FRACTIONAL LANGEVIN EQUATIONS WITH INFINITE-POINT BOUNDARY CONDITION: APPLICATION TO FRACTIONAL HARMONIC OSCILLATOR

The current study is concerned with the existence and uniqueness of the solution to the Langevin equation of two separate fractional orders. With the inﬁnite-point boundary condition, the boundary value problem is studied. The Banach contraction principle, Leray-nonlinear Schauder’s alternative, and Leray-Schauder degree theorems are all implemented. A numerical example is presented to demonstrate the accuracy of our results. In addition, as an application of our results, the mean and variance of a fractional harmonic oscillator with the undamped angular frequency of the oscillator under the eﬀect of a random force described as Gaussian colored noise are calculated.


Introduction
Given that it lacks a clear geometrical explanation, the concept of fractional calculus is not entirely apparent.In the realm of [6], a need for order has emerged as a result of the appearance of several unique forms.Even more problematic is the wide range of possible implementations.The benefits that fractional derivatives might add to the model must be carefully considered.Typically, fractional derivatives are used to represent mass transport, optical, diffusion, and other processes [8,11,24,27].Fractional-order models' benefits in simulating supercapacitor capacitances [15], temperature controllers [7], DC motors [34], and RC, LC, and RLC electric circuits [1] have been detailed with the addition of these derivatives.
When the random fluctuation force is assumed to be white noise, the Brownian motion drags the Langevin equation through to an extreme degree.The generalized Langevin equation, [28], describes the motion of the item if the random oscillation force is not white noise.Overall, the fractional order differential equation models are widely utilized nowadays as an alternative to conventional differential equations because they can more accurately represent experimental data and area measurement [22,30,33,36].A crucial differential equation in applied mathematics, physics, and other branches of science and engineering is the generalized Langevin equation.Mainardi and Pironi [18] have created and presented it.The significance of their technique is that it models the Brownian motion more accurately than the traditional one based on the classical Langevin equation, because it includes the retarding effects owing to hydrodynamic back-flow, i.e. the additional mass and the Basset memory drag.The two fluctuation-dissipation theorems and fractional calculus methods were used to produce analytical formulas for the correlation functions (both for the random force and the particle velocity).
The contributions [2,25,26,31] and else have studied several properties and results to the solution of the fractional Langevin equation using multi-point and multi-strip boundary conditions.The uniqueness of solution and other features for boundary value problems of generalized Langevin equation have attracted a great deal of attention from many writers over the last several decades, as evidenced by epitome [23,29,32,35,37] and the extensive list of references offered therein.
The generalized Langevin equation GLE can be used to analyze anomalous diffusive phenomena connected with physical or biological processes.Some recent articles on anomalous diffusion may be found in the literature [4] and a huge number of references given therein.We recall that anomalous diffusion is the phenomena that occurs most commonly in disordered or fractal media and in which the mean squared displacement (the variance) is proportional to a power of fractional order rather than linear in time (as in standard diffusion) [17].
Inspired of the previous studies, the following nonlinear fractional Langevin equation of two fractional orders is considered supplemented with the infinite-point boundary conditions where u(t) represents the position of a particle of mass m = 1 at time t ∈ [0, 1], λ ∈ R is frictional memory kernel, c D α and c D γ are the Caputo's fractional derivatives of orders 0 < α ≤ 1 and 1 Although numerous research have been conducted on fractional Langevin equations under multi-point boundary conditions, as far as we are aware, just Li et al [16] has investigated it under an infinite-point boundary conditions and provided various novel existence results of solution utilizing Leray-nonlinear Schauder's alternative and Leray-Schauder degree theory.However, it has been proved that their given outcomes count on solution form incorporates boundary values.It suggests that their method need more conclusive results in order to be more useful.
In more details, we note that their unique solution of the linear boundary value problem of fractional Langevin differential equation subject to the infinite-point boundary conditions (1.2) was given as β i ξ i > 0, which contains the boundary values u(1) and u(ξ i ), i ∈ N, even though we can insert the values of these boundary values after obtaining the form of u(t).This means that the solution above is not in the final form.
To be out of these criticisms, we resolve the boundary value problem (1.1)-(1.2) without the appearance of the boundary values u(1) and u(ξ i ), i ∈ N in the unique solution u(t).Also, we extend some restrictions on λ, β i , i ∈ N and Our analysis is carried out using three important fixed point theorems: the Banach contraction principle, Leray-nonlinear Schauder's alternative, and the Leray-Schauder degree theorems.
Furthermore, as an application, the fractional harmonic oscillator with the undamped angular frequency of the oscillator under the influence of a random force described as Gaussian colored noise was examined.The mean and variance, the two most often used statistical measures of transportation, are evaluated and plotted.

