A Novel Investigation of Non-Periodic Snap BV P in the G -Caputo Sense

: In the present paper, we consider a nonlinear fractional snap model with respect to a G -Caputo derivative and subject to non-periodic boundary conditions. Some qualitative analysis of the solution, such as existence and uniqueness, are investigated in view of ﬁxed-point theorems. Moreover, the stabilities of Ulam–Hyers and Ulam–Hyers–Rassias criterions are considered and investigated. Some numerical simulations were performed using MATLAB for understanding the theoretical results. All results in this work play an important role in understanding ocean engineering phenomena due to the huge applicability of jerk and snap in seakeeping, ride comfort, and shock response spectrum.


Introduction
The second derivative of the accelaration (fourth derivative of position) is a physical quantity called a snap or jounce, which can be modeled as: It is obvious that the model (1) can be reduced to the following equation: In fact, the terms jerk and snap are exceptionally rare for most individuals, counting physicists and engineers. Scientists jerk and snap are the third and fourth derivatives of our position with regard to time, respectively. Equation (1) contains a 4th-order derivative of the variable v 1 , and it describes a 4th-order dynamical vibration model. The corresponding fractional model is achieved by using the fractional derivative (of order less than or equal 1) instead of the standard derivative d dὶ . Many types of fractional derivatives can be used here, such as the Riemann-Liouville, Caputo and Hadamard. We prefer to use the generalized fractional derivative with respect to differentiable increasing function G. Gottlie in [1] applied the method of harmonic balance to non-linear jerk equations, which involves the third order time-derivative. In 2017, Elsonbaty et al., by applying the contraction principle, investigated the following jerk system: in which derivatives are with respect to time, and λ and β denote positive parameters with β ∈ R [2]. In 2018, Rahman et al. [3] with the help of the modified harmonic balance method, obtained a second approximate solution for a simple nonlinear 3rd-order jerk initial problem formulated as Additionally, Prakash et al. in [4], introduced an extension of the jerk system to the fractional order jerk system without any equilibrium point, given as: where α, β and γ ∈ (0, 1] are orders of fractional type. Many researchers have investigated the sufficient conditions for a wide domain of fractional nonlinear ordinary differential equations by employing methods which include standard fixed-point theorems, iterative approaches, etc. (see [5][6][7][8][9][10][11][12][13]). However, to the best of our knowledge, limited results can be found on the existence/stability of solutions for a fractional jerk system via the generalized G-Caputo derivative. In 2020, Liu et al., developed two iterative algorithms to determine the periods and then the periodic solutions of nonlinear jerk equations for two possible cases initial values unknown and initial values given [14]. The authors in the recent article [15] considered the G−fractional snap model with a constant initial conditions where the G-Caputo derivatives are illustrated by symbol c D η;G τ 1 + , and η ∈ {α, β, γ, δ} such that 0 < η ≤ 1; the increasing function G ∈ C 1 [τ 1 , τ 2 ] is such that G (ὶ) = 0, ∀ὶ ∈ [τ 1 , τ 2 ] and continuous function T ∈ C([τ 1 , τ 2 ] × R 4 ) and u 0 , u 1 , u 2 , u 3 ∈ R, but we here in this article shall use non-periodic boundary conditions that generalize many boundary and initial conditions. The authors in [16] studied the following coupled system of fractional differential equations: forὶ ∈ [τ 1 , τ 2 ] equipped with the generalized fractional integral boundary conditions v 1 (δ 1 ) = 0, v 1 (b) = τ 1 I γ 1 ;G v 1 (µ 1 ), v 2 (δ 2 ) = 0, v 2 (b) = τ 1 I γ 2 ;G v 2 (µ 2 ), (5) where G ∈ (0, 1], R τ 1 D α;G and R τ 1 D β;G denote the generalized proportional fractional derivatives of Riemann-Liouville type of order α, β ∈ (1, 2]; τ 1 I γ i ;G denote the generalized proportional fractional integrals of order γ i ∈ (0, 1), δ i , µ i ∈ (τ 1 , τ 2 ); and T i : [τ 1 , τ 2 ] × R 2 → R are continuous functions where i = 1, 2.
We consider the following problem: where the symbol c D η;G Clearly, we can write the system as follows: where This paper is organized as follows: In Section 2, we present some necessary definitions of fractional calculus and useful lemmas and some theorems about the fixed-point that are needed in the subsequent sections. In Section 3, with the help of Banach and Leary-Schauder fixed-point theorem, we give the proof of the fundamental theorems to prove the existence and uniqueness of solutions for problem (7). The stability results are extensively discussed in Section 4 in the context of the Ulam-Hyers and its generalized version, along with Ulam-Hyers-Rassias and its generalized version for solutions of the fractional Gsnap problem (7). Two significant examples, along with codes and numerical results, are presented in Section 5 in which our all outcomes are guaranteed. Those numerical examples were generated using MATLAB for understanding the theoretical results. Finally, we will give some suggestions to the reader in the conclusion Section 6.

