Recovery of a fast oscillating source in the heat equation by asymptotic of the solution

TL;DR

This paper demonstrates that a high-frequency source in a 1D heat equation can be fully recovered using partial asymptotic data of the solution, with the authors providing asymptotic analysis and solution reconstruction methods.

Contribution

The paper introduces a novel approach to recover high-frequency sources in the heat equation using incomplete asymptotic information of the solution.

Findings

Source can be recovered from two-term asymptotic data

Asymptotic solutions are constructed and justified for the original problem

Method applies to multiple recovery problems in heat equations

Abstract

Four problems about recovery of a high-frequency source in the one-dimension heat equation with homogeneous initial-boundary conditions by some information about partial asymptotic of its solution have solved. It is shown, that the source can be completely recovered from a specific data about incomplete (two-terms) asymptotic of the solution. Before formulation of each problem about recovery of a source, construction and justification of the asymptotic of the solution of original initial-boundary problem is given.

Equations

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

$${u|}_{\Gamma }=0.$$

$$⟨r_1(t,\cdot )⟩={⟨r_1(t,\tau )⟩}_{\tau }\equiv \frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}{r}_1(t,\tau )d\tau =0.$$

$$f(0,t)=f(\pi ,t)=f^{\prime \prime }(0,t)=f^{\prime \prime }(\pi ,t)=0.$$

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

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

$$u_0(x,t)=\sum _{n=1}^{\infty }\sin nx\underset{0}{\overset{t}{\int }}{e}^{-n^2(t-s)}{f}_n(s)r_0(s)ds,f_n(t)=\frac{2}{\pi }\underset{0}{\overset{\pi }{\int }}f(s,t)\sin nsds,$$

$$v_{1}(x,t,\tau)=f(x,t)\Biggl{[}\int\limits_{0}^{\tau}r_{1}(t,s)ds-\left\langle\int\limits_{0}^{\tau}r_{1}(t,s)ds\right\rangle_{\tau}\Biggr{]},$$

$$u_1(x,t)={⟨\underset{0}{\overset{\tau }{\int }}{r}_1(0,s)ds⟩}_{\tau }\cdot \sum _{n=1}^{\infty }{f}_n(0)\sin nx{e}^{-n^{2}t}.$$

$$\bigl{\|}W_{\omega}(x,t)\bigr{\|}_{C(\overline{\Pi})}=o(\omega^{-1}),\omega\to\infty.$$

$${\phi }_0\in C^1([0,T]),{\phi }_0(0)=0,$$

$${\phi }_1(t)=\frac{1}{f(x_0,0)}{⟨\underset{0}{\overset{\tau }{\int }}\frac{\partial }{\partial \tau }{\phi }_2(0,s)ds⟩}_{\tau }\cdot \sum _{n=1}^{\infty }{f}_n(0)\sin nx{e}^{-n^{2}t}.$$

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

$$\left\{\begin{array}[]{l}\frac{\partial U_{\omega}}{\partial t}+\frac{\partial W_{\omega}}{\partial t}=\frac{\partial^{2}U_{\omega}}{\partial x^{2}}+\frac{\partial^{2}W_{\omega}}{\partial x^{2}}+f(x)r(t,\omega t),\omega\gg 1,\\ \left.\bigl{(}U_{\omega}+W_{\omega}\bigr{)}\right|_{\Gamma}=0.\end{array}\right.$$

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

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

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

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

$$\left\{\begin{array}[]{l}\displaystyle\frac{\partial W_{\omega}}{\partial t}=\frac{\partial^{2}W_{\omega}}{\partial x^{2}}+\omega^{-1}\left[\frac{\partial^{2}v_{1}}{\partial x^{2}}-\frac{\partial v_{1}}{\partial t}\right],\\ \displaystyle{\left.W_{\omega}\right|_{\Gamma}=0.}\end{array}\right.$$

$$\begin{gathered} W_{\omega }(x,t)= \\ \frac{1}{\omega }\sum _{n=1}^{\infty }\sin nx\left[f_n^{\prime \prime }\underset{0}{\overset{t}{\int }}{e}^{-n^2(t-s)}p(s,\omega s)ds-f_n\underset{0}{\overset{t}{\int }}{e}^{-n^2(t-s)}{\frac{\partial }{\partial t}p(t,\tau )|}_{\begin{array}{c}t=s\\ \tau =\omega s\end{array}}ds\right]\equiv \\ {W}_{\omega ,1}(x,t)+W_{\omega ,2}(x,t), \end{gathered}$$

