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  1. %курсач (2009)
  2. \documentclass[hyperref={unicode=true}]{beamer}
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  11. \usepackage{algorithm}
  12. \usepackage[noend]{algorithmic}
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  17.  
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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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