Preliminaries and relevant lemmas
In this part, we introduce certain fractional calculus notations and terminology, as well as preliminary findings that will be used later in our proofs.We are grateful for the terminology utilized in the books [13,20].
Definition 2.1.The Riemann-Liouville fractional integral of order α > 0 for a continuous function f is defined as provided that the right-hand-side integral exists, where Γ(α) denotes the Gamma function is the Euler gamma function defined by Definition 2.2.Let n ∈ N be a positive integer and α be a positive real such that n − 1 < α ≤ n, then the fractional derivative of a function f in the Caputo sense is defined as provided that the right-hand-side integral exists and is finite.We notice that the Caputo derivative of a constant is zero.
Lemma 2.1.Let α and β be positive reals.If f is a continuous function, then we have Lemma 2.2.Let α be positive real.Then we have Let us now consider the nonlinear fractional Langevin differential equation (1.1) supplemented with the infinite-point boundary conditions (1.2), then we can state the following lemma: where Proof.From Lemmas 2.1, 2.2 and 2.3 and the Definition 2.1, it follows that followed by operating I α on both sides, we find that By inserting the boundary condition u(0) = 0 in (2.2) gives c 2 = 0 and also by inserting the boundary condition c Du(0) = 0 in (2.1) gives c 0 = 0. Using the third boundary condition in (1.2), gives Substituting the above values in (2.2) to obtain the desired results.Conversely, inserting the formula of u(t) into the left hand side of (1.1) with using the second relation of Lemma 2.1 and the second and third relations of Lemma 2.2 to obtain the right hand side of (1.1).Also, it is not difficult to see that the solution u(t) satisfies all conditions of (1.2).
In the proofs of our main results for problem (1.1)-(1.2),we use the Banach contraction principle for providing sufficient conditions to the uniqueness of solution and Leray-Schauder degree theorem and nonlinear alternative Leray-Schauder theorem for providing sufficient conditions to the existence of solution.
Definition 2.3.Let (E, d) be a Banach space.Then a map T : E → E is called a contraction mapping on E if there exists r ∈ [0, 1) such that d(T(t), T(s)) ≤ rd(t, s) for all t, s ∈ E.
Lemma 2.5 (Banach contraction principle [10]).Let (E, d) be a non-empty complete metric space with a contraction mapping T : E → E. Then T admits a unique fixed-point t * in E (i.e.T(t * ) = t * ).
Lemma 2.6 (Nonlinear alternative Leray-Schauder theorem [9]).Let E be a Banach space, C be a closed and convex subset of E, U be an open subset of C and 0 ∈ U .Suppose that the operator T : U → C is a continuous and compact map (that is, T(U ) is a relatively compact subset of C).Then either Lemma 2.7 (Leray-Schauder degree theorem [5,19]).

