attract most interest (see, for instance, [6], [7], [8]). Especially, studying IVPs and BVPs for the sub-diffusion, fractional wave equations are well-studied (see [9], [10], [11]). BVPs for mixed type equations are also an interesting target for many authors (see [12]- [16]).
Introducing a generalized Riemann-Liouville fractional derivatives (it is called Hilfer's derivative) has opened a new gate in the research of fractional calculus ( [17]- [19]). Therefore, one can find several works devoted to studying this operator in various problems [20], [21]. We also note that in 1968, M. M. Dzhrbashyan and A. B. Nersesyan introduced the following integral-differential operator [24] D σn 0x g(x) = I 1−γn which is more general than Hilfer's operator. Here I α 0x and D α 0x are the Riemann-Liouville fractional integral and the Riemann-Liouville fractional derivative of order α respectively (see Definition 2.1), σ n ∈ (0, n] which is defined by There are some works [25], [26], related with this operator. New wave of researches involving this operator might appear due to the translation of original work [24] in FCAA [27]. In addition, from announcing the concept of hyper-Bessel fractional differential derivative by I. Dimovski [28], several articles have been published dedicated to studying problems containing this type of operators (see [29]- [33]). For instance, fractional diffusion equation and wave equation were widely investigated in different domains in [22]- [23].
In this work, we investigate a boundary value problem for a mixed equation involving the sub-diffusion equation with Caputo-like counterpart of a hyper-Bessel fractional differential operator and the fractional wave equation with Hilfer's bi-ordinal derivative in a rectangular domain. The theorem about the uniqueness and existence of the solution is proved.
The rest of the paper is organized as follows: In Preliminaries section we provide necessary information on Mittag-Leffler functions (Section 2.1), hyper-Bessel functions (Section 2.2.), bi-ordinal Hilfer's fractional derivatives (Section 2.3) and on differential equation involving biordinal Hilfer's fractional derivatives (Section 2.4). Auxiliary result is formulated in Theorem 2.2. In Section 3, we formulate the main problem and state our main result in Theorem 3.1. In Appendix one can find detailed arguments of the proof of Theorem 2.1.

Preliminaries
In this section we present some definitions and auxiliary results related to generalized Hilfer's derivative and fractional hyper-Bessel differential operator which will be used in the sequel. We start recalling the definition of the Mittag-Leffler function.

Important properties of the Mittag-Leffler function
The two parameter Mittag-Leffler (M-L) function is an entire function given by Lemma 2.1 (see [3]) Let α < 2, β ∈ R and πα 2 < µ < min{π, πα}. Then the following estimate holds Here and in the rest of the paper, M denotes a positive constant. In [34], the following estimate for E δ,1 (−t δ ) in the the form g δ (t) ≤ E δ,1 (−t δ ) ≤ f δ (t) was given without proof, where Recently, the lower and upper bounds of the Kilbas-Saigo function, which is a more general form of E δ,1 (−t δ ), were announced. Also the following preposition about the bounds for two parameter M-L function is given in [35].
M-L function can be determined by Wright-type function as a special case E α,β (z) = e β,1 α,0 (z). So, we can record some properties of M-L function which can be reduced from the Wright-type function's properties.

Regularized
Caputo-like counterpart of the hyper-Bessel fractional differential operator where Γ(α) is Euler's gamma-function. Definition 2.2. The Erdelyi-Kober (E-K) fractional integral of a function f (t) ∈ C µ with arbitrary parameters δ > 0, γ ∈ R and β > 0 is defined as ( [8]) which can be reduced up to a weight to I q a+ f (t) (Riemann-Liouville fractional integral) at γ = 0 and β = 1, and Erdelyi-Kober fractional derivative of µ is the weighted space of continuous functions defined as Definition 2.3. Regularized Caputo-like counterpart of the hyper-Bessel fractional differential operator for θ < 1, 0 < α ≤ 1 and t > a ≥ 0 is defined in terms of the E-K fractional order operator or in terms of the hyper-Bessel differential (R-L type) operator where From (4) for a = 0 we obtain the definition presented in ( [32]) and also Caputo FDO is the particular case of Caputo-like counterpart hyper-Bessel operator at θ = 0.
Theorem 2.1. Assume that the following conditions hold: . Then, in Ω = {0 < x < 1, a < t < T }, the problem of finding the solution of the equation satisfying the conditions has a unique solution given by where p = 1 − θ and In fact, for a = 0 Theorem 2.1 implies the result of ( [29]) (see Theorem 3.1). For the detailed proof of Theorem 2.1 see appendix.

Differential equation involving bi-ordinal Hilfer's derivative
Let us consider the following problem: Find a solution of the equation satisfying the initial conditions where f (t) is a given function f ∈ L 1 (0, ∞), and λ, ξ 0 , ξ 1 = const. Theorem 2.
Considering above evaluations and after applying the inverse Laplace transform to (14), we can write the solution of (10)- (12) in the form (13). Theorem 2.2 is proved.

