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  1. %курсач (2009)
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  31. \floatname{algorithm}{Алгоритм}
  32.  
  33. \newtheorem{rudefine}{Defined}
  34. \newtheorem{rutheorem}{Theorem}
  35. \newtheorem{rutask}{Problem}
  36.  
  37. \newcommand{\ZZ}[1][n]{\mathbb{Z}_#1}
  38.  
  39. \usetheme{Madrid}
  40. \useoutertheme{shadow}
  41. \title{Inverse problems for quasiparabolic and quasihyperbolic equations}
  42. \date{2017}
  43. \author{Akimova Ekaterina}
  44. \institute[Foo and Bar]{
  45.   \begin{tabular}[h]{cc}
  46.  
  47. Novosibirsk State University
  48.  
  49.   \end{tabular}      
  50. }
  51. %\titlegraphic{\includegraphics[width=0.3\textwidth]{newlogo}}
  52.  
  53. \begin{document}
  54.     \begin{frame}
  55.         \titlepage
  56.     \end{frame}
  57. \begin{frame}
  58.         \frametitle{Quasiparabolic and quasihyperbolic equations}
  59.  
  60.                      \[
  61. u_{ttt}+u_{xx}+\mu u=f(x,t)+q(t)h(x,t),
  62. \]
  63. \[
  64. u_{tttt}+u_{xx}+\mu u=f(x,t)+q(t)h(x,t).
  65. \]
  66.         Case 1:
  67.         \begin{itemize}
  68.             \item $u(0,t)=0$ -  redefinition condition,
  69.             \item $u(1,t)=u_x(0,t)=0$ - basic conditions.
  70.         \end{itemize}
  71.         Case 2:
  72.         \begin{itemize}
  73.             \item $u_x(0,t)=0$ - redefinition condition,
  74.             \item $u(0,t)=u(1,t)=0$ - basic conditions.
  75.         \end{itemize}
  76.         Case 3:
  77.         \begin{itemize}
  78.             \item $u(0,t)=0$ - redefinition condition,
  79.             \item $u_x(0,t)=u_x(1,t)=0$ - basic conditions.
  80.         \end{itemize}
  81.     \end{frame}
  82.  
  83.    
  84.     \begin{frame}
  85.         \frametitle{Formulation of the problem for quasiparabolic equations}
  86.          
  87.         $Q=(0,1)\times(0,T)$-rectangle,
  88.  
  89.         $\mu(x,t)$,
  90.         $ f (x,t)$,
  91.         $ h(x,t)$  - prescribed function, defined in $\overline{Q}$.
  92.         \begin{rutask}
  93.             Find function $ \ u(x,t)$ and $\ q(t)$, connected by the equation  
  94.         \[
  95.         u_{ttt}+u_{xx}+\mu(x,t)u=f(x,t)+q(t)h(x,t)
  96.         \]
  97.         in area Q.
  98.         Conditions are satisfied for the function $u(x,t)$
  99.         \[
  100.         u(x,0)=u_t(x,0)=u(x,T)=0,  x \in [0,1]
  101.         \]
  102.  and one of the group:
  103.  
  104.             1. $u(0,t)=0$ -  redefinition condition,
  105.            
  106. $u(1,t)=u_x(0,t)=0$ - basic conditions.
  107.                
  108.             2. $u_x(0,t)=0$ -redefinition condition,
  109.             $u(0,t)=u(1,t)=0$ - basic conditions.
  110.        
  111.             3. $u(0,t)=0$ - redefinition condition,
  112.             $u_x(0,t)=u_x(1,t)=0$ - basic conditions.
  113.             \end{rutask}
  114.     \end{frame}
  115.  
  116. \begin{frame}
  117.         \frametitle{Case 1}
  118. $u(0,t)=0$ --- redefinition condition,
  119.  
  120. $u(1,t)=u_x(0,t)=0$ --- basic conditions.
  121.                      \[
  122. q(t)=\frac{u_{xx}(0,t)-f(0,t)}{h(0,t)}, h(0,t)\neq 0;
  123. \]
  124. Notation:
  125. \[
  126. f_1(x,t)=f(x,t)-\frac{f(0,t)h(x,t)}{h(0,t)},
  127. \]
  128. \[
  129. h_1(x,t)=\frac{h(x,t)}{h(0,t)}.
  130. \]
  131.  
