Recovery of a fast oscillating free term in the wave equation by asymptotics of the solution

TL;DR

This paper demonstrates that a high-frequency free term in a 1D wave equation can be fully recovered using partial asymptotic data of the solution, advancing inverse problem techniques.

Contribution

It introduces methods to recover a high-frequency free term from incomplete asymptotic information of the wave solution, with rigorous asymptotic analysis.

Findings

Complete recovery of the free term from three-term asymptotics

Construction and justification of solution asymptotics

Application to inverse problems in wave equations

Abstract

Three problems about recovery of a high-frequency free term in the one-dimension wave equation with homogeneous initial-boundary conditions by some information about partial asymptotics of its solution have been solved. It is shoun, that the free term can be completely recovered from a specific data about incomplete (three-terms) asymptotics of the solution. Before formulation of the each problem about recovery of free term, construction and justification of the asymptotics of the solution of original initial-boundary problem are given.

Equations

$$\frac{{\partial }^{2}u}{\partial t^2}=\frac{{\partial }^{2}u}{\partial x^2}+f(x,t)r(t,\omega t),(x,t)\in \Pi ,$$

$${u|}_{t=0}=0,{\frac{\partial u}{\partial t}|}_{t=0}=0,$$

$${u|}_{x=0,\pi }=0.$$

$$f,f_{x^2}^{\prime \prime },f_{x^{2}t}^{\prime \prime \prime },f_t^{\prime },f_{t^2}^{\prime \prime },f_{t^3}^{\prime \prime \prime },$$

$$r_0(t)=⟨r(t,\tau ){⟩}_{\tau }=\frac{1}{2\pi }{\int }_0^{2\pi }r(t,\tau )d\tau ,$$

$$r_1(t,\tau )=r(t,\tau )-r_0(t)\equiv \{r_1(t,\tau ){\}}_{\tau }.$$

$$f_n(t)=\frac{1}{\pi }\underset{0}{\overset{\pi }{\int }}f(s,t)\sin nsds,f_{2,n}(t)=\frac{1}{\pi }\underset{0}{\overset{\pi }{\int }}\frac{{\partial }^{2}f(s,t)}{\partial s^2}\sin nsds,$$

$$\begin{gathered} {\rho }_0(t,\tau )=\underset{0}{\overset{\tau }{\int }}\left(\underset{0}{\overset{p}{\int }}{r}_1(t,s)ds-{⟨\underset{0}{\overset{\tau }{\int }}{r}_1(t,s)ds⟩}_{\tau }\right)dp- \\ {⟨\underset{0}{\overset{\tau }{\int }}\left(\underset{0}{\overset{p}{\int }}{r}_1(t,s)ds-{⟨\underset{0}{\overset{\tau }{\int }}{r}_1(t,s)ds⟩}_{\tau }\right)dp⟩}_{\tau }\equiv {\left\{{\int }_0^{\tau }{\left\{{\int }_0^p{r}_1(t,s)ds\right\}}_{p}dp\right\}}_{\tau }. \end{gathered}$$

$$a_{1,n}=f_{2,n}(0),a_{2,n}=f_{2,n}^{\prime }(0),$$

$$b_0={\rho }_0(0,0),b_1={\rho }_{0\tau }(0,0),b_2(x)=b_0{f}_t(x,0)+b_{3}f(x,0),b_3={\rho }_{0t}(0,0),$$

$$R_{0,n}(t)=\frac{1-\cos nt}{n},R_{1,n}(t)=\frac{t}{n}-\frac{\sin nt}{n^2},$$

$$u_{\omega }(x,t)=U_{\omega }(x,t)+W_{\omega }(x,t),\omega \gg 1,$$

$$U_{\omega}(x,t)=u_{0}(x,t)+\omega^{-1}u_{1}(x,t)+\omega^{-2}\bigl{[}u_{2}(x,t)+v_{2}(x,t,\omega t)\bigr{]},\omega\gg 1,$$

$$u_0(x,t)=\sum _{n=1}^{\infty }\frac{\sin nx}{n}\underset{0}{\overset{t}{\int }}\sin n(t-s)f_n(s)r_0(s)ds,$$

$$u_1(x,t)=-b_1\left(\sum _{n=1}^{\infty }{a}_{1,n}\frac{\sin nx}{n}{R}_{1,n}(t)+t\cdot f(x,0)\right),$$