$$f_n^{\prime \prime }=\frac{1}{\pi }\underset{0}{\overset{\pi }{\int }}{f}^{\prime \prime }(s)\sin nsds,p(t,\tau )=\underset{0}{\overset{\tau }{\int }}{r}_2(t,s)ds-{⟨\underset{0}{\overset{\tau }{\int }}{r}_2(t,s)ds⟩}_{\tau }.$$

$$\Vert W_{\omega ,1}{\Vert }_{C(\overline{\Pi })}=o({\omega }^{-1}),\Vert W_{\omega ,2}{\Vert }_{C(\overline{\Pi })}=o({\omega }^{-1}),\omega \to \infty ,$$

$$f^{\prime \prime }(x)=\sum _{n=1}^{\infty }{f}_n^{\prime \prime }\sin nx,x\in \mathbb{R}.$$

$$f_n^{\prime \prime }=\frac{a_n}{n},\sum _{n=1}^{\infty }{a}_n^2<\infty .$$

$${\Vert \underset{0}{\overset{t}{\int }}{e}^{-n^2(t-s)}p(s,\omega s)ds\Vert }_{C([0,1])}=o(1),\omega \to \infty .$$

$$\left|\underset{t_0}{\overset{t}{\int }}{e}^{-n^2(t-s)}p(s,\omega s)ds\right|<\frac{\epsilon }{3},t_0=\max (0,t-\delta ).$$

$$\left|\underset{0}{\overset{t-\delta }{\int }}{e}^{-n^2(t-s)}p(s,\omega s)ds\right|<\frac{\epsilon }{3}.$$

$$\begin{gathered} \underset{0}{\overset{t-\delta }{\int }}{e}^{-n^2(t-s)}p(s,\omega s)ds= \\ \sum _{j=0}^{m-1}\left[\underset{t_j}{\overset{t_{j+1}}{\int }}{e}^{-n^2(t-s)}p(s,\omega s)ds-\underset{t_j}{\overset{t_{j+1}}{\int }}{e}^{-n^2(t-t_j)}p(t_j,\omega s)ds\right]+ \\ \sum _{j=0}^{m-1}\underset{t_j}{\overset{t_{j+1}}{\int }}{e}^{-n^2(t-t_j)}p(t_j,\omega s)ds=S_1+S_2. \end{gathered}$$

$$|S_1|<\frac{\epsilon }{6}.$$

$$\sum _{j=0}^{m-1}\underset{t_j}{\overset{t_{j+1}}{\int }}{e}^{-n^2(t-t_j)}p(t_j,\omega s)ds=\frac{1}{\omega }{e}^{-n^2(t-t_j)}\left[\underset{0}{\overset{\omega t_{j+1}}{\int }}p(t_j,\tau )d\tau -\underset{0}{\overset{\omega t_j}{\int }}p(t_j,\tau )d\tau \right]$$

Full text

517.955.8

. . , . . , . .

$\displaystyle f(x,t)r(t,\omega t)$ ($\displaystyle r$ – $\displaystyle 2\pi$- , $\displaystyle\omega\gg 1$) - . ( , $\displaystyle f,r$ , , ) . :

1. $\displaystyle r(t,\tau)$;
2. $\displaystyle f(x,t)\equiv f(x)$;
3. $\displaystyle f(x)$ $\displaystyle r_{1}(t,\tau)=r(t,\tau)-r_{0}(t)$, $\displaystyle r_{0}(t)=(2\pi)^{-1}\int\limits_{0}^{2\pi}r(t,\tau)d\tau$ , , 4) $\displaystyle f(x)$ $\displaystyle r(t,\tau)$. 4) , $\displaystyle f(x)$ $\displaystyle N$ . 1)-3) : 1) ( $\displaystyle\omega\to\infty$) , $\displaystyle x=x_{0}$, 2) $\displaystyle t=t_{0}$, 3) 1), 2) $\displaystyle x=x_{0},t=t_{0}$. 4) 3) , , $\displaystyle u_{0}(x,t)$ $\displaystyle N-1$ $\displaystyle x=x_{1},x_{2},...,x_{N-1}$ $\displaystyle t=t_{0}$. , $\displaystyle\alpha_{j}(t),t\in(t_{0}-\delta,t_{0}+\delta)\equiv I_{\delta},\delta>0,j=\overline{1,N}$, $\displaystyle\alpha_{j}(t)=u_{0}(x_{j},t),t\in I_{\delta}$.

