Abstract
By considering a metric space with partially ordered sets, we employ the coupled fixed point type to scrutinize the uniqueness theory for the Langevin equation that included two generalized orders. We analyze our problem with four-point and strip conditions. The description of the rigid plate bounded by a Newtonian fluid is provided as an application of our results. The exact solution of this problem and approximate solutions are compared.
1. Introduction
The fractional calculus concept is not absolutely intuitive, where it has no clear geometrical interpretation. Several distinct forms have appeared, to the point that the necessity for order has developed in the field [1, 2]. The variety of potential implementations is even more difficult. One has to think closely about what the inserting of fractional derivatives in the model can provide. Fractional derivatives are generally inserted for modeling processes of mass transport, optics, diffusion, etc. [3, 4]. With the inserting of these derivatives, fractional-order models have described its advantages when modeling supercapacitor capacitances [5] and controllers for temperature [6], DC motors [7], or RC, LC, and RLC electric circuits [8]. Through the present paper, we deem the generalized Langevin differential equation of two generalized different orders:
where , , and are the Caputo generalized derivatives with and , and the continuously differentiable function is given. This equation is supplemented with the strip and four-point conditions:
where and . The third boundary condition, which appears as a linear combination of nonlocal point and Riemann-Liouville fractional integral condition on an arbitrary segment , can be explained as a gathering of the values of the obscure function at local point , and nonlocal point ) is proportionate to the strip contribution of the unbeknown function. The investigation of the fractional Langevin equation (1) together with four-point and strip conditions (2) makes our problem new especially when applying coupled immutable point type in the case of an existing mixed monotone mapping.
The main goal of our research is to investigate the solution uniqueness for our case ((1) and (2)) by virtue of applying coupled immutable point type in a metric space with partially ordered set in the case of an existing mixed monotone mapping. It is worth pointing out, as far as we know, that no contribution till now studied the uniqueness of solution for the generalized Langevin differential equation (1) by using coupled immutable point type in a metric space with partially ordered set in the case of an existing mixed monotone mapping except Fazli and Nieto [9]. In a metric space with partially ordered set, the coupled immutable point type in the case of an existing mixed monotone mapping was established at first by Bhaskar and Lakshmikantham [10] and extended by many authors, for instance, [11–13]. The prominence of this process rises from the reality that it is a deductive process that accords convergent sequences to the unique solution of our case ((1) and (2)).
The Brownian motion exceedingly draws through the Langevin equation when the random fluctuation force is submitted to be white noise. If the random oscillation force is not white noise, the object motion is depicted by the generalized Langevin equation [14]. Overall, the ordinary differential equations cannot precisely characterize experimental data and area measurement; as an alternative approach, fractional-order differential equation models are extremely used today [15–17].
The generalized Langevin equation is a substantial differential equation in applied mathematics, physics, and other areas of science and engineering. It has been developed and presented by Mainradi and Pironi [18]. With multipoint and multistrip boundary conditions, [19–21] investigated some properties and results to the solution of fractional Langevin equation. The uniqueness of solution and other properties for boundary value problems of the generalized Langevin equation have drawn a plentiful attention from diversified contributors within the previous decades, check for epitome [22–24] and the spacious roster of references presented therein. Analytical expressions of the correlation functions have been obtained using the two fluctuation-dissipation theorems and fractional calculus approaches.
It is worth mentioning that the Langevin equation is extremely applied to characterize the development of physical phenomena in fluctuating environments. However, for the systems in complex media, the ordinary order to the Langevin equation does not give the true depiction of the dynamics. One of the most important possible generalizations to the Langevin equation is by replacing the derivative of positive integer order by a derivative of fractional order which gives rise to a fractional Langevin equation (see [25] and the references therein).
The Hyers-Ulam-Rassias along with Hyers-Ulam (HU) stability results for fractional Langevin equation has been studied in [26]. An explicit solution to nonhomogeneous fractional delayed Langevin equations has been given in [27]. The stochastic nonlinear fractional Langevin equation with a multiplicative noise has been studied by [28]. The hybrid Sturm-Liouville-Langevin equations with new versions of Caputo fractional derivatives have been investigated in [29].
2. Preliminaries
The authors in [10] inserted the next basic connotations of coupled immutable point type and mixed monotone mapping.
