Using the Hilfer–Katugampola fractional derivative in initial-value Mathieu fractional differential equations with application to a particle in the plane

We examine a class of nonlinear fractional Mathieu equations with a damping term. The equation is an important equation of mathematical physics as it has many applications in various fields of the physical sciences. By utilizing Schauder’s fixed-point theorem, the existence arises of solutions for the proposed equation with the Hilfer–Katugampola fractional derivative, and an application is additionally examined. Two examples guarantee the obtained results.


Introduction
During recent years, fractional Calculus draws increasing attention due to its applications in various applicable fields such as physics, mechanics, chemistry, engineering, etc. The reader interested in the subject should refer to the papers [1-10]. In the literature, one can find that there are many definitions of fractional derivatives [11][12][13][14][15].
In [16], the authors introduced a new generalized derivative involving exponential functions in their kernels that, upon considering limiting cases, converges to classical derivatives. They solved Cauchy linear fractional-type problems within this derivative. One of the generalizations of the well-known Riemann-Liouville and the Hadamard fractional integrals was introduced by Katugampola ( [17][18][19]) in a new fractional integral operator given by Matar et al. investigated the existence and uniqueness of solutions for a p-Laplacian boundary value problem defined by a semilinear fractional system that involves Caputo-Katugampola fractional derivatives (C-KFD) [20]. The Mathieu equation (ME) is an important equation of mathematical physics as it has many applications in several fields of the physical sciences [21][22][23][24][25][26][27][28][29][30][31]. In 1868, Émile Léonard Mathieu introduced for the fist time the second-order differential equation and had encountered them while studying vibrating elliptical drumheads, of the form D 2 y(t) + a -2b cos(2t) y(t) = 0, where D 2 y := dy 2 dt 2 , a and b are real or complex constants [32]. The solution of Equation (1) is built in the form where p is a periodical function with period π and σ is the so-called characteristic index depending on the values of a and b. The function y(t) = exp(-iσ t)p(-t), represents the second solution. In 2010, Rand et al. studied for the first time ME in fractional settings and used the method of harmonic balance to obtain both a lower-and a higher-order approximation for the transition curves [33]. Ebaid  for t, e i ∈ (a, b], where ρ D α,β is H-KFD of order 0 < α < 1 and type 0 ≤ β ≤ 1, and ρ I 1-γ is a generalized fractional derivative of order (1γ ) with γ = α + βαβ and ρ > 0 [35]. Here, w : (a, b] × R → R is a given continuous function, e i , (i = 0, 1, . . . , m) are prefixed points satisfying a < e 1 ≤ e 2 ≤ · · · ≤ e m < b, and q i are real numbers. They also established the existence of solutions by using Krasnoselskii's fixed-point theorem.
Our objective in this work is to study the existence and uniqueness of solutions of the Mathieu fractional differential equation (MFDE) with H-KFD, for t ∈ I 0 = [0, T], with the initial condition for e i ∈ (0, T], where p(t) = a -2b cos(2t), a, b are real constants, ρ D α,β is H-KFD of order α, type 0 ≤ β ≤ 1, and ρ I 1-γ is a generalized fractional derivative of order (1γ ), here γ = α + βαβ, ρ > 0. The map w : I 0 × R 2 → R is a given continuous function, e i for i = 0, 1, . . . , m are prefixed points satisfying 0 < e 1 ≤ e 2 ≤ · · · ≤ e m < T, and q i are real numbers. Also, we consider the existence and uniqueness of Problem (3) and (4).
The plan of the work is as follows. In Sect. 2 we begin with some definitions and lemmas that will be used to prove our main result. In Sect. 3, we prove the existence and uniqueness of the solution. In Sect. 4, we provide two examples to illustrate our main results. We show an application to examine the validity of our theoretical results on the fractional-order representation of the motion of a particle along a straight line in Sect. 5.

Essential preliminaries
For the convenience of the reader, we present here some basic definitions and lemmas, which are used throughout this paper.
] be a finite interval on the half-axis R + , C(J) be the Banach space of all continuous functions from J into R + with the norm y C = max t∈J |y(t)| and the parameters ρ > 0, 0 ≤ γ < 1.
(1) The weighted space C γ ,ρ (J) of continuous functions y on (a, b] is defined by with the norm where C 0,ρ (J) = C(J).
(2) Let δ ρ = (t ρ d dt ). For n ∈ N, we denote by C n δ ρ ,γ (J) the Banach space of functions y that are continuously differentiable on J, with operator δ ρ , up to order (n -1) and that have the derivative δ n ρ y of order n on (a, b] such that δ n ρ y ∈ C γ ,ρ (J), that is, where n ∈ N, with the norms , For n = 0, we have C 0 δ ρ ,γ (J) = C γ ,ρ (J).
Properties 1 ( [17]) We recall some properties of ρ D α,β a + as follows: P1) The operator ρ D α,β a + can be written as a + is an interpolator of the following fractional derivatives: • Hilfer (ρ → 1), First, we state the following key lemma.
The following key theorems are used in the remainder of the paper.
if and only if y satisfies the following equation Theorem 14 (Banach's fixed-point theorem [38]) Let Y be a nonempty closed subset of a Banach space X and F : Y → Y be a contraction operator. Then, there is a unique y ∈ Y with F(y) = y.