.  .  [1, 2], ( ). , [1] 1) $\displaystyle f(x)r(t)$, $\displaystyle r$ ( [1] : + ); - , $\displaystyle x=x_{0}$. , . . , ( $\displaystyle x=x_{0}$) ( ). . . 1)-4). , 1) [3], $\displaystyle f(x,t)$ $\displaystyle t$.

, ( ) ( , ). , : , ; ; [4]–[7] .

, , , . . , . .

, , , $\displaystyle\overline{\Pi}$. ( ) – [8]. - [9]-[10] . .

1

$\displaystyle 1.1^{\circ}$

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

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$\displaystyle f(x,t)$ $\displaystyle r(t,\tau)$, $\displaystyle(x,t)\in[0,\pi]\times[0,T]\equiv S_{0}$ $\displaystyle(t,\tau)\in[0,T]\times[0,\infty)\equiv S_{1}$, . $\displaystyle r(t,\tau)$ – 2$\displaystyle\pi$- $\displaystyle\tau$; $\displaystyle r(t,\tau)=r_{0}(t)+r_{1}(t,\tau)$, $\displaystyle r_{1}(t,\tau)$ :

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$\displaystyle f\in C^{3,0}(S_{0}),r_{0}\in C^{0}([0,T]),r_{1}\in C^{1+\alpha,0}(S_{1}),\alpha\in(0,1),$

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$\displaystyle C^{l,m}(S)$, $\displaystyle l,m$ – , . $\displaystyle r(t,\tau)$, , , , ( ). $\displaystyle C(\overline{\Pi})$ ( $\displaystyle\omega^{-1}$) (1.1)-(1.2).

(1.1)-(1.2) :

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(1.6)-(1.8) .

**   1****.**

$\displaystyle u_{\omega}(x,t)$* (1.1)-(1.2) (1.4)-(1.8),*

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*  *.

($\displaystyle N$- $\displaystyle N$) (1.1)-(1.2) $\displaystyle C(\overline{\Pi})$ ( ) $\displaystyle r(t,\tau)$ $\displaystyle f(x)$, (1.3) ( ) . , .

$\displaystyle 1.2^{\circ}$

• (1.1)-(1.2) , .$\displaystyle 1^{\circ}$, $\displaystyle f(x,t)$, $\displaystyle r(t,\tau)$ ( ) . $\displaystyle x_{0}\in(0,\pi)$ – , $\displaystyle f(x_{0},t)\neq 0$. $\displaystyle\varphi_{0}(t),\varphi_{1}(t),\varphi_{2}(t,\tau)$, :

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$\displaystyle\varphi_{2}\in C^{1+\alpha,1}(S_{2})$ $\displaystyle 2\pi$- $\displaystyle\tau$ : $\displaystyle\left\langle\varphi_{2}(t,\cdot)\right\rangle=0,$

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– 1 – $\displaystyle r(t,\tau)$ ( ), $\displaystyle u_{\omega}(x,t)$ (1.1)-(1.2)

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**   2****.**

$\displaystyle\varphi_{0},\varphi_{1}$* $\displaystyle\varphi_{2}$ $\displaystyle x_{0}$, , $\displaystyle r$ ( ), $\displaystyle u_{\omega}(x,t)$ (1.1)-(1.2) (1.11). $\displaystyle r$ .*

$\displaystyle 1.3^{\circ}$

1.9**.**

$\displaystyle u_{\omega}$ (1.4) (1.1)-(1.2):

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(1.5),

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$\displaystyle\omega^{-k},\;(k=0,1)$, $\displaystyle\langle...\rangle_{\tau}$ $\displaystyle\tau=\omega t$. :

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, (1.16) : $\displaystyle{\left.v_{1}\right|_{x=0,\pi}=0}$. , , (1.14)-(1.16) $\displaystyle u_{0},v_{1}$ $\displaystyle u_{1}$ (1.6), (1.7) (1.8) . .