Let be a partial order relation on a nonempty set which is reflexive, antisymmetric, and transitive. We refer the pair to a partially ordered set. Let be an element in , and then is an upper bound (lower bound) for a subset if () for each . If there are lower and upper bounds for , then is called bounded partially ordered set. Let and be two elements in , and then and are said to be comparable if either or (or both, in case of ).
Let us recall the next definitions related to coupled immutable point type and mixed monotone mapping:
Definition 1. Consider that is partially ordered and mapping . It is called that the mapping has mixed monotone property if is nonincreasing with respect to and nondecreasing with respect to for each and in , and this means that
Definition 2. Let , and then is called coupled fixed point of if the two relations and are satisfied.
The next coupled immutable point types represent the fundamental outcomes of the contribution [10].
Theorem 3. Consider the partially ordered . Postulate that is a complete metric space. Assume that is a continuous mapping that has mixed monotone property on . Let such that If satisfy the inequalities, Assume either (i) is a continuous mapping or(ii)The set satisfies at least one of the following properties: (a)If the sequence is a nondecreasing, then for all (b)If the sequence is a nonincreasing, then for all Then, there are that satisfy the equalities Let us introduce the next partial order on the space
Theorem 4 (addendum to the presumptions of Theorem 3). Assume that for each , there is a pair that is comparable to and , and then, has a unique coupled immutable pair .
Theorem 5 (addendum to the presumptions of Theorem 3). Assume that each pair of elements of has a lower or an upper bound in . Then, . Furthermore, where
Next, let us render sundry famous definitions and identities for fractional calculus. For additional specifics, check [30, 31].
Definition 6. A generalized integral for Riemann-Liouville has the integral form where is Gamma function and , provided that the integral exists.
Definition 7. Suppose and are positively real with , the Caputo derivative of has the integral form provided that the integral exists. We remark that where is constant.
Lemma 8. Let and . If , then we have
Lemma 9. Let be positive integer and . Then, we have
Lemma 10. The generalized Langevin equation (1) with the conditions in (2) has a unique representation of the solution if and only if the function is a solution of the integral equation
Proof. In view of Lemmas 8 and 9, we can find Inserting the condition in (17) gives , and also inserting the boundary condition in (16) gives . Using the third boundary condition in (2) gives Substituting these values of , , and in the equation (17), we acquire the desirable results. Conversely, by aiding of the third identity of Lemma 8 and the second identity of Lemma 9, one can see that the solution satisfies the fractional differential equation (16). Also, it is easy to see that the unique solution satisfies all boundary conditions in (17).
3. Main Results
Presume that is partially ordered where ; it is evident that is a complete metric space of all continuous functions endowed with the distance
Distinctly, if the sequence is a nondecreasing in and converges to and the sequence is a nonincreasing in and converges to , it follows that and , for all .
Define the space , and then, it is a complete metric space endowed with the distance
Furthermore, the set is partially ordered if we acquaint the next ordered relation in
For any , the functions and are also in and are lower bound and upper bound of and , respectively. Therefore, for every , there exists a that is comparable to and .
Previously starting and showing the fundamental outcomes, we insert the next presumptions: Assume that
() The function is a jointly continuous
() The function satisfies
where is the Lipschitz constant
For the sake of computational convenience, we set
Our mainly investigation is based on the sign of the value of , so we introduce the results in two ways when and when as in the following two subsections.
3.1. In the Case of
Consider the following two operators:
Definition 11. An element is called a coupled lower and upper solution of the boundary value problems (1) and (2) if
Theorem 12. Through the accompanying presumptions () and (), if the problems (1) and (2) have a coupled upper and lower solutions and where and and are defined as in (23) and (24), respectively, then it has a unique solution in .
Proof. Consider the operator
where and are defined as in (27) and (28), respectively. The continuity of the operator comes according to the assumption (), so and it is well defined. Now, let such that , and then for each and aiding of the assumption (), we have
Thus, with fix , we find that
which leads to , and thus, is monotonously nondecreasing in .
Again, let such that , and then for each and aiding of the assumption (), we have
Thus, with fix , we find that
which leads to , and thus, is monotonously nonincreasing in . Therefore, has the mixed monotone property.
For each with and , we have
For each with and , we have
Therefore, for , we have
Now, we have the partially ordered and the continuous operator which has mixed monotone property. Thus, we emphasize that there exists a coupled lower and upper solution for the problems (1) and (2) such that and . So, for every , there exists a that is comparable to and . These mean that all the assumptions of Theorems 3 and 4 are satisfied. Therefore, has a unique coupled fixed point in ; that is, the boundary value problems (1) and (2) have the unique solution . This coupled solution, according to Theorem 5, can be obtained as
This ends the proof.