$$v_2(x,t,\tau )=f(x,t){\rho }_0(t,\tau ),$$

$$u_{2}(x,t)=-\sum_{n=1}^{\infty}\frac{\sin{nx}}{n}\biggl{(}b_{0}a_{2,n}R_{1,n}(t)+b_{0}a_{1,n}R_{0,n}(t)+b_{3}a_{1,n}R_{1,n}(t)\biggr{)}-b_{0}f(x,0)-t\cdot b_{2}(x)$$

$$\bigl{\|}W_{\omega}(x,t)\bigr{\|}_{C(\Pi)}=o(\omega^{-2}),\omega\to\infty.$$

$${\phi }_0\in C^2([0,T]),{\phi }_0(0)=0,{\phi }_0^{\prime }(0)=0;$$

$${\phi }_1(t)=-b_1\left(\sum _{n=1}^{\infty }{a}_{1,n}\frac{\sin n{x}_0}{n}{R}_{1,n}(t)+t\cdot f(x_0,0)\right),$$

$$\varphi_{2}(t)=\sum_{n=1}^{\infty}\frac{\sin{nx_{0}}}{n}\biggl{(}b_{0}a_{2,n}R_{1,n}(t)+b_{0}a_{1,n}R_{0,n}(t)+b_{3}a_{1,n}R_{1,n}(t)\biggr{)}-b_{0}f(x_{0},0)-t\cdot b_{2}(x_{0}),$$

$$r_1(t,\tau )=\frac{1}{f(x_0,t)}\frac{{\partial }^2}{\partial {\tau }^2}\chi (t,\tau ).$$

$$\left\|u_{\omega}(x_{0},t)-\left[\varphi_{0}(t)+\frac{1}{\omega}\varphi_{1}(t)+\frac{1}{\omega^{2}}\bigl{(}\varphi_{2}(t)+\chi(t,\omega t)\bigr{)}\right]\right\|_{C([0,T])}=o(\omega^{-2}),\;\omega\to\infty.$$

$$u_{\omega}(x,t)=u_{0}(x,t)+\omega^{-1}u_{1}(x,t)+\omega^{-2}\bigl{[}u_{2}(x,t)+v_{2}(x,t,\omega t)\bigr{]}+\omega^{-3}v_{3}(x,t,\omega t)+Z_{\omega}(x,t),$$

$$\left\{\begin{array}[]{l}\frac{\partial^{2}u_{0}}{\partial t^{2}}+\frac{\partial^{2}v_{2}}{\partial\tau^{2}}+\omega^{-1}\left[\frac{\partial^{2}u_{1}}{\partial t^{2}}+2\frac{\partial^{2}v_{2}}{\partial\tau\partial t}+\frac{\partial^{2}v_{3}}{\partial\tau^{2}}\right]+\\ \qquad\omega^{-2}\left[\frac{\partial^{2}u_{2}}{\partial t^{2}}+\frac{\partial^{2}v_{2}}{\partial t^{2}}+2\frac{\partial^{2}v_{3}}{\partial\tau\partial t}\right]+\omega^{-3}\frac{\partial^{2}v_{3}}{\partial t^{2}}+\frac{\partial^{2}Z_{\omega}}{\partial t^{2}}=\\ \qquad\qquad\frac{\partial^{2}u_{0}}{\partial x^{2}}+\omega^{-1}\frac{\partial^{2}u_{1}}{\partial x^{2}}+\omega^{-2}\left[\frac{\partial^{2}u_{2}}{\partial x^{2}}+\frac{\partial^{2}v_{2}}{\partial x^{2}}\right]+\omega^{-3}\frac{\partial^{2}v_{3}}{\partial x^{2}}+\\ \qquad\qquad\qquad\frac{\partial^{2}Z_{\omega}}{\partial x^{2}}+f(x,t)r(t,\tau),\\ \left[u_{0}+\omega^{-1}u_{1}+\omega^{-2}(u_{2}+v_{2})+\omega^{-3}v_{3}+Z_{\omega}\right]_{t,\tau=0}=0,\\ \left[\frac{\partial u_{0}}{\partial t}+\omega^{-1}\left(\frac{\partial u_{1}}{\partial t}+\frac{\partial v_{2}}{\partial\tau}\right)+\omega^{-2}\left(\frac{\partial u_{2}}{\partial t}+\frac{\partial v_{2}}{\partial t}+\frac{\partial v_{3}}{\partial\tau}\right)+\omega^{-3}\frac{\partial v_{3}}{\partial t}+\frac{\partial Z_{\omega}}{\partial t}\right]_{t,\tau=0}=0,\\ \left[u_{0}+\omega^{-1}u_{1}+\omega^{-2}(u_{2}+v_{2})+\omega^{-3}v_{3}+Z_{\omega}\right]_{x=0,\pi}=0,\;\tau=\omega t.\end{array}\right.$$

