scholarly journals Analysis of a Coupled System of Nonlinear Fractional Langevin Equations with Certain Nonlocal and Nonseparated Boundary Conditions

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Zaid Laadjal ◽  
Qasem M. Al-Mdallal ◽  
Fahd Jarad
Keyword(s):  
Boundary Conditions ◽  
Coupled System ◽  
Fractional Integrals ◽  
Langevin Equations ◽  
Uniqueness Of Solutions ◽  
Caputo Fractional Derivatives ◽  
Nonseparated Boundary Conditions ◽  
New Type ◽  
Predictor Corrector ◽  
Derivatives Of

In this article, we use some fixed point theorems to discuss the existence and uniqueness of solutions to a coupled system of a nonlinear Langevin differential equation which involves Caputo fractional derivatives of different orders and is governed by new type of nonlocal and nonseparated boundary conditions consisting of fractional integrals and derivatives. The considered boundary conditions are totally dissimilar than the ones already handled in the literature. Additionally, we modify the Adams-type predictor-corrector method by implicitly implementing the Gauss–Seidel method in order to solve some specific particular cases of the system.

scholarly journals Sequential Riemann–Liouville and Hadamard–Caputo Fractional Differential Systems with Nonlocal Coupled Fractional Integral Boundary Conditions

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 174
Author(s):  
Chanakarn Kiataramkul ◽  
Weera Yukunthorn ◽  
Sotiris K. Ntouyas ◽  
Jessada Tariboon
Keyword(s):  
Boundary Conditions ◽  
Fractional Integral ◽  
Integral Boundary Conditions ◽  
Contraction Mapping Principle ◽  
Uniqueness Of Solutions ◽  
Caputo Fractional Derivatives ◽  
Integral Boundary ◽  
Banach Contraction ◽  
Fractional Differential ◽  
Banach Contraction Mapping Principle

In this paper, we initiate the study of existence of solutions for a fractional differential system which contains mixed Riemann–Liouville and Hadamard–Caputo fractional derivatives, complemented with nonlocal coupled fractional integral boundary conditions. We derive necessary conditions for the existence and uniqueness of solutions of the considered system, by using standard fixed point theorems, such as Banach contraction mapping principle and Leray–Schauder alternative. Numerical examples illustrating the obtained results are also presented.

scholarly journals RIEMANN–LIOUVILLE FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS WITH FRACTIONAL NONLOCAL MULTI-POINT BOUNDARY CONDITIONS

Fractals ◽  
2021 ◽  
pp. 2240002
Author(s):  
BASHIR AHMAD ◽  
BADRAH ALGHAMDI ◽  
RAVI P. AGARWAL ◽  
AHMED ALSAEDI
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Existence And Uniqueness ◽  
Weighted Space ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Fractional Integrals ◽  
Existence Theory ◽  
Uniqueness Of Solutions

In this paper, we investigate the existence and uniqueness of solutions for Riemann–Liouville fractional integro-differential equations equipped with fractional nonlocal multi-point and strip boundary conditions in the weighted space. The methods of our study include the well-known tools of the fixed point theory, which are commonly applied to establish the existence theory for the initial and boundary value problems after converting them into the fixed point problems. We also discuss the case when the nonlinearity depends on the Riemann–Liouville fractional integrals of the unknown function. Numerical examples illustrating the main results are presented.

A Study of Coupled Systems of ψ-Hilfer Type Sequential Fractional Differential Equations with Integro-Multipoint Boundary Conditions

2021 ◽  
Vol 5 (4) ◽  
pp. 162
Author(s):  
Ayub Samadi ◽  
Cholticha Nuchpong ◽  
Sotiris K. Ntouyas ◽  
Jessada Tariboon
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Fractional Differential Equations ◽  
Fixed Point Theorems ◽  
Coupled System ◽  
Coupled Systems ◽  
Uniqueness Of Solutions ◽  
Multipoint Boundary Conditions ◽  
Fractional Differential ◽  
Sequential Fractional Differential Equations