3.2. In the Case of
Consider the following two operators:
Definition 13. The pair is called a coupled lower and upper solution of the boundary value problems (1) and (2) if
Theorem 14. Through the accompanying presumptions and , if the problems (1) and (2) have coupled lower and upper solutions and where and and are defined as in (25) and (26), respectively, then it has a unique solution in .
Proof. Consider the operator where and are defined as in (23) and (24), respectively. The remnant of proof is identical to the proof of the former theorem.
4. Motion of an Immersed Plate
Consider now the rigid plate of mass immersed in a Newtonian fluid of infinite extent and connected by a massless spring of stiffness to a fixed point. We assume that the small motions of the spring do not disturb the fluid and that the area of the plate, , is sufficiently large as to produce in the fluid adjacent to the plate the velocity field and stresses developed in the preceding section. For the previous problem, Bagley and Torvik [32] have found the differential equation describing the displacement of the plate to be where is the - fractional derivative of order , is the fluid density, and is the viscosity. The equation above was then known as the Bagley and Torvik equation. Thus, the fractional derivative is established to become visible in the differential equation which depicts the motion of a simple, physical system depending on familiar mechanical and fluid components. Furthermore, its existence may be anticipated in any system distinguished by localized motion in a viscous fluid. Such is the case for oscillations of a polymeric material. Bagley and Torvik in their work [32] believe this accounts for the success of a fractional derivative in modeling these materials.
By replacing - fractional derivative by the Caputo fractional derivative with noting that , we deduce that equation (42) is equivalent to equation (1) with the boundary conditions in (2) and , , , and . Thus, equation (42) can be rewritten as
Since , then the is continuous and satisfies the Lipschitz condition with .
Let the rigid plate of mass and area immersed in the Newtonian fluid, e.g., water with the values of viscosity and density and at 40°C and connected by a massless spring of stiffness to a fixed point which implies that .
By carrying out Mathematica 11 software, it is easy to compute and where and are defined as in (23) and (24), respectively. According to Theorem 12, since , we have to choose which implies that there is a unique solution of our problem (43).
Now, we are seeking to compute the exact solution. For this, apply the Laplace transform to the problem (43) to get where
It is known that which implies that
where is the complementary error function defined as
It is easy to see that
which lead to for all . Also, it is not difficult to show
and from the continuity of and the definition of the Caputo derivative. The last condition in (2) gives
In order to determine the values of minimum and maximum solutions , we solve both inequalities in Definition 11 with taking into account the definitions of the two operators and as in (27) and (28), respectively, and we find that and for all where
In view of the behavior of the function on the interval , numerically, we get that it is increasing on which yields that . Conclusion, we can take the minimum and maximum solutions . Under the assumptions of Theorem 12 and all of which have been fulfilled, there exists a unique solution for the problem (43) in the interval . With the help of Theorem 5, we can determine the unique solution to this problem as where
In Figures 1 and 2, we graph the exact solution with blue color, lower approximate solution with yellow color, and upper approximate solution with green color. We note that the lower and upper solutions take the same behavior slightly different and intersect the exact solution at and in Figure 1 and and in Figure 2, respectively. Therefore, we expect that they intersect at the same point in the higher order.
5. Conclusion
Here, some transactions are presented with the uniqueness of solution for four-point and strip generalized Langevin equation that included two generalized orders in distinct intervals. We solve the problem with the properties of the fractional calculus and coupled immutable point type. It turned out that there are two cases to apply our theorems. We took each case into consideration and gave an illustrative example for each one. The fractional differential equation, due to Bagley and Torvik which describes the motion of the rigid plate immersed in a Newtonian fluid of infinite extent and connected by a massless spring of stiffness to a fixed point, is taken as an application of our results. It is shown in Figures 1 and 2 that there is a small difference between the exact and approximate solutions.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no competing interests.
Authors’ Contributions
All authors contributed equally and significantly in writing this paper. All authors have read and agreed to the final version of the manuscript.
Acknowledgments
This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia, under grant no. KEP-PhD-60-130-42. The authors, therefore, acknowledge with thanks DSR for their technical and financial support.