  132.     \end{frame}
  133.     \begin{frame}
  134.         \frametitle{Problem 1}
  135.                     Notation: $v(x,t)=u_{xx}(x,t)$, $f_2(x,t)=f_{1xx}(x,t)$, $h_2(x,t)=h_{1xx}(x,t)$,
  136.          $f_1(x,t)=f(x,t)-\frac{f(0,t)h(x,t)}{h(0,t)}$, $h_1(x,t)=\frac{h(x,t)}{h(0,t)}$, $\beta(t)=-h_{1x}(1,t)$ and $\alpha(t)=h_1(1,t)$.
  137.  
  138. Consider the problem
  139.         \[
  140.         \begin{cases}
  141.          v_{ttt}(x,t)+v_{xx}(x,t)+\mu v(x,t)=f_2(x,t)+ h_2(x,t)v(0,t), \\
  142.         v(1,t)- \alpha (t) v(0,t)=0, x\in[0,1], t\in[0,T],\\
  143.         v_x(0,t)+\beta (t)v(0,t)=0, x\in[0,1], t\in[0,T].
  144.         \end{cases}
  145.         \]
  146.            
  147.     \end{frame}
  148.  
  149.     \begin{frame}
  150.         \frametitle{Existence theorem 1}
  151.                       $h(x,t)\in C^2(\overline{Q})$, and conditions are met when $ t\in [0,T]$, $x \in [0,1]$:
  152. \[
  153. h(0,t)\neq 0,
  154. h(1,t)\equiv0,
  155. \mu\textless 0,
  156. -\frac{h_x(1,t)}{h(0,t)}\leqslant0,
  157. \]
  158. \[
  159. \mu^3+\left(4\left(\max_{\overline {Q}}\left|\frac{h_{xx}(x,t)}{h(0,t)}\right|\right)^{2}-\mu\right)\left(\max\limits_{\overline{Q}}\left|\frac{h_{xx}(x,t)}{h(0,t)}\right|\right)^2\leqslant0,
  160. \]
  161. \[
  162. h_t(0,t)h_x(x,t)-h_{xt}(x,t)h(0,t)\geqslant0.
  163. \]
  164. Then inverse problem I with redefinition condition $u(0,t)=0$ and with basic conditions $u(1,t)=u_x(0,t)=0$ has a solution $\{u(x,t), q(t)\}$ such that $u(x,t) \in L_2(Q)$, $u_{xx}(x,t) \in L_2(Q)$, $u_{ttt}(x,t) \in L_2(Q)$, $q(t) \in L_2([0,T])$, for any function  $f(x,t)$, such that $f(x,t), f_{xx}(x,t)\in L_2(Q)$,
  165. \[
  166. f(1,t)-\frac{f(0,t)h(1,t)}{h(0,t)}=0,
  167. f_x(1,t)-\frac{f(0,t)h_x(1,t)}{h(0,t)}=0.
  168. \]
  169.     \end{frame}
  170.  
  171.     \begin{frame}
  172.         \frametitle{Case 2}
  173. $u_x(0,t)=0$ --- redefinition condition,
  174.  
  175. $u(0,t)=u(1,t)=0$ --- basic conditions.
  176.                      \[
  177. q(t)=\frac{u_{xxx}(0,t)-f_x(0,t)}{h_x(0,t)}, h_x(0,t)\neq 0;
  178. \]
  179. Notations:
  180. \[
  181. \tilde{f_1}(x,t)=f(x,t)-\frac{f_x(0,t)h(x,t)}{h_x(0,t)},
  182. \]
  183. \[
  184. \tilde{h_1}(x,t)=\frac{h(x,t)}{h_x(0,t)}.
  185. \]
  186.  
  187.     \end{frame}
  188.     \begin{frame}
  189.         \frametitle{Problem 2}
  190.                      Notations: $v(x,t)=u_{xx}(x,t)$, $\tilde{f}_2(x,t)=\tilde{f}_{1xx}(x,t)$, $\tilde{h}_2(x,t)=\tilde{h}_{1xx}(x,t)$,                 $\tilde{f}_1(x,t)=f(x,t)-\frac{f_x(0,t)h(x,t)}{h_x(0,t)}$, $\tilde{h}_1(x,t)=\frac{h(x,t)}{h_x(0,t)}$,
  191.         $\tilde{\alpha}(t)=\tilde{h}_1(1,t)$ и $\tilde{\beta}(t)=-\tilde{h}_{1}(0,t)$.
  192.  
  193.         Consider the problem:
  194.         \[
  195.         \begin{cases}
  196.          v_{ttt}(x,t)+v_{xx}(x,t)+\mu v(x,t)=\tilde{f_2}(x,t)+\tilde{ h_2}(x,t)v(0,t), \\
  197.         v(1,t)-\tilde{\alpha} (t) v_x(0,t)=0, x\in[0,1], t\in[0,T],\\
  198.         v(0,t)+\tilde{\beta }(t)v_x(0,t)=0, x\in[0,1], t\in[0,T].