$$\displaystyle\displaystyle\left\{\begin{array}[]{l}\frac{\partial^{2}u_{0}}{\partial t^{2}}=\frac{\partial^{2}u_{0}}{\partial x^{2}}+f(x,t)r_{0}(t),\\ \left.u_{0}\right|_{t=0}=0,\;\left.\frac{\partial u_{0}}{\partial t}\right|_{t=0}=0,\\ \left.u_{0}\right|_{x=0,\pi}=0,\end{array}\right.$$

$$\displaystyle\displaystyle\left\{\begin{array}[]{l}\frac{\partial^{2}v_{2}}{\partial\tau^{2}}=f(x,t)r_{1}(t,\tau),\\ v_{2}(x,t,\tau+2\pi)=v_{2}(x,t,\tau),\\ \left\langle v_{2}(x,t,\tau)\right\rangle_{\tau}=0,\end{array}\right.$$

$$\displaystyle\displaystyle\left\{\begin{array}[]{l}\frac{\partial^{2}u_{1}}{\partial t^{2}}=\frac{\partial^{2}u_{1}}{\partial x^{2}},\\ \left.u_{1}\right|_{t=0}=0,\;\left.\frac{\partial u_{1}}{\partial t}\right|_{t=0}=-\left.\frac{\partial v_{2}}{\partial\tau}\right|_{t,\tau=0},\\ \left.u_{1}\right|_{x=0,\pi}=0,\end{array}\right.$$

$$\displaystyle\displaystyle\left\{\begin{array}[]{l}\frac{\partial^{2}v_{3}}{\partial\tau^{2}}=-2\frac{\partial^{2}v_{2}}{\partial\tau\partial t},\\ v_{3}(x,t,\tau+2\pi)=v_{3}(x,t,\tau),\\ \left\langle v_{3}(x,t,\tau)\right\rangle_{\tau}=0,\end{array}\right.$$

$$\displaystyle\displaystyle\left\{\begin{array}[]{l}\frac{\partial^{2}u_{2}}{\partial t^{2}}=\frac{\partial^{2}u_{2}}{\partial x^{2}},\\ \left.u_{2}\right|_{t=0}=-\left.v_{2}\right|_{t=0},\;\left.\frac{\partial u_{2}}{\partial t}\right|_{t=0}=-\left[\frac{\partial v_{3}}{\partial\tau}+\frac{\partial v_{2}}{\partial t}\right]_{t,\tau=0},\\ \left.u_{2}\right|_{x=0,\pi}=0,\end{array}\right.$$

$$v_3(x,t,\tau )=2f_t(x,t){\rho }_1(t,\tau )+2f(x,t){\rho }_{1t}(t,\tau ),$$

$${\rho }_1(t,\tau )={⟨\underset{0}{\overset{\tau }{\int }}{\rho }_0(t,s)ds⟩}_{\tau }-\underset{0}{\overset{\tau }{\int }}{\rho }_0(t,s)ds.$$

Full text

517.955.8

. ., . .

• . , ( ) . [1, 2].

( ., [3]–[6]) . [7] – [9], , . [3]–[9] .

$\displaystyle f(x,t)r(t,\omega t)$, $\displaystyle r(t,\tau)$ $\displaystyle\tau$, $\displaystyle\omega$ :

1. $\displaystyle r(t,\tau)$;
2. $\displaystyle f(x,t)\equiv f(x)$;
3. $\displaystyle f,r$ $\displaystyle r_{0}$ $\displaystyle r(t,\tau)$ $\displaystyle\tau$;
4. – $\displaystyle f$ $\displaystyle r$. , ( ) . , .

, , 1, 2 1 , 1), [10]. [10] [11], 1 , .

1

1.1

$\displaystyle\Pi$ : $\displaystyle{\Pi=\{(x,t):0\leq x\leq\pi;0\leq t\leq T\}},$ $\displaystyle T>0$. $\displaystyle\Pi$ - $\displaystyle\omega$:

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$\displaystyle f(x,t)$ $\displaystyle\Pi$ $\displaystyle\Pi$

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$\displaystyle x=0,x=\pi$, (1.4) $\displaystyle\Pi$ $\displaystyle x$. , (1.4) $\displaystyle x=0,\pi$ $\displaystyle f$ $\displaystyle f^{\prime\prime}_{x^{2}}$.

