Session II.1 - Computational Dynamics
Poster
Numerical method for solving special Cauchy problem for the second order integro-differential equation
Sayakhat Karakenova
Kh. Dosmukhamedov Atyrau University, Kazakhstan - This email address is being protected from spambots. You need JavaScript enabled to view it.
In the present communication we conserved the special Cauchy problem for the second order integro-differential equation $$ \frac{d^2x}{dt^2}=A_1(t){\frac{dx}{dt}} +A_2(t) x(t)+\varphi_1(t) {\int_{0}^{T}} \psi_1(\tau) f_1(\tau,\dot{x}(\tau))d\tau +\varphi_2(t) {\int_{0}^{T}} \psi_2(\tau) f_2(\tau,x(\tau))d\tau+g(t) \;(1) $$
where the $A_1(t), A_2(t), \varphi_1(t), \varphi_2(t),\psi_1(\tau), \psi_2(\tau)$ are continuous on $f:[0,T] \times \mathbb{R}^n \rightarrow \mathbb{R}^n$, is continuous.
A solution to equation (1) is continuosly differentable on [0,T] function $x(t)\in C([0,T],R^n)$, which satisfies equation for any $t \in [0,T]$.
Equation (1) adduce to special Cauchy problem by the Dzhumabaev parametrization method. An iterative method is proposed to solve a special Cauchy problem. The iterative method is implemented numerically.
This research is funded by the Committee of Science of the Ministry of Science and Higher Education of the Republic of Kazakhstan (Grant No. AP15473218).