Session II.1 - Computational Dynamics
Poster
Numerical solution of a semi-periodic initial problem of the fourth order hyperbolic type using a generalized operator and functional parametrization
Zhanibek Tokmurzin
K.Zhubanov Aktobe Regional University, Kazakhstan - This email address is being protected from spambots. You need JavaScript enabled to view it.
The domain $\Omega=[0,T]\times[0,\omega]$ we consider the following semi-periodic initial boundary value problem for a fourth order system of partial differential equations: $$ \frac{\partial^4 u}{\partial t^3 \partial x} =A_1 (t,x) \frac{\partial^3 u}{\partial t^2 \partial x}+A_2 (t,x) \frac{\partial^3 u}{\partial t^3 }+A_3 (t,x) \frac{\partial^2 u}{\partial t^2}+A_4 (t,x) \frac{\partial^2 u}{\partial t \partial x}+$$ $$ +A_5 (t,x) \frac{\partial u}{\partial t}+A_6 (t,x)\frac{\partial u}{\partial x} +A_7 (t,x)u+f(t,x), $$ $$ u(0,x)=\varphi_1 (x), \quad x\in[0,\omega], $$ $$ \frac{\partial u(t,x)}{\partial t}\Big|_{t=0} =\varphi_2 (x), \quad x\in[0,\omega], )$$ $$ \frac{\partial^2 u(t,x)}{\partial^2 t}\Big|_{t=0} =\frac{\partial^2 u(t,x)}{\partial^2 t}\Big|_{t=T}, \quad x\in[0,\omega], $$ $$ u(t,0)=\psi (x), \quad t\in[0,T], $$ where $u(t,x)=col(u_1 (t,x),u_2 (t,x),…,u_n (t,x))$ is the unknown function; the $n\times n$-matrices $A_i (t,x), (i=\overline{1,7}),$ and $n$- vector function $f(t,x)$ are continuous on $ \Omega$; $n$ vector-function $\psi(t)$ are continuously three times differentiable on $[0,T]$; the $n$ vector-functions $\varphi_1 (x)$ and $\varphi_2 (x)$ are continuously differentiable on $[0,\omega]$.
Reducing the order of the problem by two times, we reduce it to the Cauchy problem for a system of n first-order ordinary differential equations. It is shown that the Cauchy problem has a solution using the method of generalized operations or the method of functional parameterization. Using numerical methods for one variable of the Cauchy problem, solutions of the Cauchy problem are obtained.
This research is funded by the Committee of Science of the Ministry of Science and Higher Education of the Republic of Kazakhstan (Grant No. AP14971198).