Preliminaries
Some primitive notions, definitions and notations, which will be utilized throughout the manuscript, are recalled here. Let G : [τ 1 , τ 2 ] → R be increasing via G (ὶ) = 0, ∀ὶ. We start this part by defining G−fractional integrals and derivatives. In all notations of this section, we set The qth G−integral for an integrable function v : [τ 1 , τ 2 ] → R with respect to G is illustrated as follows ( [17]): where By applying the Algorithm 1, we can obtain the qth G−integral (8). Let n ∈ N and G, v ∈ C n [τ 1 , τ 2 ] be such that G has the same properties mentioned above. The qth-G-fractional derivative of v is defined by in which n = [q] + 1 [18]. The qth G-Caputo derivative of v is defined by in which n = [q] + 1 for q / ∈ N, n = q for q ∈ N [19]. In other words, This derivative gives the Caputo-Hadamard derivative and the Caputo derivative when G(ὶ) = lnὶ and G(ὶ) =ὶ, respectively. The qth G-Caputo derivative of the function v is specified as ( [19], Theorem 3) By using the MATLAB function in Algoritm 2, we can get the qth G-Caputo derivative (9). The composition rules for the above G-operators are recalled in this lemma.
Theorem 1 (Banach fixed-point theorem). Let V = (V, · V ) be a Banach space, and let Ψ : V → V be a contraction mapping on a closed ball that is, there exists a positive real number < 1, such that

Existence-Uniqueness Results
Here, we analyze the existence properties of solutions and their uniqueness for the proposed fractional non-periodic snap problem. We require the following lemma, which specifies the corresponding integral equation. Hereafter, we assume are continuously differentiable real-valued functions on (τ 1 , τ 2 ). Lemma 3. Let γ, β, α, δ ∈ (0, 1], λ, µ, ν, ξ = 1, and (H1) be held. If g is a real-valued continuous function on [τ 1 , τ 2 ], then the solution of the linear fractional non-periodic snap problem is given by Proof. Consider v(ὶ) satisfying the snap problem (10). The continuity of g and with c 0 ∈ R. The differentiability of I δ;G τ + 1 g and to both sides of (12). Using the boundary condition, we deduce that Similarly, applying the γ-th integral operator I γ;G The boundary condition gives the following: Next, we apply the β-th integral operator I The boundary condition c D α;G Finally, we apply the integral operator I α;G The last boundary condition v(τ 1 ) = λ 0 v(τ 2 ) can be applied to get Therefore, we now see that v(ὶ) fulfills (11), and the proof is ended.
In the next result, our goal is to verify the unique solution of the fractional non-periodic snap problem (7) by using Banach fixed-point theorem.
Consider the space The following notations are useful: , , , , In virtue of Lemma 3, we can use the integral solution of the fractional non-periodic snap problem (7) to define a operator Ψ : V → V as the following. where The following hypothesis is strongly needed for the contraction principle of the operator Ψ.
Theorem 3. Let (H1) and (H2) be held. Then, the fractional non-periodic snap problem (7) admits a unique solution on [τ 1 , Proof. Define the compact subset B ς of the Banach space V as To apply the Banach Theorem, we verify that ΨB ς ⊂ B ς , i.e., Ψ maps B ς into itself, where Ψ is defined by (16). For any v ∈ B ς , we obtain In addition, by (13) and (14), we have The last equality From the above inequalities, we obtain This implies that Ψv V ≤ ς, which means that Ψv ∈ B ς . In particular, we notice that In the next step, the contractive property of the operator Ψ is investigated. Let v,v ∈ V; then, we estimate Additionally, we have From the above inequalities, we obtain Since ∆ < 1, Ψ is a contraction on V, and Banach fixed-point theorem guarantees the existence of a unique fixed point for Ψ, and accordingly, the existence of a unique solution for the fractional non-periodic snap problem (7) and the proof are ended.
The next existence property for possible solutions of the fractional non-periodic snap problem (7) is checked based on hypotheses of Leary-Shauder fixed-point theorem. We assume the following hypotheses: (H3) The function T ∈ C([τ 1 , τ 2 ] × R 4 ). Moreover, there exist a monotonic increasing finite real-valued function h∈ C[0, ∞) and a finite real-valued function ∈ C[τ 1 , τ 2 ], such that Proof. Let Ψ : V → V be defined as in (16) and B be any bounded convex closed subset in V. We show that the hypotheses of Leary-Schauder fixed-point theorem can be applied on the operator Ψ. Hence, the proof consists of several steps.
Step 1 : ] The continuity of the operator Ψ is obtained by applying the dominated convergence theorem and noting that the function T is jointly continuous.
Therefore, by Leray-Schauder fixed-point theorem, Ψ admits at least one fixed point v ∈ U ς as a solution of the fractional non-periodic snap problem (7), and this finishes the proof.

