The Langevin Equation in Terms of Generalized Liouville–Caputo Derivatives with Nonlocal Boundary Conditions Involving a Generalized Fractional Integral

In this paper, we establish sufficient conditions for the existence of solutions for a nonlinear Langevin equation based on Liouville-Caputo-type generalized fractional differential operators of different orders, supplemented with nonlocal boundary conditions involving a generalized integral operator. The modern techniques of functional analysis are employed to obtain the desired results. The paper concludes with illustrative examples.

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Solving a linear fractional equation with nonlocal boundary conditions based on multiscale orthonormal bases method in the reproducing kernel space

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2019-0291 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Wei Jiang ◽

Zhong Chen ◽

Ning Hu ◽

Yali Chen

Keyword(s):

Differential Equations ◽

Boundary Conditions ◽

Fractional Differential Equations ◽

Reproducing Kernel ◽

Nonlocal Boundary ◽

Nonlocal Boundary Conditions ◽

Kernel Space ◽

Reproducing Kernel Space ◽

Orthonormal Bases ◽

Fractional Differential

AbstractIn recent years, the study of fractional differential equations has become a hot spot. It is more difficult to solve fractional differential equations with nonlocal boundary conditions. In this article, we propose a multiscale orthonormal bases collocation method for linear fractional-order nonlocal boundary value problems. In algorithm construction, the solution is expanded by the multiscale orthonormal bases of a reproducing kernel space. The nonlocal boundary conditions are transformed into operator equations, which are involved in finding the collocation coefficients as constrain conditions. In theory, the convergent order and stability analysis of the proposed method are presented rigorously. Finally, numerical examples show the stability, accuracy and effectiveness of the method.

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On the solvability of the Lyapunov equation for nonselfadjoint differential operators of order 2m with nonlocal boundary conditions

Annales Polonici Mathematici ◽

10.4064/ap77-1-7 ◽

2001 ◽

Vol 77 (1) ◽

pp. 79-104

Author(s):

Aris Tersenov

Keyword(s):

Boundary Conditions ◽

Differential Operators ◽

Nonlocal Boundary ◽

Lyapunov Equation ◽

Nonlocal Boundary Conditions

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Existence of solutions for a system of fractional differential equations with coupled nonlocal boundary conditions

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2018-0024 ◽

2018 ◽

Vol 21 (2) ◽

pp. 423-441 ◽

Cited By ~ 28

Author(s):

Bashir Ahmad ◽

Rodica Luca

Keyword(s):

Differential Equations ◽

Boundary Conditions ◽

Fractional Differential Equations ◽

Existence Of Solutions ◽

Nonlocal Boundary ◽

Nonlocal Boundary Conditions ◽

Fractional Integrals ◽

Nonlinear Alternative ◽

Coupled Boundary Conditions ◽

Fractional Differential

AbstractWe study the existence of solutions for a system of nonlinear Caputo fractional differential equations with coupled boundary conditions involving Riemann-Liouville fractional integrals, by using the Schauder fixed point theorem and the nonlinear alternative of Leray-Schauder type. Two examples are given to support our main results.

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