$\displaystyle Q$ - : $\displaystyle Q=\{(t,\tau)\in[0,T]\times[0,\infty)\}$. $\displaystyle r(t,\tau)$ , $\displaystyle Q$, $\displaystyle 2\pi$- $\displaystyle\tau$. $\displaystyle r_{0}(t)$ $\displaystyle\tau$ ( ):

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$\displaystyle r_{1}(t,\tau)$ – :

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, $\displaystyle r_{0}\in C([0,\pi])$ $\displaystyle 2\pi$- $\displaystyle\tau$ $\displaystyle\frac{\partial^{k}r_{1}}{\partial t^{k}}\in C(Q)$, $\displaystyle k=\overline{0,3}$. $\displaystyle r(t,\tau)$, ( ).

, :

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(1.1)-(1.3) :

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**   1****.**

$\displaystyle u_{\omega}(x,t)$* (1.1)-(1.3) (1.8)-(1.13), $\displaystyle W_{\omega}$*

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1.2 1

, (1.1)–(1.3) , .$\displaystyle 1.1$, $\displaystyle f(x,t)$, $\displaystyle r(t,\tau)$ ( ) . $\displaystyle x_{0}\in(0,\pi)$, $\displaystyle f(x_{0},t)\neq 0,t\in[0,T]$, $\displaystyle\varphi_{0}(t)$ $\displaystyle\chi(t,\tau)$, :

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$\displaystyle\chi(t,\tau)$ – $\displaystyle Q$, $\displaystyle 2\pi$- $\displaystyle\tau$ , $\displaystyle\frac{\partial^{k+2}\chi}{\partial t^{k}\partial\tau^{2}},k=\overline{0,3},$ $\displaystyle C(Q)$. $\displaystyle\varphi_{1}(t)$ $\displaystyle\varphi_{2}(t)$:

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$\displaystyle\rho_{0}(t,\tau)$ (1.7)

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1 $\displaystyle r(t,\tau)$ ( ), $\displaystyle u_{\omega}(x,t)$ (1.1)-(1.3) ( $\displaystyle f(x,t)$ .1.1)

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$\displaystyle x_{0}$ $\displaystyle\varphi_{0},\varphi_{1},\varphi_{2},\chi$ , .

**   2****.**

$\displaystyle\chi,\varphi_{0}$* $\displaystyle x_{0}$, , 1 , . . $\displaystyle r$ ( ), $\displaystyle u_{\omega}(x,t)$ (1.1)-(1.3) (1.18).*

*  *.

$\displaystyle r_{0}(t)=\left\langle r(t,\tau)\right\rangle_{\tau}$ , $\displaystyle r_{1}(t,\tau)$ (1.17).

1.3

1.14**.**

(1.1)-(1.3) :

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$\displaystyle u_{i},v_{i}$ $\displaystyle Z_{\omega}$ .

$\displaystyle u_{\omega}$ (1.1)-(1.3), :

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$\displaystyle\omega^{-k},\;(k=\overline{0,3})$. $\displaystyle\langle...\rangle$ $\displaystyle\tau=\omega t$, :

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(1.23)–(1.39) $\displaystyle u_{0},u_{1},u_{2},v_{2}$ (1.10), (1.11), (1.12) (1.13) , . (1.35)

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(1.19) (1.23)–(1.39) , $\displaystyle Z_{\omega}(x,t)$

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(1.12), (1.40) $\displaystyle p_{i}(x,t,\tau)$ 2$\displaystyle\pi$- $\displaystyle\tau$ , , $\displaystyle\left.p_{i}(x,t,\tau)\right|_{x=0,\pi}=0,\left.c_{i}(x)\right|_{x=0,\pi}=0,i=1,2$.

(1.41)

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$\displaystyle W_{\omega}(x,t)=Z_{\omega}(x,t)+\omega^{-3}v_{3}(x,t,\omega t)$, 1 , :

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, , $\displaystyle f$ $\displaystyle r$, $\displaystyle p_{2}(x,t,\tau),c_{1}(x),c_{2}(x)$ ,

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$\displaystyle(x,t)\in\overline{\Pi}$. , $\displaystyle n_{0}\in\mathbb{N}$

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. $\displaystyle\varepsilon$ – . $\displaystyle t_{0}>0$, $\displaystyle x\in[0,\pi],t\in[0,t_{0}]$, $\displaystyle\omega>0$

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$\displaystyle[0,t],t\in[t_{0},T]$, $\displaystyle m$ $\displaystyle[t_{j},t_{j+1}),j=0,1,\ldots,m-1$,

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$\displaystyle m$ , $\displaystyle\omega>0$

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,

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, $\displaystyle p_{1,n}(s,\tau)$ , $\displaystyle\omega_{0}>0$, $\displaystyle\omega>\omega_{0}$

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(1.43)–(1.45) (1.42). 1 .

