On non‐instantaneous impulsive fractional differential equations and their equivalent integral equations

2021 ◽  
Author(s):  
Arran Fernandez ◽  
Sartaj Ali ◽  
Akbar Zada
Keyword(s):  
Differential Equations ◽  
Integral Equations ◽  
Fractional Differential Equations ◽  
Impulsive Fractional Differential Equations ◽  
Fractional Differential

scholarly journals Convergence Analysis of Implicit Euler Method for a Class of Nonlinear Impulsive Fractional Differential Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Yuexin Yu ◽  
Sheng Zhen
Keyword(s):  
Differential Equations ◽  
Convergence Analysis ◽  
Integral Equations ◽  
Fractional Differential Equations ◽  
Euler Method ◽  
First Order ◽  
Implicit Euler Method ◽  
Impulsive Fractional Differential Equations ◽  
Fractional Differential ◽  
Theoretical Results

For a class of nonlinear impulsive fractional differential equations, we first transform them into equivalent integral equations, and then the implicit Euler method is adapted for solving the problem. The convergence analysis of the method shows that the method is convergent of the first order. The numerical results verify the correctness of the theoretical results.

scholarly journals Analysis of Implicit Type Nonlinear Dynamical Problem of Impulsive Fractional Differential Equations

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-15 ◽  
Author(s):  
Naveed Ahmad ◽  
Zeeshan Ali ◽  
Kamal Shah ◽  
Akbar Zada ◽  
Ghaus ur Rahman
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Point Theorem ◽  
Fractional Differential Equations ◽  
Nonlocal Boundary ◽  
Dynamical Problem ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Impulsive Fractional Differential Equations ◽  
Fractional Differential

We study the existence, uniqueness, and various kinds of Ulam–Hyers stability of the solutions to a nonlinear implicit type dynamical problem of impulsive fractional differential equations with nonlocal boundary conditions involving Caputo derivative. We develop conditions for uniqueness and existence by using the classical fixed point theorems such as Banach fixed point theorem and Krasnoselskii’s fixed point theorem. For stability, we utilized classical functional analysis. Also, an example is given to demonstrate our main theoretical results.

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