Fractional Langevin Equations with Nonseparated Integral Boundary Conditions

Abstract

In this paper, we discuss the existence of solutions for nonlinear fractional Langevin equations with nonseparated type integral boundary conditions. The Banach fixed point theorem and Krasnoselskii fixed point theorem are applied to establish the results. Some examples are provided for the illustration of the main work.

Lie Symmetry Analysis, Exact Solutions, and Conservation Laws for the Generalized Time-Fractional KdV-Like Equation

“…The most important part of fractional calculus is devoted to the fractional differential equations (FDEs); in the literature, there are diverse definitions for fractional derivative including Riemann-Liouville derivative, Caputo derivative, and Conformable derivative, but the most popular one is Riemann-Liouville derivative. The fractional derivative has attracted the attention of many researchers in different areas such as viscoelasticity, vibration, economic, biology, and fluid mechanics (see [1][2][3][4][5][6][7][8][9][10][11]). Unfortunately, it is almost difficult to solving and detecting all solutions of nonlinear partial differential equations (PDEs) which renders it a challenging problem, because of this, an interesting advance has been made, and some methods for solving this type of equations have been discussed; among them are subequation method, homotopy perturbation method, the first integral method, and Lie group method (see [12][13][14][15][16][17][18]).…”

Existence Solution for Coupled System of Langevin Fractional Differential Equations of Caputo Type with Riemann–Stieltjes Integral Boundary Conditions

“…Analytical expressions of the correlation functions were obtained using the two fluctuation-dissipation theorems and fractional calculus approaches. The fractional Langevin equation has been received the attention of many scientists due to its extremely useful applications in different fields of science and has been dealt with under different conditions (see [8][9][10][11][12][13][14]).…”

New Existence Results for Nonlinear Fractional Integrodifferential Equations

“…Fractional differential equations are relevant in many fields of science, such as chemistry, fluid systems, and electromagnetic; for more details about the theory of fractional differential equations and their applications, we invite the readers to see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] and the references therein. Some physical applications of fractional differential equations include viscoelasticity, Schrodinger equation, fractional diffusion equation, and fractional relaxation equation; for more details, we refer the readers to [17].…”