2.

1.14 (1.1)–(1.3) $\displaystyle r(t,\tau)$ ( ) (1.8), (1.9)–(1.13),

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, $\displaystyle r(t,\tau)$ , $\displaystyle u_{\omega}$ – (1.1)-(1.3). (1.14),(1.18) $\displaystyle t\in[0,T]$

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(1.46) $\displaystyle\omega$, $\displaystyle\tau$ $\displaystyle t$,

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.$\displaystyle 1.1$ $\displaystyle\frac{\partial^{2}v_{2}(x,t,\tau)}{\partial\tau^{2}}=f(x,t)r_{1}(t,\tau)$. (1.48),

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.1.1 $\displaystyle u_{0}(x,t)$ (1.10),

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, $\displaystyle K(t,s)$ . (1.47), (1.50)

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$\displaystyle r_{0}(t)$. (1.49) $\displaystyle r_{1}$ :

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, $\displaystyle\chi$, .1.1 .

$\displaystyle r(t,\tau)=r_{0}(t)+r_{1}(t,\tau)$ ( ), 1, (1.1)-(1.3), (1.8)–(1.13). , $\displaystyle u_{\omega}(x,t)$ (1.18). :

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(1.50),(1.51) $\displaystyle\frac{\partial^{2}}{\partial t^{2}}u_{0}(x_{0},t)=\varphi_{0}^{\prime\prime}(t)$. $\displaystyle u_{0}(x_{0},0)=\varphi_{0}(0)=0,\left.\frac{\partial u_{0}(x_{0},t)}{\partial t}\right|_{t=0}=\varphi^{\prime}_{0}(0)=0$, $\displaystyle u_{0}(x_{0},t)=\varphi_{0}(t)$. $\displaystyle r_{1}$ (1.49) (1.27). $\displaystyle x=x_{0}$ , $\displaystyle\chi(t,\tau)$ $\displaystyle v_{2}(x,t,\tau)$ $\displaystyle 2\pi$- $\displaystyle\tau$. $\displaystyle v_{2}(x_{0},t,\tau)=\chi(t,\tau)$. , , (1.11) (1.15) $\displaystyle u_{1}(x_{0},t)=\varphi_{1}(t)$, (1.13) (1.16) $\displaystyle u_{2}(x_{0},t)=\varphi_{2}(t)$. 2 .

2

2.1

$\displaystyle\Pi$ Q - , . $\displaystyle 1.1$. (1.1)-(1.3) $\displaystyle f(x,t)\equiv f(x)$. , $\displaystyle f\in C^{2}([0,\pi]),f(0)=f(\pi)=0$, $\displaystyle r(t,\tau)=r_{0}(t)+r_{1}(t,\tau)$ – 2$\displaystyle\pi$- $\displaystyle\tau$, $\displaystyle r_{0}\in C([0,T])$ – , $\displaystyle r_{1}\in C^{\alpha,0}_{2\pi}(Q),\alpha\in(0,1)$111 $\displaystyle C^{\alpha,0}_{2\pi}(Q)$ $\displaystyle Q$ $\displaystyle v(t,\tau))2\pi$- $\displaystyle\tau$, $\displaystyle(t,\tau)\in Q$ $\displaystyle t$ $\displaystyle\alpha$ .. :

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$\displaystyle f_{n}$ - $\displaystyle f(x)$ ( . .$\displaystyle 1.1$).

**   3****.**

\displaystyle\bigl{\|}u_{\omega}-u_{0}\bigr{\|}_{C(\Pi)}=o(1),\;\omega\to\infty,**

$\displaystyle u_{\omega}$* - (1.1)–(1.3).*

2.2 2

(1.1)–(1.3). , $\displaystyle r(t,\tau)$ , .$\displaystyle 2.1$ $\displaystyle r_{0}\in C^{2}([0,T])$, $\displaystyle f(x,t)\equiv f(x)$, .$\displaystyle 2.1$, .

, $\displaystyle\Lambda_{n}(t),n\in\mathbb{N},t\in[0,T]$ – , (2.1).

