A Pseudo-Spectral Scheme for Systems of Two-Point Boundary Value Problems with Left and Right Sided Fractional Derivatives and Related Integral Equations

2021 ◽  
Vol 128 (1) ◽  
pp. 21-41
Author(s):  
I. G. Ameen ◽  
N. A. Elkot ◽  
M. A. Zaky ◽  
A. S. Hendy ◽  
E. H. Doha
Keyword(s):  
Boundary Value Problems ◽  
Integral Equations ◽  
Fractional Derivatives ◽  
Boundary Value ◽  
Point Boundary ◽  
Related Integral ◽  
Left And Right ◽  
Pseudo Spectral

scholarly journals A New Reproducing Kernel Approach for Nonlinear Fractional Three-Point Boundary Value Problems

2020 ◽  
Vol 4 (4) ◽  
pp. 53
Author(s):  
Mehmet Giyas Sakar ◽  
Onur Saldır
Keyword(s):  
Exact Solutions ◽  
Numerical Solution ◽  
Boundary Value Problems ◽  
Legendre Polynomials ◽  
Fractional Derivatives ◽  
Reproducing Kernel ◽  
Boundary Value ◽  
Point Boundary ◽  
Shifted Legendre Polynomials ◽  
Kernel Approach

In this article, a new reproducing kernel approach is developed for obtaining a numerical solution of multi-order fractional nonlinear three-point boundary value problems. This approach is based on a reproducing kernel, which is constructed by shifted Legendre polynomials (L-RKM). In the considered problem, fractional derivatives with respect to α and β are defined in the Caputo sense. This method has been applied to some examples that have exact solutions. In order to show the robustness of the proposed method, some examples are solved and numerical results are given in tabulated forms.

scholarly journals New Lyapunov-type inequalities for fractional multi-point boundary value problems involving Hilfer-Katugampola fractional derivative

2021 ◽  
Vol 7 (1) ◽  
pp. 1074-1094
Author(s):  
Wei Zhang ◽  
◽  
Jifeng Zhang ◽  
Jinbo Ni
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Boundary Value Problems ◽  
Integral Equations ◽  
Fractional Differential Equations ◽  
Boundary Value ◽  
Contraction Principle ◽  
Point Boundary ◽  
Fractional Differential ◽  
Banach’S Contraction Principle

<abstract><p>In this paper, we present new Lyapunov-type inequalities for Hilfer-Katugampola fractional differential equations. We first give some unique properties of the Hilfer-Katugampola fractional derivative, and then by using these new properties we convert the multi-point boundary value problems of Hilfer-Katugampola fractional differential equations into the equivalent integral equations with corresponding Green's functions, respectively. Finally, we make use of the Banach's contraction principle to derive the desired results, and give a series of corollaries to show that the current results extend and enrich the previous results in the literature.</p></abstract>

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