  199.         \end{cases}
  200.         \]
  201.            
  202.     \end{frame}
  203.  
  204.     \begin{frame}
  205.         \frametitle{Existence theorem 2}
  206.                      $h(x,t)\in C^2(\overline{Q})$ and conditions are met when $ t\in [0,T]$, $x \in [0,1]$:
  207. \[
  208. h_x(0,t)\neq 0,
  209. h(1,t)\equiv0,
  210. \mu\textless 0,
  211. -\frac{h(0,t)}{h_x(0,t)}\leqslant0,
  212. \]
  213. \[
  214. \mu^3+\left(4\left(\max_{\overline {Q}}\left|\frac{h_{xx}(x,t)}{h_x(0,t)}\right|\right)^2-\mu\right)
  215. \left(\max\limits_{\overline{Q}}\left|\frac{h_{xx}(x,t)}{h_x(0,t)}\right|\right)^2\leqslant0,
  216. \]
  217. \[
  218. h_{xt}(0,t)h(x,t)-h_{t}(x,t)h_x(0,t)\leqslant0.
  219. \]
  220. Then inverse problem I with redefinition condition $u_x(0,t)=0$ and with basic conditions $u(0,t)=u(1,t)=0$ has a solution $\{u(x,t), q(t)\}$ such that $u(x,t) \in L_2(Q)$, $u_{xx}(x,t) \in L_2(Q)$, $u_{ttt}(x,t) \in L_2(Q)$, $q(t) \in L_2([0,T])$,
  221.  for any function $f(x,t)$  such that
  222. $f(x,t), f_{xx}(x,t)\in L_2(Q)$:
  223. \[
  224. f(1,t)-\frac{f_x(0,t)}{h_x(0,t)}h(1,t)=0,
  225. f(0,t)-\frac{f_x(0,t)}{h_x(0,t)}h(0,t)=0.
  226. \]
  227.  
  228.     \end{frame}
  229. \begin{frame}
  230.         \frametitle{Problem 3}
  231. $u(0,t)=0$ --- redefinition condition,
  232.  
  233. $u_x(0,t)=u_x(1,t)=0$ --- basic conditions.
  234.                      \[
  235. q(t)=\frac{u_{xx}(0,t)-f(0,t)}{h(0,t)}, h(0,t)\neq 0;
  236. \]
  237. Notations:
  238. \[
  239. \hat{f_1}(x,t)=f(x,t)-\frac{f(0,t)h(x,t)}{h(0,t)},
  240. \]
  241. \[
  242. \hat{h_1}(x,t)=\frac{h(x,t)}{h(0,t)}.
  243. \]
  244.  
  245.     \end{frame}
  246.  
  247.     \begin{frame}
  248.         \frametitle{Problem 3}
  249.                      Notations: $v(x,t)=u_{xx}(x,t)$, $\hat{f}_2=\hat{f}_{1xx}$, $\hat{h}_2=\hat{h}_{1xx}$, $\hat{f}                        _1(x,t)=f(x,t)-\frac{f(0,t)h(x,t)}{h(0,t)}$, $\hat{h}_1(x,t)=\frac{h(x,t)}{h(0,t)}$, $\hat{\alpha}(t)=\hat{h}_{1x}              (0,t)$ и $\hat{\beta}(t)=\hat{h}_{1x}(1,t)$.
  250.  
  251.         Consider the problem:
  252.         \[
  253.         \begin{cases}
  254.         v_{ttt}(x,t)+v_{xx}(x,t)+\mu v(x,t)=\hat{f_2}(x,t)+\hat{ h_2}(x,t)v(0,t),\\
  255.         v_x(0,t)-\hat{\alpha} (t)  v(0,t)=0, x\in[0,1], t\in[0,T],\\
  256.         v_x(1,t)+\hat{\beta }(t)v(0,t)=0, x\in[0,1], t\in[0,T].
  257.         \end{cases}
  258.         \]
  259.            
  260.     \end{frame}
  261.  
  262.     \begin{frame}
  263.         \frametitle{Existence theorem 3}
  264.          $h(x,t)\in C^4(\overline{Q})$ and conditions are met when $t\in[0,T]$:
  265. \[
  266. h(0,t)\neq0,
  267. h_x(1,t)\equiv 0,
  268. \mu\textless 0,
  269. \frac{h_x(0,t)}{h(0,t)}\geqslant0,
  270. \]
  271. \[
  272. h_{xt}(0,t)h(0,t)-h_t(0,t)h_x(0,t)\geqslant0,
  273. \]
  274. \[
  275. h_{xttt}(0,t)h(0,t)+h_{xtt}(0,t)h_t(0,t)-h_{ttt}(0,t)h_x(0,t)-h_{tt}(0,t)h_{xt}\geqslant0.