**   1****.**

$\displaystyle t_{0}\in(0,T]$, $\displaystyle|r_{0}(t_{0})|>|r_{0}(0)|$, $\displaystyle c_{0}>0$ $\displaystyle n_{0}\in\mathbb{N}$, $\displaystyle n\geq n_{0}$ $\displaystyle\Lambda_{n}(t_{0})>\frac{c_{0}}{n^{2}}$.

$\displaystyle r_{0}=const\neq 0$* $\displaystyle t_{0,1}=2\pi\frac{l_{0}}{m_{0}}$, $\displaystyle l_{0},m_{0}\in\mathbb{N}$ – , $\displaystyle c_{1}>0$ $\displaystyle n_{1}\in\mathbb{N}$, $\displaystyle n\geq n_{1},n\neq sm_{0},s\in\mathbb{N}$, $\displaystyle\Lambda_{n}(t_{0,1})>\frac{c_{0}}{n^{2}}$.*

*  *.

$\displaystyle r_{0}\equiv 0$, , $\displaystyle\Lambda_{n}(t)\equiv 0$ $\displaystyle n\in\mathbb{N}$.

$\displaystyle M_{0}$ $\displaystyle n\in\mathbb{N}$, $\displaystyle\Lambda_{n}(t_{0})=0$.

(1.1)–(1.3) $\displaystyle f$

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2 $\displaystyle f$, .2.1, $\displaystyle u_{\omega}(x,t)$ (1.1)–(1.3) :

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**   4****.**

$\displaystyle t_{0}$, $\displaystyle|r_{0}(t_{0})|>|r_{0}(0)|$. $\displaystyle M_{0}=\emptyset$ , $\displaystyle f_{n}=\frac{\psi_{n}}{\Lambda_{n}},n\in\mathbb{N}$. $\displaystyle M_{0}\neq\emptyset$, , $\displaystyle\psi_{n}=0,n\in M_{0}$, $\displaystyle f_{n}=\frac{\psi_{n}}{\Lambda_{n}},n\notin M_{0}$, $\displaystyle f_{n}$ – $\displaystyle n\in M_{0}$.

2.3

3.

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$\displaystyle\varepsilon$ – . , , .2.1 $\displaystyle f,r_{1}$, , $\displaystyle W_{\omega}$, $\displaystyle(x,t)\in{\Pi},{\omega}>0$. - , $\displaystyle n_{0}$ , $\displaystyle\omega>0,(x,t)\in[0,\pi]\times[0,T]$

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$\displaystyle t_{0}>0$ , $\displaystyle(x,t)\in[0,\pi]\times[0,t_{0}]$ $\displaystyle\omega>0$

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$\displaystyle t\in[t_{0},T]$ $\displaystyle[0,t]$ $\displaystyle m$ $\displaystyle[t_{j},t_{j+1}),j=\overline{0,m-1},$ :

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1 $\displaystyle m$ , $\displaystyle(x,t)\in[0,\pi]\times[t_{0},T]$ $\displaystyle\omega>0$

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$\displaystyle\left\langle r_{1}(t,\tau)\right\rangle_{\tau}=0$ $\displaystyle\omega_{0}$ , $\displaystyle m,t\in[t_{0},T],$ $\displaystyle\omega>\omega_{0}$

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(2.6), (2.7) $\displaystyle\omega_{0}>0,$ $\displaystyle\omega>\omega_{0}$

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(2.4), (2.8) 3 .

1.

$\displaystyle t_{0}\in(0,T]$ , $\displaystyle|r_{0}(t_{0})|>|r_{0}(0)|$.

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, $\displaystyle c_{1}$ $\displaystyle N_{1}$, $\displaystyle n>\mathbb{N_{1}}$

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$\displaystyle r_{0}=const\neq 0$. $\displaystyle r_{0}(0)=1$.

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. .

4.

$\displaystyle t_{0}\in(0,T]$ , $\displaystyle|r_{0}(t_{0})|>|r_{0}(0)|$. , $\displaystyle f$, .$\displaystyle 2.2$, . 3 (2.1), (2.3)

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$\displaystyle\psi_{n}$ $\displaystyle\psi_{n}=\alpha_{n}n^{-5}$, $\displaystyle{\alpha_{n}}\in l_{2}$. $\displaystyle M_{0}=\emptyset$ (2.9) 1,

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, $\displaystyle f$ ,

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$\displaystyle f_{n}$ (2.10). $\displaystyle f$ , , 2. $\displaystyle M_{0}\neq\emptyset$ $\displaystyle f$, , , $\displaystyle n\in M_{0}$ $\displaystyle\psi_{n}=0$. $\displaystyle f_{n}=0,n\in M_{0}$ $\displaystyle f_{n}=\frac{\psi_{n}}{\Lambda_{n}},n\notin M_{0}$. . ( $\displaystyle r_{0}=const\neq 0$) .