Formulation of a problem and main result
Let us consider the following equation in a domain Ω = Ω 1 ∪ Ω 2 ∪ Q. Here is the regularized Caputo-like counterpart of the hyper-Bessel operator defined as in (3), D (γ,β)µ t is the bi-ordinal Hilfer's derivative defined as in (8).
Problem. Find a solution of (15) in Ω, satisfying regularity conditions and the boundary-initial conditions as well as the gluing conditions here ϕ(x) is a given function.
The key motivation to formulate this problem is a possible application in diffusion-wave processes, which will be described by the mixed type equation as Eq. (15) [38]. Moreover, fractional derivatives used in the mixed equation have more general and also specific character. Therefore, we separately studied related Cauchy problems for ordinary differential equations of fractional order.
The intention of this paper is to prove the uniqueness and existence of the solution to the problem (15)- (20), as stated in the following theorem.
Proof: First we introduce new notations: here τ (x) and ψ(x) are unknown functions to be found later.
Using the method of separation of variables for solving the homogeneous equation corresponding to (15), i.e. searching solution as u(x, t) = T (t)X(x) and considering (16) and (17) in homogeneous case, yield the following problem: It is obvious that (23) is a Sturm-Liouville problem on finding eigenvalues and eigenfunctions and it has the following solution: Using the fact that the system of eigenfunctions {X k } in (24) forms an orthogonal basis in L 2 (0, 1) [39], we look for the solution u(x, t) and given function f (x, t) in the form of series expansions as follows: where u k (t) is unknown function to be found, f k (t) is known and given by Substituting (25) and (26) into equation (15) in Ω 1 and considering initial condition (21) gives the following fractional differential equation where τ k is the coefficient of series expansion of τ (x) in terms of orthogonal basis (23), i.e., After finding the solution of this problem, then considering (24) we can write the solution of (15) in Ω 1 satisfying the conditions (16), (17) and (22) stated in (5). Now by using the solution (5), we evaluate (t − a) 1−(1−θ)α u t (x, t): Considering above given evaluations we obtain the following relation on Q deduced from Ω 1 as t → a+: Now we establish another relation on Q which will be reduced from Ω 2 . According to variable separation method, considering (25), (26) and initial conditions (18), (21), we obtain the following problem finding a solution of equation and satisfying the initial conditions It is obvious that (13) is the solution for above-given problem. Hence, using the solution (13) and taking into account (25) we write the solution of (15) in Ω 2 satisfying (18) and (21) as where W k (t) in the following formulas After substituting (29) and (22) into gluing condition (19) and substituting (30), (27) into the gluing condition (20), we obtain the following the system of linear algebraic equations with respect to τ k and ψ k : From (31), we find ψ k and τ k : here The system of linear equations (31) is equivalent to the considered problem in terms of existing the solution. For that reason, if ∆ = 0 for any k, (31) has only one solution or the considered problem's solution is unique if it exists. Therefore, we show that ∆ is not equal to zero for any k.
By using Lemma 2.4, the behavior of ∆ at k → ∞ can be written as: This proves that ∆ = 0 for sufficiently large k. According to Theorem 3.1 and using Proposition 2.1 we can show that This proves the uniqueness of the solution of the considered problem. Moreover, one can write an upper bound of 1 ∆ by using the last evaluation: Now we find an estimate for B by using Lemma 2.1: Using the last result and (34) we estimate B ∆ ϕ k : here ϕ 1k = 2 1 0 ϕ ′ (x) sin(kπx)dx. Now let us find the upper bound of C: Here we imply that f x (x, t) ∈ L 1 (0, a) for convergence of the last integral.
In [29], the uniform convergence of series of u(x, t) and u xx (x, t) showed for t > 0. Similarly, for t > a, we obtain the following estimate: Since |τ k | < ∞ and f (·, t) ∈ C 3 [0, 1], then the above series converges and hence, by the Weierstrass M-test the series of u(x, t) is uniformly convergent in Ω 1 .
Thus, the series in the expression of u xx (x, t) is bounded by a convergent series which is uniformly convergent according to the Weierstrass M-test. Then, the series of C (t θ ∂ ∂t ) α u(x, t) which can be written by has uniform convergence which can be showed in the same way to the uniform convergence of the series of u xx (x, t) (see [29]). Now we need to show that the series of u(x, t) and its derivatives should converge uniformly in Ω 2 by using (28). We estimate Consider estimates of the Mittag-Leffler function (see Lemma 2.1) where f 1k (t) = for all t >t > 0, 0 ≤ x ≤ 1.
Using the equation in Ω 2 , we write D Finally, considering the Weierstrass M-test, the above arguments prove that Fourier series in (5) and (28) converge uniformly in the domains Ω 1 and Ω 2 . This is the proof that the considered problem's solution exists in Ω. Theorem 3.1 is proved.
This proves that solution (5) satisfies the equation We would like to note that using the result of this work, one can consider FPDE with the Bessel operator considering local [40] and non-local boundary value problems [11]. In that case the Fourier-Bessel series will play an important role. The other possible applications are related with the consideration of more general operators in space variables. For instance, in [41], very general positive operators have been considered, and the results of this paper can be extended to that setting as well.