(1.13)-(1.16)

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(1.17) (1.7)

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1 :

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, . , $\displaystyle f\in C^{3}([0,\pi])$ $\displaystyle f^{(2k)}(0)=f^{(2k)}(\pi)=0,k=0,1,$ $\displaystyle f^{\prime\prime}(x)$ , $\displaystyle 2\pi$- $\displaystyle C^{3}$ $\displaystyle x\in\mathbb{R}$. ,

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$\displaystyle f^{\prime\prime}\in C^{1}(\mathbb{R})$,

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• , $\displaystyle n\in\mathbb{N}$

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. $\displaystyle\varepsilon$ – . $\displaystyle\delta>0$ , $\displaystyle t\in[0,1],n\in\mathbb{N}$ $\displaystyle\omega>0$

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$\displaystyle t_{0}=t-\delta>0$ $\displaystyle n_{0}$ , $\displaystyle n\geq n_{0},t\geq\delta$ $\displaystyle\omega>0$

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$\displaystyle[0,t-\delta],t>\delta,$ $\displaystyle m$ $\displaystyle[t_{j},t_{j+1}),j=0,1,\ldots,m-1$.

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$\displaystyle m$ , $\displaystyle n\leq n_{0}$ $\displaystyle\omega>0$

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,

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, $\displaystyle p(t,\tau)$, , $\displaystyle\omega_{0}>0$, $\displaystyle\omega>\omega_{0}$

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(1.22) – (1.25) (1.21). 1 .

2.

1.9 , $\displaystyle r(t,\tau)$ ( ) (1.1)-(1.2) , :

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$\displaystyle r(t,\tau)$ 1 $\displaystyle u_{\omega}$ – (1.1)-(1.2). (1.9),(1.11) $\displaystyle t\in[0,1]$

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(1.26) $\displaystyle\omega$, $\displaystyle\tau$ ,

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. $\displaystyle 1^{\circ}$ $\displaystyle\frac{\partial v_{1}(x,t)}{\partial\tau}=f(x,t)r_{1}(t,\tau)$. (1.28),

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$\displaystyle u_{0}(x,t)$, . $\displaystyle 1^{\circ}$, (1.6), $\displaystyle t$ ,

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, . 1 $\displaystyle f$, $\displaystyle K(t,s)$ . (1.27), (1.30) II

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$\displaystyle r_{0}(t)$. (1.29) $\displaystyle r_{1}$ :

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, $\displaystyle\varphi_{1}$, $\displaystyle C^{1+\alpha,0}(S)$.

$\displaystyle r(t,\tau)=r_{0}(t)+r_{1}(t,\tau)$ . $\displaystyle 1^{\circ}$, 1, , , (1.1)-(1.2), (1.4),(1.5). , $\displaystyle u_{\omega}(x,t)$ (1.11). :

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2 , $\displaystyle r_{0}(t)$ (1.32). (1.30),(1.32) $\displaystyle\frac{\partial}{\partial t}u_{0}(x_{0},t)=\varphi^{\prime}_{0}(t)$. , $\displaystyle u_{0}(x_{0},0)=\varphi_{0}(0)=0$, $\displaystyle u_{0}(x_{0},t)=\varphi_{0}(t)$. $\displaystyle r_{1}$ (1.29). (1.15) $\displaystyle x=x_{0}$ , $\displaystyle\varphi_{2}(t,\tau)$ $\displaystyle v_{1}(x,t,\tau)$, $\displaystyle 2\pi$- $\displaystyle\tau$, $\displaystyle v_{1}(x_{0},t,\tau)=\varphi_{2}(t,\tau)$. , (1.8),

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.