3

3.1 3

(1.1)-(1.3) $\displaystyle f(x,t)\equiv f(x)\in C^{4}([0,\pi])$, $\displaystyle f^{(2k)}(0)=f^{(2k)}(\pi)=0,k=\overline{0,1}$, $\displaystyle r$ ( ), $\displaystyle r_{0}\in C^{2}([0,\pi])$. , $\displaystyle r_{0}$ , , $\displaystyle t_{0}\in(0,T]$, $\displaystyle|r_{0}(t_{0})|>|r_{0}(0)|$ ( $\displaystyle r_{0}(t)\equiv const$ – . 2), $\displaystyle f$ $\displaystyle r_{1}$ . $\displaystyle\Lambda_{n}=\Lambda_{n}(t_{0}),n\in\mathbb{N}$ – , (2.1). , $\displaystyle M_{0}$ $\displaystyle n$, $\displaystyle\Lambda_{n}(t_{0})=0$, . $\displaystyle Q$ $\displaystyle 2\pi$- $\displaystyle\chi(t,\tau),$ $\displaystyle\frac{\partial^{k+2}\chi}{\partial t^{k}\partial\tau^{2}},k=\overline{0,3}$, $\displaystyle C(Q)$. $\displaystyle\psi\in C^{7}([0,\pi]),\psi^{(2j)}(0)=\psi^{(2j)}(\pi)=0,j=\overline{0,3}$, $\displaystyle x_{0}\in(0,\pi)$, $\displaystyle\widetilde{f}(x_{0})\neq 0$,

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$\displaystyle\varphi_{0}(t),\varphi_{1}(t),\varphi_{2}(t)$, . $\displaystyle\varphi_{0}(t)$ :

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$\displaystyle\varphi_{1},\varphi_{2}$ .$\displaystyle 1.2$ $\displaystyle f(x,t)$ $\displaystyle\widetilde{f}(x)$,

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3 $\displaystyle f$ $\displaystyle r$, , $\displaystyle u_{\omega}(x,t)$ (1.1)–(1.3) :

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1, 2 .

**   5****.**

$\displaystyle r_{0}(t),\varphi_{0}(t),\psi(x),\chi(t,\tau)$* $\displaystyle x_{0},t_{0}$ . 3 , . . ( .3.1 ) $\displaystyle f$ $\displaystyle r_{1}$, $\displaystyle u_{\omega}(x,t)$ (1.1)-(1.3) $\displaystyle f(x,t)\equiv f(x)$ (3.2) (3.3). $\displaystyle f(x)=\widetilde{f}(x)$ (3.1), $\displaystyle r_{1}(t,\tau)=(f(x_{0}))^{-1}\frac{\partial^{2}}{\partial\tau^{2}}\chi(t,\tau)$.*

4

$\displaystyle 4.1^{\circ}$ 4

(1.1)-(1.3), $\displaystyle f(x,t)\equiv f(x)$ $\displaystyle r(t,\tau)$ . , $\displaystyle r(t,\tau)$ – ( ) $\displaystyle r_{0}\in C^{2}([0,\pi])$, $\displaystyle f(x)=\sum\limits_{n=1}^{N}f_{n}\sin nx$ – $\displaystyle N$ $\displaystyle f_{n}$. , , $\displaystyle t_{0}\in(0,T),x_{j}\in(0,\pi),j=\overline{0,N-1},$ $\displaystyle x_{i}\neq x_{k}$ $\displaystyle i\neq k$, $\displaystyle\varphi_{0}(t),\chi(t,\tau)$ $\displaystyle\alpha_{j}(t)$, : $\displaystyle\varphi_{0}$ $\displaystyle\chi$ – , .1.2, $\displaystyle\varphi_{0}\in C^{4}([0,T)$

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:

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$\displaystyle f(x)$, $\displaystyle b_{0},b_{1},b_{3},R_{0,n}$ $\displaystyle R_{1,n}$ – , .1.1.

4 $\displaystyle f(x)$ $\displaystyle r(t,\tau)$, , $\displaystyle u_{\omega}(x,t)$ (1.1)–(1.3)

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.