In this paper, the existence and uniqueness of solutions for a coupled system of ψ-Hilfer type sequential fractional differential equations supplemented with nonlocal integro-multi-point boundary conditions is investigated. The presented results are obtained via the classical Banach and Krasnosel’skiĭ’s fixed point theorems and the Leray–Schauder alternative. Examples are included to illustrate the effectiveness of the obtained results.

scholarly journals Multi-Point and Anti-Periodic Conditions for Generalized Langevin Equation with Two Fractional Orders

2019 ◽  
Vol 3 (4) ◽  
pp. 51 ◽  
Author(s):  
Ahmed Salem ◽  
Balqees Alghamdi
Keyword(s):  
Langevin Equation ◽  
Contraction Mapping Principle ◽  
Generalized Langevin Equation ◽  
Uniqueness Of Solutions ◽  
Caputo Fractional Derivatives ◽  
New Class ◽  
Banach Contraction ◽  
Derivatives Of ◽  
Fractional Orders ◽  
Banach Contraction Mapping Principle

With anti-periodic and a new class of multi-point boundary conditions, we investigate, in this paper, the existence and uniqueness of solutions for the Langevin equation that has Caputo fractional derivatives of two different orders. Existence of solutions is obtained by applying Krasnoselskii–Zabreiko’s and the Leray–Schauder fixed point theorems. The Banach contraction mapping principle is used to investigate the uniqueness. Illustrative examples are provided to apply of the fundamental investigations.

scholarly journals Existence Theorems for Mixed Riemann–Liouville and Caputo Fractional Differential Equations and Inclusions with Nonlocal Fractional Integro-Differential Boundary Conditions

2019 ◽  
Vol 3 (2) ◽  
pp. 21 ◽  
Author(s):  
Sotiris K. Ntouyas ◽  
Ahmed Alsaedi ◽  
Bashir Ahmad
Keyword(s):  
Boundary Conditions ◽  
Existence And Uniqueness ◽  
Fractional Derivatives ◽  
Existence Theorems ◽  
Uniqueness Of Solutions ◽  
Caputo Fractional Derivatives ◽  
New Class ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential ◽  
Differential Equations And Inclusions

In this paper, we discuss the existence and uniqueness of solutions for a new class of single and multi-valued boundary value problems involving both Riemann–Liouville and Caputo fractional derivatives, and nonlocal fractional integro-differential boundary conditions. Our results rely on modern tools of functional analysis. We also demonstrate the application of the obtained results with the aid of examples.

scholarly journals Study on Implicit-Type Fractional Coupled System with Integral Boundary Conditions

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 300
Author(s):  
Longfei Lin ◽  
Yansheng Liu ◽  
Daliang Zhao
Keyword(s):  
Boundary Conditions ◽  
Topological Degree ◽  
Fractional Derivatives ◽  
Existence Result ◽  
Degree Theory ◽  
Integral Boundary Conditions ◽  
Coupled System ◽  
Topological Degree Theory ◽  
Caputo Fractional Derivatives ◽  
Integral Boundary

This paper is concerned with a class of implicit-type coupled system with integral boundary conditions involving Caputo fractional derivatives. First, the existence result of solutions for the considered system is obtained by means of topological degree theory. Next, Ulam–Hyers stability and generalized Ulam–Hyers stability are studied under some suitable assumptions. Finally, one example is worked out to illustrate the main results.

scholarly journals Iterative positive solutions to a coupled fractional differential system with the multistrip and multipoint mixed boundary conditions

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Xiaodi Zhao ◽  
Yuehan Liu ◽  
Huihui Pang
Keyword(s):  
Boundary Conditions ◽  
Positive Solutions ◽  
Mixed Boundary ◽  
Coupled System ◽  
Mixed Boundary Conditions ◽  
Monotone Iterative Technique ◽  
Monotone Iterative ◽  
Fractional Differential System ◽  
Fractional Differential ◽  
Derivatives Of

Abstract Using the monotone iterative technique, we investigate the existence of iterative positive solutions to a coupled system of fractional differential equations supplemented with multistrip and multipoint mixed boundary conditions. It is worth mentioning that the nonlinear terms of the system depend on the lower fractional-order derivatives of the unknown functions and the boundary conditions involve the combination of the multistrip fractional integral and the multipoint value of the unknown functions in $[0,1]$ [ 0 , 1 ] .

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