Numerical treatment of non-linear Volterra integro-differential equation by using Runge-Kutta methods

2019 ◽  
Author(s):  
Atefa J. Saleh ◽  
Rawaa E. Esa ◽  
Ali F. Jameel
Keyword(s):  
Differential Equation ◽  
Numerical Treatment ◽  
Integro Differential Equation ◽  
Non Linear ◽  
Runge Kutta

ANALYSIS OF ELASTIC IMPACT ON THIN SHELL STRUCTURES

2008 ◽  
Vol 22 (09n11) ◽  
pp. 1349-1354 ◽  
Author(s):  
SHIUH-CHUAN HER ◽  
CHING-CHUAN LIAO
Keyword(s):  
Differential Equation ◽  
Contact Force ◽  
Thin Shell ◽  
Time History ◽  
Dynamic Responses ◽  
Low Velocity Impact ◽  
Integro Differential Equation ◽  
Impact Process ◽  
Non Linear ◽  
The Impact

In this paper, a solution method for the response of a thin shell structure subjected to low velocity impact by a sphere is presented. The governing equation of the impact process is obtained by simultaneously solving the equations of motions for the sphere and shell. The derivation is based on the explicit expression of the displacement of the mid-surface of the shell under a single impulse load acting normal to apex of the shell. Incorporating the theory of convolution and Hertz contact law, a non-linear integro-differential equation in terms of the indentation of the contact, for the impact process is derived. The non-linear integro-differential equation is solved by the numerical scheme of Runge-Kutta method to obtain the time history of the contact force at the impact point of the shell. The contact force is then applied on the apex of the shell, the dynamic responses of the shell including the displacement and stress are obtained by the finite element method. The results are validated with the experimental test and numerical calculation published in the literatures. The effects of the radius and velocity of the impactor on the impact response is investigated through parametric study.

Approximate controllability of impulsive fractional integro-differential equation with state-dependent delay in Hilbert spaces

2018 ◽  
Vol 36 (2) ◽  
pp. 603-622 ◽  
Author(s):  
Yong Zhou ◽  
S Suganya ◽  
M Mallika Arjunan ◽  
B Ahmad
Keyword(s):  
Differential Equation ◽  
Hilbert Spaces ◽  
Approximate Controllability ◽  
Semigroup Theory ◽  
Sufficient Conditions ◽  
Integro Differential Equation ◽  
State Dependent ◽  
State Dependent Delay ◽  
Non Linear ◽  
Approximately Controllable

Abstract In this paper, the problem of approximate controllability for non-linear impulsive fractional integro-differential equation with state-dependent delay in Hilbert spaces is investigated. We study the approximate controllability for non-linear impulsive integro-differential systems under the assumption that the corresponding linear control system is approximately controllable. By utilizing the methods of fractional calculus, semigroup theory, fixed-point theorem coupled with solution operator, sufficient conditions are formulated and proved. Finally, an example is provided to illustrate the proposed theory.

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