  276. \]
  277. Then inverse problem I with redefinition condition $u(0,t)=0$ with basic conditions  
  278. $u_x(0,t)=u_x(1,t)=0$ has a solution $\{u(x,t), q(t)\}$ such that $u(x,t) \in L_2(Q)$, $u_{xx}(x,t) \in L_2(Q)$,
  279. $u_{ttt}(x,t) \in L_2(Q)$, $q(t) \in L_2([0,T])$  for any function $f(x,t)$such that $f(x,t), f_{xx}(x,t)\in L_2(Q)$:
  280. \[
  281. f_x(0,t)-\frac{f(0,t)}{h(0,t)}h_x(0,t)=0,
  282. \]
  283. \[
  284. f_x(1,t)-\frac{f(0,t)}{h(0,t)}h_x(1,t)=0.
  285. \]
  286.     \end{frame}
  287.  
  288.  
  289.     \begin{frame}
  290.         \frametitle{Formulation of the problem for hyperbolic equations}
  291.          
  292.         $Q=(0,1)\times(0,T)$-rectangle,
  293.  
  294.         $\mu(x,t)$,
  295.         $ f (x,t)$,
  296.         $ h(x,t)$  -  prescribed function, defined in $\overline{Q}$.
  297.         \begin{rutask}
  298.             Find function $ \ u(x,t)$ and $\ q(t)$, connected by the equation  
  299.         \[
  300.         u_{tttt}+u_{xx}+\mu(x,t)u=f(x,t)+q(t)h(x,t)
  301.         \]
  302. in area Q.
  303.         Conditions are satisfied for the function
  304.          $u(x,t)$ conditions
  305.         \[
  306.         u(x,0)=u_t(x,0)=u_{tt}(x,0)=u_t(x,T)=0,  x \in [0,1],
  307.         \]
  308. and one of the group:
  309.  
  310.             1. $u(0,t)=0$ -  redefinition condition ,
  311.            
  312. $u(1,t)=u_x(0,t)=0$ -basic condition.
  313.                
  314.             2. $u_x(0,t)=0$ - redefinition,
  315.             $u(0,t)=u(1,t)=0$ - basic.
  316.        
  317.             3. $u(0,t)=0$ - redefinition,
  318.             $u_x(0,t)=u_x(1,t)=0$ - basic.
  319.         \end{rutask}
  320.     \end{frame}
  321.  
  322.  
  323.  
  324.  
  325.     \begin{frame}
  326.         \frametitle{Existence theorem  4}
  327.                   $h(x,t)\in C^6(\overline{Q})$, $\mu$, $\beta(t)=-\frac{h_x(1,t)}{h(0,t)}$. Conditions are met for this functions and for  
  328. $t\in [0,T]$:  
  329. \[
  330. h(0,t)\neq 0,    
  331. \mu\geqslant0,
  332. \beta(T)\leqslant 0,
  333. \beta^{''}(t)\geqslant 0,
  334. \beta_{2}(T)=\beta_{2}^{'}(T)=0,
  335. \]
  336. \[
  337. \beta^{'}(t)(A-t)-\beta(t)\geqslant 0,
  338. -\beta^{'}(t)(A-t)-\beta(t)\leqslant 0,
  339. \]
  340. \[
  341. -3\beta^{'''}(T)+(A-T)\beta^{''''}(T)\leqslant 0,
  342. -4\beta^{''''}(t)+(A-t)\beta ^{'''''}(t)\geqslant 0.
  343. \]
  344. Then inverse problem II  with redefinition condition: $u(0,t)=0$ with basic conditions $u(1,t)=u_x(0,t)=0$  has a solution $\left\{
  345. u(x,t),q(t)\right\}$  such that $u(x,t)\in{L_2(Q)}$, $q(t)\in {L_2([0,T])}$,
  346. $u_{tttt}(x,t)\in{L_2(Q)}$, $u_{xx}(x,t)\in{L_2(Q)}$,$v(x,t)\in{L_2(Q)}$,$v_{ttt}(x,t)\in{L_2(Q)}$ and $v_{xx}(x,t)\in{L_2(Q)}$ for any function $f(x,t)$  such that $f(x,t),f_{xx}(x,t), f_{xxt}(x,t)\in L_2(Q)$:
  347. \[
  348. f(1,t)-\frac{f(0,t)h(1,t)}{h(0,t)}=0,
  349. f_x(1,t)-\frac{f(0,t)h_x(1,t)}{h(0,t)}=0.
