Abstract:By employing Shauder fixed-point theorem, this work tries to obtain the existence results for the solution of a nonlinear Langevin coupled system of fractional order whose nonlinear terms depend on Caputo fractional derivatives. We study this system subject to Stieltjes integral boundary conditions. A numerical example explaining our result is attached.
“…Fractional differential equations (FDEs) have gained a lot of attention in recent years due to their numerous applications in engineering, physics, biology, chemistry, and other fields (see, for instance, [1][2][3][4][5][6], and the references therein for more information on the boundary value issues of FDEs and inclusions subject to diverse boundary conditions). Differential inclusion and differential equations are thought to be particularly helpful when studying dynamical systems and stochastic processes (see [7][8][9][10] for some recent related results).…”
In this paper, we study coupled nonlinear Langevin fractional problems with different orders of μ-Caputo fractional derivatives on arbitrary domains with nonlocal integral boundary conditions. In order to ensure the existence and uniqueness of the solutions to the problem at hand, the tools of the fixed-point theory are applied. An overview of the main results of this study is presented through examples.
“…Fractional differential equations (FDEs) have gained a lot of attention in recent years due to their numerous applications in engineering, physics, biology, chemistry, and other fields (see, for instance, [1][2][3][4][5][6], and the references therein for more information on the boundary value issues of FDEs and inclusions subject to diverse boundary conditions). Differential inclusion and differential equations are thought to be particularly helpful when studying dynamical systems and stochastic processes (see [7][8][9][10] for some recent related results).…”
In this paper, we study coupled nonlinear Langevin fractional problems with different orders of μ-Caputo fractional derivatives on arbitrary domains with nonlocal integral boundary conditions. In order to ensure the existence and uniqueness of the solutions to the problem at hand, the tools of the fixed-point theory are applied. An overview of the main results of this study is presented through examples.
“…Due to the importance of the Caputo version, there are many versions established as generalization of it, such as Caputo-Katugampola, Caputo-Hadamard, Caputo-Fabrizio, etc. Furthermore, it is drown attention of huge number of contributors to study physical and mathematical modelings contain it and its related versions, see [9][10][11][12][13] and references cited therein.…”
This manuscript investigates the existence, uniqueness and Ulam–Hyers stability (UH) of solution to fractional differential equations with non-instantaneous impulses on an arbitrary domain. Using the modern tools of functional analysis, we achieve the required conditions. Finally, we provide an example of how our results can be applied.
“…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][16][17].…”
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.
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