2

$\displaystyle 2.1^{\circ}$

$\displaystyle\Pi$ $\displaystyle\Gamma$ - , $\displaystyle\S 1$. (1.1)-(1.2) $\displaystyle f(x,t)\equiv f(x)$. , $\displaystyle f\in C^{1}([0,\pi]),f(0)=f(\pi)=0$, $\displaystyle r(t,\tau)=r_{0}(t)+r_{1}(t,\tau)$, $\displaystyle r_{0}\in C([0,T]),r_{1}\in C^{\alpha,0}(S),\alpha\in(0,1),$ $\displaystyle r_{1}(t,\tau)$ – 2$\displaystyle\pi$- $\displaystyle\tau$

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: $\displaystyle f_{n}=n^{-1}b_{n}$, $\displaystyle\sum_{n=1}^{\infty}b_{n}^{2}<\infty,$ - , $\displaystyle u_{0}(x,t)$ - (1.1)-(1.2) $\displaystyle r=r_{0}.$

**   3****.**

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

$\displaystyle u_{\omega}$* - (2.1), (2.2).*

$\displaystyle 2.2^{\circ}$

2 . $\displaystyle r_{0}\in C^{1},$ $\displaystyle f\in C^{2}([0,\pi])$ $\displaystyle f(0)=f(\pi)$, $\displaystyle t_{0}\in(0,1]$ $\displaystyle u_{0}(x,t_{0})\equiv\psi(x)$ :

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$\displaystyle t>0$ :

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, (1.1)-(1.2) $\displaystyle f(x,t)\equiv f(x)$. , $\displaystyle r(t,\tau)$ $\displaystyle\S 1$, $\displaystyle f\in C^{1}([0,\pi])$ . $\displaystyle t_{0}\in(0,T]$, $\displaystyle r_{0}(t_{0})\neq 0$. $\displaystyle\Lambda_{n}\equiv\Lambda_{n}(t_{0}),n=1,2,\ldots$ ( . (2.1)).

. * $\displaystyle\Lambda_{n}$ . , $\displaystyle n_{0}$ $\displaystyle a_{0}>0$, $\displaystyle n\geq n_{0},\;\Lambda_{n}\geq a_{o}n^{-2}$.*

$\displaystyle t_{0}$ $\displaystyle\psi(x)$, (2.2) ( , ). :

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2 $\displaystyle f\in C^{1}([0,\pi]),f(0)=f(\pi)=0$, $\displaystyle u_{\omega}(x,t)$ (1.1)-(1.2) :

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$\displaystyle\Lambda_{n}$ $\displaystyle n_{1},n_{2},...,n_{s}$, $\displaystyle M_{0}$ – .

**   4****.**

1. 2 , $\displaystyle M_{0}$ . $\displaystyle f=\sum_{n=1}^{\infty}f_{n}\sin{nx}$, $\displaystyle f_{n}=\frac{\psi_{n}}{\Lambda_{n}}$.

2. $\displaystyle M_{0}$ , 2 , $\displaystyle\psi_{n_{j}}=0,n_{j}\in M_{0}.$ $\displaystyle f_{n}=\frac{\psi_{n}}{\Lambda_{n}},n\not\in M_{0},$ $\displaystyle f_{n}$ $\displaystyle n\in M_{0}$ - .

$\displaystyle t_{0}$, $\displaystyle f$. $\displaystyle r_{0}(0)\neq 0$, $\displaystyle t_{0}$ (0,1], $\displaystyle r_{0}(t),t\in[0,1]$. $\displaystyle r_{0}(0)=0$, $\displaystyle t_{0}\in(0,1)$ , $\displaystyle r_{0}(t_{0})\neq 0$ $\displaystyle r_{0}(t)$ $\displaystyle t\in[0,t_{0}]$ .

$\displaystyle 2.3^{\circ}.$** **

3

$\displaystyle W_{\omega}(x,t)=u_{\omega}(x,t)-u_{0}(x,t)=\sum_{n=1}^{\infty}f_{n}\sin{nx}\int\limits_{0}^{t}e^{-n^{2}(t-s)}r_{1}(s,\omega s)ds=\sum_{n=1}^{\infty}f_{n}\sin{nx}\int\limits_{0}^{t}e^{-n^{2}(t-s)}r_{1}(s,\omega s)ds+\sum_{n=n_{0}+1}^{n_{0}}f_{n}\sin{nx}\int\limits_{0}^{t}e^{-n^{2}(t-s)}r_{1}(s,\omega s)ds\equiv S_{\omega,1}+S_{\omega,2},n_{o}\in\mathbb{N}$

$\displaystyle\varepsilon$ - . , $\displaystyle\int\limits_{0}^{t}e^{-n^{2}(t-s)}r_{1}(s,\omega s)ds=O(n^{-2}),n\to\infty$, $\displaystyle t\in[0,1],\omega>0$, $\displaystyle n_{0}$ , $\displaystyle\omega>0$

\displaystyle\bigl{\|}S_{\omega,2}\bigr{\|}_{C(\overline{\Pi})}<\frac{\varepsilon}{2}.