Main result
In the following, we present a significant lemma to show the principal theorems.
on both sides of Eq. (20) and Lemma 7, we have By Lemma 9 and property (P1) of operator ρ D α,β a + , we obtain that is, equation (3) holds. To this end, applying ρ I 1-γ 0 + of both sides of Eq. (15): then, applying Lemma 5 and Theorem 6, we obtain and we can write By Lemma 9 and since 1 - Substituting t = e i and multiplying by q i in Eq. (15), we obtain then, From Eqs. (22) and (24), we obtain The proof is completed.
Now, we will prove our first existence result for the problem (3) and (4) that is based on Shauder's fixed-point theorem.
Theorem 18 Assume the following hypotheses hold.
Then, the problem (3) and (4) has at least one solution.
Proof To prove the existence result, we will transform the problem (3) and (4) into a fixedpoint problem. We define the operator F : Note that Fu ∈ C 1-γ ,ρ (I 0 ), ∀u ∈ C 1-γ ,ρ (I 0 ). Since the problem (3) and (4) is equivalent to the fractional integral equation (26), the fixed points of F are solutions of the problem (3) and (4). We establish that F satisfies the assumption of Schauder's fixed-point Theorem 15. This could be proved through several steps.
Step 1. We prove that F is a continuous operator. For any bounded set Y ⊂ C 1-γ ,ρ (I 0 ) there exists ζ > 0 such that Let (y n ) n∈N ∈ Y be a real sequence such that lim n→∞ y ny C 1-γ ,ρ = 0.
Then, for each t ∈ I 0 : Then, Lebesgue's dominated convergence theorem asserts that Consequently, F is continuous.
Step 2. Let ζ ≥ M 0 1-M 1 , we will show that F(Y) ⊂ Y. From (A2) and for each t ∈ I 0 , we have We obtain that Then, F(Y) ⊂ Y.
Step 3. We show that F is uniformly bounded. For any y ∈ Y, it follows that which implies that F(Y) is uniformly bounded.
Step 4. We prove the equicontinuity of F . Let y ∈ Y and t 1 , t 2 ∈ I 0 with t 1 < t 2 . Therefore, We deduce that as |t 2t 1 | → 0, which implies that F(Y) is equicontinuous. Thus, by the Arzelà-Ascoli theorem, the operator F is completely continuous. By the Schauder fixed-point theorem the operator F has a fixed point y ∈ Y. Now, we will use the Banach contraction principle to prove the uniqueness of the solution.
Proof Let y, z ∈ C 1-γ ,ρ I 0 , be such that Thus, we have Then, for all t ∈ I 0 , we have Thus, Due to Eq. (28), the operator F is a contraction mapping. Using the Banach contraction mapping theorem, we deduce that F has a unique fixed point that is the unique solution of the problem (3) and (4).

Some illustrative examples
Now, we illustrate some examples that guarantee our main results. In this case, we use a computational technique for checking the solutions of MFDEs with the H-KFD problem (3) and (4), and applying a tiny step h by the implicit trapezoidal PI rule, which, as we will see, usually shows excellent accuracy [39].

An application of a particle in the plane
Linear motion is the most basic of all motions. According to Newton's first law of motion, objects that do not experience any net force will continue to move in a straight line with a constant velocity until they are subjected to a net force.
Here, in this section, we consider an application to examine the validity of our theoretical results on the fractional-order representation of the motion of a particle along a straight line. In this case, we consider a constrained motion of a particle along a straight line re- strained by two linear springs with equal spring constants (stiffness coefficient) under an external force and fractional damping along the t-axis (Fig. 3).
The springs, unless subjected to a force, are assumed to have free length (unstretched length) and resist a change in length. The motion of the system along the t-axis is independent of the initial spring tension. The springs are anchored on the t-axis at t = -1 and t = 1, and the vibration of the particle in this example is restricted to the t-axis only.
The vibration of the system is represented by a system of equations with the first equation having a similar form to simple harmonic oscillator, which cannot produce instability. Hence, the existence solution of the system depends on the following equation represented as MFDEs with H-KFD where for t ∈ (0, 2]. Consider particular values of the parameters = 1.5 m and θ = 0.5. It is clear that ρ = 3 > 0, α = 0.75 ∈ (0, 1), β = 0.4 ∈ [0, 1], The general integral solution of (33) is the fractional integral equation . Table 3 shows the numerical results of for = 1, 1.25, 1.5, and 1.75. These results are shown graphically in Fig. 4. Therefore, all conditions of Theorem 19 hold. Thus, the MFDE with H-KFD (33) has a solution.

Conclusion
Over the last several years, the study of FME has drawn increasing attention due to its applications in various fields of the physical sciences, in applied mathematics, and in many  engineering fields. To the best of the authors' knowledge, there are no paper studies on ME with H-KFD. Motivated by the importance of these equations, we investigated the existence and uniqueness of solutions for MFDEs associated to H-KFDs. The Schauder fixed-point theorem was the key of our analysis to establish the existence of solutions.
However, by adding un extra condition, we succeeded in obtaining a unique solution by using the Banach fixed-point theorem. Finally, we present two examples with application to validate our main theoretical results. We were able to produce a computational technique for checking our problem and two algorithms for numerical approximation of solutions with excellent accuracy.   (gamma(gammavar) * rho^(gammavar-1)-...