Solvability of Fractional Differential Equation with Nonlocal Boundary Conditions at Resonance

2016 ◽  
Vol 45 (4) ◽  
pp. 625-638 ◽  
Author(s):  
Do Huy Hoang ◽  
Le Cong Nhan ◽  
Le Xuan Truong
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Fractional Differential Equation ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
At Resonance ◽  
Fractional Differential

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.

scholarly journals Existence of Positive Solutions for Fractional Differential Equation with Nonlocal Boundary Condition

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Hongliang Gao ◽  
Xiaoling Han
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Fixed Point ◽  
Fractional Differential Equation ◽  
Positive Solutions ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Existence Of Positive Solutions ◽  
Fractional Differential ◽  
The Fixed Point Theorem

By using the fixed point theorem, existence of positive solutions for fractional differential equation with nonlocal boundary conditionD0+αu(t)+a(t)f(t,u(t))=0,0<t<1,u(0)=0,u(1)=∑i=1∞αiu(ξi)is considered, where1<α≤2is a real number,D0+αis the standard Riemann-Liouville differentiation, andξi∈(0,1),  αi∈[0,∞)with∑i=1∞αiξiα-1<1,a(t)∈C([0,1],[0,∞)),  f(t,u)∈C([0,1]×[0,∞),[0,∞)).

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