scholarly journals Caputo-type fractional boundary value problems for differential equations and inclusions with multiple fractional derivatives

2017 ◽  
Vol 2017 (1) ◽  
pp. 1-22
Keyword(s):  
Differential Equations ◽  
Boundary Value Problems ◽  
Fractional Derivatives ◽  
Boundary Value ◽  
Differential Equations And Inclusions ◽  
Fractional Boundary Value Problems

scholarly journals On the boundary value problems of piecewise differential equations with left-right fractional derivatives and delay

2021 ◽  
Vol 26 (6) ◽  
pp. 1087-1105
Author(s):  
Yuxin Zhang ◽  
Xiping Liu ◽  
Mei Jia
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Point Theorem ◽  
Fractional Derivatives ◽  
Boundary Value ◽  
State Variables ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Existence And Multiplicity

In this paper, we study the multi-point boundary value problems for a new kind of piecewise differential equations with left and right fractional derivatives and delay. In this system, the state variables satisfy the different equations in different time intervals, and they interact with each other through positive and negative delay. Some new results on the existence, no-existence and multiplicity for the positive solutions of the boundary value problems are obtained by using Guo–Krasnoselskii’s fixed point theorem and Leggett–Williams fixed point theorem. The results for existence highlight the influence of perturbation parameters. Finally, an example is given out to illustrate our main results.

scholarly journals On fractional boundary value problems involving fractional derivatives with Mittag-Leffler kernel and nonlinear integral conditions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mohammed S. Abdo ◽  
Thabet Abdeljawad ◽  
Saeed M. Ali ◽  
Kamal Shah
Keyword(s):  
Boundary Value Problems ◽  
Existence And Uniqueness ◽  
Fractional Derivatives ◽  
Fractional Integral ◽  
Fixed Point Theorems ◽  
Boundary Value ◽  
Integral Conditions ◽  
Caputo Fractional Derivatives ◽  
Derivatives Of ◽  
Fractional Boundary Value Problems

AbstractIn this paper, we consider two classes of boundary value problems for nonlinear implicit differential equations with nonlinear integral conditions involving Atangana–Baleanu–Caputo fractional derivatives of orders $0<\vartheta \leq 1$ 0 < ϑ ≤ 1 and $1<\vartheta \leq 2$ 1 < ϑ ≤ 2 . We structure the equivalent fractional integral equations of the proposed problems. Further, the existence and uniqueness theorems are proved with the aid of fixed point theorems of Krasnoselskii and Banach. Lastly, the paper includes pertinent examples to justify the validity of the results.

scholarly journals An Adaptive Pseudospectral Method for Fractional Order Boundary Value Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Mohammad Maleki ◽  
Ishak Hashim ◽  
Majid Tavassoli Kajani ◽  
Saeid Abbasbandy
Keyword(s):  
Boundary Value Problems ◽  
Fractional Derivatives ◽  
Singular Integrals ◽  
Boundary Value ◽  
Pseudospectral Method ◽  
Interpolation Polynomial ◽  
Algebraic Equations ◽  
Collocation Points ◽  
Volterra Integrodifferential Equation ◽  
Fractional Boundary Value Problems

An adaptive pseudospectral method is presented for solving a class of multiterm fractional boundary value problems (FBVP) which involve Caputo-type fractional derivatives. The multiterm FBVP is first converted into a singular Volterra integrodifferential equation (SVIDE). By dividing the interval of the problem to subintervals, the unknown function is approximated using a piecewise interpolation polynomial with unknown coefficients which is based on shifted Legendre-Gauss (ShLG) collocation points. Then the problem is reduced to a system of algebraic equations, thus greatly simplifying the problem. Further, some additional conditions are considered to maintain the continuity of the approximate solution and its derivatives at the interface of subintervals. In order to convert the singular integrals of SVIDE into nonsingular ones, integration by parts is utilized. In the method developed in this paper, the accuracy can be improved either by increasing the number of subintervals or by increasing the degree of the polynomial on each subinterval. Using several examples including Bagley-Torvik equation the proposed method is shown to be efficient and accurate.

scholarly journals Notes on Local and Nonlocal Intuitionistic Fuzzy Fractional Boundary Value Problems with Caputo Fractional Derivatives

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Ali El Mfadel ◽  
Said Melliani ◽  
M’hamed Elomari
Keyword(s):  
Boundary Value Problems ◽  
Existence And Uniqueness ◽  
Fractional Derivatives ◽  
Fixed Point Theorems ◽  
Boundary Value ◽  
Caputo Fractional Derivatives ◽  
Existence And Uniqueness Results ◽  
Intuitionistic Fuzzy ◽  
Uniqueness Results ◽  
Fractional Boundary Value Problems

In this paper, we investigate the existence and uniqueness results of intuitionistic fuzzy local and nonlocal fractional boundary value problems by employing intuitionistic fuzzy fractional calculus and some fixed-point theorems. As an application, we conclude this manuscript by giving an example to illustrate the obtained results.

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