, $\displaystyle\left\langle r_{1}(t,\tau)\right\rangle_{\tau}=0$, $\displaystyle\omega_{0}$ , $\displaystyle n_{0}$ $\displaystyle\omega>\omega_{0}$

\displaystyle\bigl{\|}\sum_{n=1}^{n_{0}}f_{n}\sin{nx}\int\limits_{0}^{t}e^{-n^{2}(t-s)}r_{1}(s,\omega s)ds\bigr{\|}_{C(\overline{\Pi})}<\frac{\varepsilon}{2}.

, $\displaystyle\omega>\omega_{0}$

\displaystyle\bigl{\|}W_{\omega}\bigr{\|}_{C(\overline{\Pi})}<\varepsilon.

.

4.

, $\displaystyle f\in C^{1}([0,\pi])$ . 3 (2.3), (2.4)

$\displaystyle\sum_{n=1}^{\infty}f_{n}\Lambda_{n}\sin{nx}=\sum_{n=1}^{\infty}\psi_{n}\sin{nx}$

(2.2) $\displaystyle b_{n}>0,n=1,2,\ldots$, $\displaystyle n$

$\displaystyle\left|\psi_{n}\right|\leq b_{n}n^{-4},\sum_{n=1}^{\infty}|b_{n}|^{2}<\infty,$

$\displaystyle n\geq n_{0}$

$\displaystyle\left|f_{n}\right|\leq c_{n}n^{-2},\;c_{n}=b_{n}a_{0},$

$\displaystyle f\in C^{1}([0,\pi])$. $\displaystyle\Lambda_{n}\neq 0$,

$\displaystyle f=\sum_{n=1}^{\infty}f_{n}\sin{nx}\in C^{1}([0,\pi]),\;f_{n}=\frac{\psi_{n}}{\Lambda_{n}}.$

4 .

3

$\displaystyle 3.1^{\circ}$ 3

(1.1)-(1.2) $\displaystyle f(x,t)\equiv f(x),f\in C^{3}([0,\pi]),f^{(2k)}(0)=f^{(2k)}(\pi)=0,k=0,1$, $\displaystyle r=r_{0}+r_{1}$, ( ). , $\displaystyle r_{0}$ , $\displaystyle f$ $\displaystyle r_{1}$ . $\displaystyle t_{0}\in(0,T]$, $\displaystyle\Lambda_{n}=\Lambda_{n}(t_{0}),n=1,2,\ldots$ – , (2.1). , $\displaystyle\Lambda_{n}\neq 0,n=1,2,\ldots$ . $\displaystyle 2.2^{\circ}$. : $\displaystyle\psi(x),\varphi_{0}(t),\varphi_{1}(t)$ $\displaystyle\varphi_{2}(t,\tau)$ . $\displaystyle\psi\in C^{6}([0,\pi]),\psi^{(2n)}(0)=\psi^{(2n)}(\pi)=0,n=0,1,2.$

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$\displaystyle\widetilde{f}\in C^{3}([0,\pi]),\widetilde{f}^{(2k)}(0)=\widetilde{f}^{(2k)}(\pi)=0,k=0,1.$ $\displaystyle x_{0}\in(0,\pi)$ – , $\displaystyle\widetilde{f}(x_{0})\neq 0$. $\displaystyle\varphi_{0}(t)$

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$\displaystyle K(t,s)$ – , $\displaystyle\S 1$, $\displaystyle\varphi_{1},\varphi_{2}$ .$\displaystyle 1.2^{\circ}$ ( $\displaystyle\varphi_{1}$ $\displaystyle\varphi_{2}$) $\displaystyle f(x)$ $\displaystyle\widetilde{f}(x)$. (3.2) $\displaystyle r_{0},\psi,\varphi_{0},\varphi_{1}$ $\displaystyle\varphi_{2}$.

1, 2 .

