Fractional Differential Equations with Nonlocal Integral and Integer–Fractional-Order Neumann Type Boundary Conditions

2015 ◽  
Vol 13 (5) ◽  
pp. 2365-2381 ◽  
Author(s):  
Bashir Ahmad ◽  
Sotiris K. Ntouyas ◽  
Jessada Tariboon
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Neumann Type ◽  
Fractional Differential ◽  
Type Boundary

scholarly journals New Sequential Fractional Differential Equations with Mixed-Type Boundary Conditions

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Haiyan Zhang ◽  
Yaohong Li ◽  
Jingbao Yang
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Fractional Differential Equations ◽  
Mixed Type ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Integral Boundary ◽  
Fractional Differential ◽  
Type Boundary ◽  
Sequential Fractional Differential Equations

In this paper, we introduce new sequential fractional differential equations with mixed-type boundary conditions CDq+kCDq−1ut=ft,ut,CDq−1ut,t∈0,1,α1u0+β1u1+γ1Iruη=ε1,η∈0,1,α2u′0+β2u′1+γ2Iru′η=ε2, where q∈1,2 is a real number, k,r>0,αi,βi,γi,εi∈ℝ,i=1,2,CDq is the Caputo fractional derivative, and the boundary conditions include antiperiodic and Riemann-Liouville fractional integral boundary value cases. Our approach to treat the above problem is based upon standard tools of fixed point theory and some new inequalities of norm form. Some existence results are obtained and well illustrated through the aid of examples.

scholarly journals Existence and uniqueness of positive solutions for a class of nonlinear fractional differential equations with mixed-type boundary value conditions

2018 ◽  
Vol 24 (1) ◽  
pp. 73-94 ◽  
Author(s):  
Fang Wang ◽  
Lishan Liu ◽  
Debin Kong ◽  
Yonghong Wu
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Positive Solutions ◽  
Fractional Differential Equations ◽  
Mixed Type ◽  
Index Theory ◽  
Nonlinear Fractional Differential Equations ◽  
Fixed Point Index Theory ◽  
Fractional Differential ◽  
Type Boundary

In this article, we study a class of nonlinear fractional differential equations with mixed-type boundary conditions. The fractional derivatives are involved in the nonlinear term and the boundary conditions. By using the properties of the Green function, the fixed point index theory and the Banach contraction mapping principle based on some available operators, we obtain the existence of positive solutions and a unique positive solution of the problem. Finally, two examples are given to demonstrate the validity of our main results.

Adjoint Fractional Differential Expressions and Operators

2007 ◽  
Author(s):  
Igor Podlubny ◽  
YangQuan Chen
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Fractional Differential ◽  
And Control

In this article we present the notions of adjoint differential expressions for fractional-order differential expressions, adjoint boundary conditions for fractional differential equations, and adjoint fractional-order operators. These notions are based on new formulas obtained for various types of fractional derivatives. The introduced notions can be used in many fields of modelling and control of real dynamical systems and processes.

scholarly journals On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions

2021 ◽  
Vol 19 (1) ◽  
pp. 760-772
Author(s):  
Ahmed Alsaedi ◽  
Bashir Ahmad ◽  
Badrah Alghamdi ◽  
Sotiris K. Ntouyas
Keyword(s):  
Nonlinear System ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Fractional Differential Equations ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Numerical Examples ◽  
Multipoint Boundary ◽  
Multipoint Boundary Conditions ◽  
Fractional Differential

Abstract We study a nonlinear system of Riemann-Liouville fractional differential equations equipped with nonseparated semi-coupled integro-multipoint boundary conditions. We make use of the tools of the fixed-point theory to obtain the desired results, which are well-supported with numerical examples.

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

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.

Analysis of a New Class of Impulsive Implicit Sequential Fractional Differential Equations

2020 ◽  
Vol 21 (6) ◽  
pp. 571-587 ◽  
Author(s):  
Akbar Zada ◽  
Sartaj Ali ◽  
Tongxing Li
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Sufficient Conditions ◽  
Uniqueness Of Solutions ◽  
Integer Order ◽  
New Class ◽  
Proposed Model ◽  
Fractional Differential ◽  
Sequential Fractional Differential Equations

AbstractIn this paper, we study an implicit sequential fractional order differential equation with non-instantaneous impulses and multi-point boundary conditions. The article comprehensively elaborate four different types of Ulam’s stability in the lights of generalized Diaz Margolis’s fixed point theorem. Moreover, some sufficient conditions are constructed to observe the existence and uniqueness of solutions for the proposed model. The proposed model contains both the integer order and fractional order derivatives. Thus, the exponential function appearers in the solution of the proposed model which will lead researchers to study fractional differential equations with well known methods of integer order differential equations. In the last, few examples are provided to show the applicability of our main results.

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