  350. \]
  351.     \end{frame}
  352.  
  353.            \begin{frame}        
  354.         \frametitle{Existence theorem  5}
  355.  
  356.  $h(x,t)\in C^2(\overline{Q})$, $\tilde{\beta}(t)=-\frac{h(0,t)}{h_x(0,t)}$.  Conditions are met for this functions and for
  357. $t\in [0,T]$:  
  358. \[
  359. h_x(0,t)\neq 0,    
  360. \mu\geqslant0,
  361. \tilde{\beta}(T)\geqslant 0,
  362. \tilde{\beta}^{'}(t)\geqslant 0,
  363. \]
  364. \[
  365. \tilde{\beta}(t)-\tilde{\beta}^{'}(t)\leqslant 0.
  366. \]
  367. Then inverse problem II with redefinition condition: $u_x(0,t)=0$ with basic conditions $u(0,t)=u(1,t)=0$  has a solution $\left\{
  368. u(x,t),q(t)\right\}$  such that $u(x,t)\in{L_2(Q)}$, $q(t)\in {L_2([0,T])}$,
  369. $u_{tttt}(x,t)\in{L_2(Q)}$, $u_{xx}(x,t)\in{L_2(Q)}$,$v(x,t)\in{L_2(Q)}$,$v_{ttt}(x,t)\in{L_2(Q)}$ and $v_{xx}(x,t)\in{L_2(Q)}$ for any function $f(x,t)$  such that $f(x,t),f_{xx}(x,t),f_{xxt}(x,t)\in L_2(Q)$:
  370. \[
  371. f(0,t)-\frac{f_x(0,t)}{h_x(0,t)}h(0,t)=0,
  372. f(1,t)-\frac{f_x(0,t)}{h_x(0,t)}h(1,t)=0.
  373. \]
  374.     \end{frame}
  375.  
  376.  
  377.  
  378. \begin{frame}
  379.         \frametitle{Existence theorem  6}
  380. Пусть $h(x,t)\in C^2(\overline{Q})$. Conditions are met for this functions and for
  381. $t\in [0,T]$:  
  382. \[
  383. h(0,t)\neq 0,  
  384. h_x(1,t)=0,
  385. \frac{h_x(0,T)}{h(0,T)}\geqslant 0,
  386. \]
  387. \[
  388. h_x(0,t)h(0,t)-(h_{xt}(0,t)h(0,t)+h_t(0,t)h_x(0,t))(A-t)\geqslant 0.  
  389. \]
  390. Then inverse problem II  with redefinition condition: $u(0,t)=0$  with basic conditions $u_x(0,t)=u_x(1,t)=0$  has a solution $\left\{u(x,t),q(t)\right\}$  such that $u(x,t)\in{L_2(Q)}$, $q(t)\in {L_2([0,T])}$, $u_{tttt}(x,t)\in{L_2(Q)}$, $u_{xx}(x,t)\in{L_2(Q)}$,$v(x,t)\in{L_2(Q)}$,$v_{ttt}(x,t)\in{L_2(Q)}$ and $v_{xx}(x,t)\in{L_2(Q)}$ for any function $f(x,t)$  such that $f(x,t),f_{xx}(x,t), f_{xxt}(x,t)\in L_2(Q)$:
  391. \[
  392. f_x(0,t)-\frac{f(0,t)}{h(0,t)}h_x(0,t)=0,
  393. f_x(1,t)-\frac{f(0,t)}{h(0,t)}h_x(1,t)=0.
  394. \]
  395.  
  396.     \end{frame}
  397.  
  398.  
  399.  
  400. \begin{frame}
  401.         \frametitle{Conclusion}
  402.         We have studied inverse problems for quasiparabolic and quasi-hyperbolic equations
  403.         \begin{itemize}
  404.             \item $u_{ttt}+u_{xx}+\mu(x,t)u=f(x,t)+q(t)h(x,t)$,
  405.             \item $u_{tttt}+u_{xx}+\mu(x,t)u=f(x,t)+q(t)h(x,t)$,
  406.         \end{itemize}
  407. We have proved the existence of a solution $\left\{q(t),u(x,t)\right\}$ and defined the right parts for three different cases.
  408.     \end{frame}
  409.  
  410. \begin{frame}
  411.         \frametitle{Conclusion}
  412. Thank you for attention
  413.  
  414.     \end{frame}
  415.  
  416.  
  417.    
  418.  
  419. \end{document}
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