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$\displaystyle\psi_{n},n=\overline{1,N}$. $\displaystyle A=(\sin nx_{j})_{n=1,j=0}^{N,N-1}$ , (4.5) - $\displaystyle\psi\equiv\psi(\varphi_{0}(t_{0}),\alpha(t_{0}))$222 $\displaystyle x_{j},j=\overline{0,N-1},$ .. $\displaystyle r_{0}(t)$ $\displaystyle r_{0}(t_{0})=1$.

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$\displaystyle f_{n},n=\overline{1,N},$

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,

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$\displaystyle K(t,s)=-\sum\limits_{n=1}^{N}nf_{n}\sin{n(t-s)}\sin nx_{0},\,f(x)=\sum\limits_{n=1}^{N}f_{n}\sin nx,f_{n}$ – (4.7). $\displaystyle C([0,T])$ $\displaystyle l(t)$ $\displaystyle S(\varphi_{0},\alpha,t_{0},\mu(t))$.

**   6****.**

$\displaystyle\varphi_{0},\chi,\alpha_{j},j=\overline{1,N-1}$, $\displaystyle t_{0},x_{k},k=\overline{0,N-1}$, , 4 , :

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*  *.

(4.10) $\displaystyle f(x)$ ; $\displaystyle r_{0}(t)$ – (4.9) $\displaystyle\mu(t)=\varphi^{\prime\prime}_{0}(t)$, $\displaystyle r_{1}(t,\tau)=(f(x_{0}))^{-1}\frac{\partial^{2}}{\partial\tau^{2}}\chi(t,\tau)$.

6.

, $\displaystyle(f,r)$ . 2 $\displaystyle u_{\omega}(x,t)$ (1.8)-(1.13). 1 3, (4.3), (4.4)

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(4.11) $\displaystyle\omega^{0}$, (2.1) $\displaystyle u_{0}$,

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$\displaystyle t=t_{0},f_{n}\Lambda_{n}(t_{0})=\psi_{n}$, (4.5). , $\displaystyle\psi$ . (4.12) $\displaystyle t$ $\displaystyle t=t_{0},f_{n}\Lambda_{n}(t_{0})=\psi_{n}$, (4.6), $\displaystyle f$. $\displaystyle r$ $\displaystyle f(x)$ 2. , $\displaystyle r_{0}(t)=S(\varphi_{0},\alpha,t_{0},\varphi^{\prime\prime}_{0}(t))$ (4.9); $\displaystyle r_{1}(t,\tau)=(f(x_{0}))^{-1}\frac{\partial^{2}}{\partial\tau^{2}}\chi(t,\tau)$. , 4 , (4.4) (4.10) .

, (4.10) . $\displaystyle f(x)$ $\displaystyle r(t,\tau)$ , . 1 $\displaystyle u_{\omega}$ (1.1)–(1.3) (1.8)-(1.13),

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[TABLE]

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, $\displaystyle u_{\omega}$ (4.3), (4.4). , 1, :

[TABLE]

. (4.1), (4.2) $\displaystyle\varphi_{1},\varphi_{2}$ 1. , , . (4.10). 6 .

$\displaystyle 4.2^{\circ}$

(1.1)–(1.3) $\displaystyle N=2$. :

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6 (4.10). (4.5)

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$\displaystyle\psi_{1}=\sin{1}-1,\psi_{2}=-\frac{1}{8}\sin{2}+\frac{1}{4}$. (4.6)

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$\displaystyle f_{1}=-1,f_{2}=1$. (4.8): $\displaystyle-\sin\frac{\pi}{2}+\sin\pi=-1\neq 0$.

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$\displaystyle r_{0}(t)=t$. (4.10):

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.

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Список литературы

• [1]

Babich P. V., Levenshtam V. B. Direct and inverse asymptotic problems high-frequency terms // Asymptotic Analysis. 2016. . 97. C. 329–336.

• [2]

.  ., .  ., .  . // . 2017. . 57, №12.

• [3]

.  ., .  ., .  . . : , 1982. 88 .

• [4]

.  . . .: , 1984.

• [5]

.  . . .: , 1994.

• [6]

.  . . : . . - , 2008.

• [7]

.  . // . 2013. . 53, №5. . 744–752.

• [8]

.  . // . 2015. . 55, №5. . 830–835.

• [9]

.  . // . . 2014. . 50, №6. . 795–806.

• [10]

.  ., .  . // . 2018. . 104, №4.

• [11]

.  . // . 1960. .15, №2. . 97–154.

, ( ).

, ( ), - ( ).