Stability Criterion
We introduce in this section many stability criteria, namely, the Ulam-Hyers and Ulam-Hyers-Rassias, with their generalizations for the solutions of the fractional non-periodic snap problem (7) on [τ 1 , τ 2 ].

Definition 1 ([21]
). The fractional non-periodic snap problem (7) is Ulam-Hyers stable if there exists a positive real number χ * T , such that for any ε > 0, and v * ∈ V satisfying there exists v ∈ V satisfying the fractional non-periodic snap problem (7) with (7) is generalized Ulam-Hyers stable if there exists a function π * T ∈ C[0, ∞) with π * T (0) = 0 such that, for any ε > 0 and v * ∈ V satisfying the inequality (19), there exists v ∈ V as a solution of the fractional non-periodic snap problem (7) with (7) is Ulam-Hyers-Rassias stable with respect to a function ϑ ∈ C[τ 1 , τ 2 ] if there exists a positive real number χ * T , such that for any ε > 0, and v * ∈ V satisfying

Definition 4. The fractional non-periodic snap problem
there exists v ∈ V satisfying the fractional non-periodic snap problem (7) with ϑ(ὶ).

Remark 1.
The relationships among these kinds of stability can be summarized by the following implications: The Ulam-Hyers stability of the fractional non-periodic snap problem (7) is investigated by the next result.
Proof. For every ε > 0 and v * ∈ V satisfying (19), we can find a function ρ ∈ C[τ 1 , Using Theorem 3, there exists a unique solution v ∈ V satisfying the fractional non-periodic snap problem (7). Then, Additionally, we have Finally, From the above inequalities, we obtain Since ∆ < 1, this shows the existence of a positive real and hence according to Definition 1, the solution of (7) is Ulam-Hyers stable. Similarly, it shows the existence of a function π * T ∈ C[0, ∞) with π * T (0) = 0 such that Hence, the solution of (7) is GUH stable.
Since ∆ < 1, this shows the existence of a positive real number Hence, according to Definition 3, the solution of (7) is Ulam-Hyers-Rassias stable. Therefore, the solution of (7) is generalized Ulam-Hyers-Rassias stable.
One can see the 2D spectrum of ∆, ∆ and ∆ * h(ς) in Figure 3. In all three cases for the order α i , we see that all requirements of Theorem 4 are fulfilled. Therefore, this guarantees that for all of three different cases by terms of the order α, the fractional non-periodic snap problem (24)

Conclusions
In this paper, we defined a new fractional mathematical model consisting of a fractional snap equation with non-periodic boundary conditions in the framework of the generalized G-operators. Thus, some investigations on the qualitative behaviors of its solutions, including existence, uniqueness and stability, were performed separately. To obtain the uniqueness of the solution, we used Banach contraction theorem, and for the general existence of at least one solution, we used the Shauder fixed-point theorem. Ulam-Hyers and Ulam-Hyers-Rassias with their generalizations were discussed and investigated. In the final step, we designed examples with different cases of the function G, such as Caputo, Caputo-Hadamard and Katugampola; and with different orders of q, we obtained some numerical results concerning the fractional non-periodic snap problem.
Author Contributions: X.W.: Actualization, methodology, formal analysis, validation, investigation, and initial draft. A.B.: Actualization, methodology, formal analysis, validation, investigation, and initial draft. N.T.: Actualization, validation, methodology, formal analysis, investigation, and initial draft. M.M.M.: Actualization, validation, methodology, formal analysis, investigation, and initial draft. M.E.S.: Actualization, methodology, formal analysis, validation, investigation, software, simulation, initial draft and was a major contributor in writing the manuscript. M.K.A.K.: Actualization, methodology, formal analysis, validation, investigation, initial draft, and supervision of the original draft and editing. X.-G.Y.: Actualization, methodology, formal analysis, validation, investigation, and initial draft. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.
Institutional Review Board Statement: Not applicable.