Existence and uniqueness results
Let E = C([0, 1], R) be the Banach space of all continuous functions from [0, 1] −→ R endowed the norm defined by Before stating and proving the main results, we introduce the following hypotheses: Assume that where L is the Lipschitz constant.
(H 3 ) There exists a positive function ω ∈ C([0, 1], R + ) and a nondecreasing function ϕ : R + → R + such that (H 4 ) There exist two positive constants η and L such that For computational convenience, we set where Lemma 3.1.Under the assumption (H 1 ), the function g(•, •) satisfies the following for all u ∈ E and t, t 1 , t 2 ∈ [0, 1] such that t 1 < t 2 .Furthermore, under the assumptions (H 1 ) and (H 2 ), it satisfies Proof.From the definition of the function g in Lemma 2.4, we have Also, by using assumption (H 2 ) with letting u, v ∈ E and t ∈ [0, 1], we can deduce that In view of Lemma 2.4, we transform problem (1.1)-(1.2) as where the operator T : E −→ E is defined by where g(•, •) is defined in Lemma 2.4.
The following theorem is devoted to provide the conditions that satisfy the assumptions of Banach contraction mapping principle to give a unique solution of the boundary value problem (1.1)-(1.2).Then, for u ∈ B r , we have By employing Lemma 3.1, we have By the hypothesis Q < 1, it follows that the operator T defined in (3. Proof.The continuity of the function f implies that the operator T : E → E defined by (3.3) is continuous.Assume that B r = {u ∈ E : u < r} be an open subset of the Banach space E with radius r > 0. First, we are in a position to prove that the operator T : E → E is completely continuous.Assume that u ∈ B r .Then, as in the proof of Theorem 3.1, we have which concludes the boundedness of the operator T .Suppose that t 1 , t 2 ∈ [0, 1] such that t 1 < t 2 , it follows that which implies, by using Lemma 3.1, that It is clear that the right-hand side of the above inequality approaches zero as t 1 → t 2 .
Since the operator T satisfies the above assumptions, it follows by the Arzela-Ascoli theorem that T : E → E is completely continuous.
According to the Leray-Schauder nonlinear alternative Lemma 2.6, the result will follow once we prove the boundedness of the set of all solution to equations u = δT u for some δ ∈ [0, 1].Let u is a solution of the equation u = δT u for some δ ∈ [0, 1], then for all t ∈ [0, 1], from the boundedness of the operator T , we have By the assumption K > 1, then there exists a constant M > 0 such that u = M .Setting the open set Ω = {u ∈ E : u < M }.
Based on the form of Ω, there is no u ∈ ∂Ω such that u = δT (u) for some δ ∈ (0, 1).Since the operator T : Ω → E is continuous and completely continuous, then by the nonlinear alternative of Leray-Schauder type Lemma 3.1, we deduce that T has a fixed point u ∈ Ω which is a solution of problem (1.1)-(1.2).This ends the proof.
Theorem 3.3.Assume that the assumptions (H 1 ) and (H 4 ) hold.Then the boundary value problem (1.1)-(1.2) has at least one solution if where A and B are given by (3.1) and (3.2), respectively.
Proof.Let us define the open ball B r ⊂ E with radius r > 0 as where r will be determined later.It is adequate to prove that the operator T : To do this, assume that u = σT u for some σ ∈ [0, 1].Then, as in the preceding results, we have provided that ηA + B < 1 which leads to η < (1 − B)/A.Now, suppose that there exists > 0 such that By the analysis above, it follows that (3.4) holds.Let us now define the continuous operator In view of the results in Theorems above, it is clear that the operator h σ : E → E is completely continuous.By the homotopy invariance of topological degree, it follows that

Fractional harmonic oscillator
In actual oscillators, damping or friction slows the system's motion.The velocity falls in proportion to the frictional force applied.While in a basic undriven harmonic oscillator the only force operating on the mass is the restoring force, in a damped harmonic oscillator there is also a frictional force acting in the opposite direction of the motion.The dynamic of a fractional treatment of a harmonic oscillator [3] with the undamped angular frequency of the oscillator ω under the influence of a random force modeled as Gaussian colored noise, whose corresponding fractional differential equation, associated with the displacement, can be written as where u(t) indicates the location of a particle with mass m = 1 at time t ∈ [0, 1], λ ∈ R is frictional memory kernel and the internal noise ρ(t) is a random force satisfying the fluctuation-dissipation theorem of a zero-mean ρ(t) = 0 and with an arbitrary correlation function C(t − t) = ρ(t)ρ(t ) .The correlation function for a free Brownian particle in one dimension can be taken as [14] C(t) = Kδ(t) where K = kT , k is a Boltzmann constant, T is the absolute temperature of the heat bath and δ(•) is a dirac delta function.
The equation (4.1) is supplemented with the infinite-point conditions (1.2).It is clear that the function f (t, u(t)) = −ω 2 u(t) + ρ(t) satisfies the assumptions (H 1 ) and (H 2 ) with L = ω 2 .According to our main results in Theorems 3.1 provided that Q < 1, the problem (4.1) has a unique solution which can be evaluate through applying Laplace transform as follows which has the mean x(t) = g λ,ω C where C is a constant can be evaluated easily by using the last condition in (1.2) as , and E µ α,γ (•) is the generalized of the Mittag-Leffler function [12,21] The two most commonly used statistical measures of transport are the mean, µ(t), and the variance, σ 2 (t), defined as Thus, the mean of displacement can be provided as and the displacement can be given as We can also evaluate the mean square displacement which provides an indication of the most likely displacement that one can expect a particle to have in a certain time.In the free particle case, the mean square displacement can be calculate by which leads to the variance of the process is given as The fractional derivative results from the shear stress-induced collective behavior of the liquid.Some of the limiting examples can be related to the fractional harmonic oscillator Langevin equation.We have two interesting constraints here.Setting the liquid's shear stress to zero (λ = 0), we can find the basic harmonic oscillator.If we exclude the oscillator's undamped angular frequency (ω = 0), we get a special super-diffusive instance of the fractional Langevin equation.

Lemma 2 . 4 .
The unique representation of the solution of the boundary value problem (1.1) and (1.2) is given by