**   5****.**

$\displaystyle r_{0},\psi,\varphi_{0},\varphi_{1},$* $\displaystyle\varphi_{2}$ $\displaystyle x_{0},t_{0}$ . ( .$\displaystyle 3.1^{\circ}$ ) $\displaystyle f$ $\displaystyle r_{1}$, $\displaystyle u_{\omega}(x,t)$ (1.1)-(1.2) $\displaystyle f(x,t)\equiv f(x)$ (1.11) (2.4). $\displaystyle f(x)=\widetilde{f}(x)$ (3.1), $\displaystyle r_{1}(t,\tau)=(f(x_{0}))^{-1}\frac{\partial}{\partial\tau}\varphi_{2}(t,\tau)$. 3 .*

4

$\displaystyle 4.1^{\circ}$ 4

$\displaystyle\Pi$ $\displaystyle\Gamma$ – , $\displaystyle\S 1$. (1.1)-(1.2), $\displaystyle f(x,t)\equiv f(x)$ $\displaystyle r(t,\tau)$ . , $\displaystyle r(t,\tau)$ – ( ), $\displaystyle f(x,t)\equiv f(x)=\sum\limits_{n=1}^{N}f_{n}\sin nx$ – $\displaystyle N$ $\displaystyle f_{n}$. , , $\displaystyle t_{0}\in(0,T),x_{j}\in(0,\pi),j=\overline{0,N-1},x_{i}\neq x_{k}$ $\displaystyle i\neq k$, $\displaystyle\varphi_{0}(t),\varphi_{2}(t,\tau),\varphi_{0}\in C^{1}([0,T]),\varphi_{0}(0)=0,\varphi_{2}\in C^{1+\alpha,0}(S),\alpha\in(0,1),\varphi_{2}(t,\tau)$ – $\displaystyle 2\pi$- $\displaystyle\tau$ $\displaystyle\alpha_{j}\in C^{1}([t_{0}-\delta,t_{0}+\delta]),\delta>0,j=\overline{1,N-1},(t_{0}-\delta,t_{0}+\delta)\subset(0,T)$. $\displaystyle\alpha(t)$ - $\displaystyle\varphi_{0}(t),\alpha_{j}(t),t\in(t_{0}-\delta,t_{0}+\delta),j=\overline{0,N-1}$. - .

, , :

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$\displaystyle f(x)$.

4 $\displaystyle r(t,\tau)$ $\displaystyle f(x)$, , $\displaystyle u_{\omega}(x,t)$ (1.1), (1.2)

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.

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$\displaystyle\psi_{n},n=\overline{1,N}$. $\displaystyle A=(\sin nx_{j})_{n=1,j=0}^{N,N-1}$ , (4.3) $\displaystyle\psi\equiv\psi(\alpha(t_{0}))$.

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

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,

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$\displaystyle r_{0}(t)$ $\displaystyle r_{0}(t_{0})=1$.

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$\displaystyle K(t,s)=-\sum\limits_{n=1}^{N}n^{2}F_{n}e^{-n^{2}(t-s)}\sin nx_{0},\,f(x)=\sum\limits_{n=1}^{N}f_{n}\sin nx$. $\displaystyle C([0,T])$ $\displaystyle l(t)$ $\displaystyle S(\alpha,t_{0},x_{0},\mu(t))$.

**   6****.**

$\displaystyle\varphi_{i},i=\overline{0,2},\alpha_{j},j=\overline{1,N-1}$, $\displaystyle t_{0},x_{k},k=\overline{0,N-1}$, , 4 , :

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$\displaystyle f(x)$* , $\displaystyle r_{0}(t)$ – (4.7) $\displaystyle\mu(t)=\varphi^{\prime}_{0}(t)$, $\displaystyle r_{1}(t,\tau)=(f(x_{0}))^{-1}\frac{\partial}{\partial\tau}\varphi_{2}(t,\tau)$.*

6 ; , . , (4.8) 4, , ( ) . 6.

$\displaystyle 4.2^{\circ}$

(1.1)-(1.2) 6 $\displaystyle N=2$. :

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(4.8). (4.3)

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$\displaystyle\psi_{1}=e^{-1},\psi_{2}=\frac{1}{16}(3+e^{-4})$. (4.4)

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$\displaystyle f_{1}=1,f_{2}=1$. (4.6): $\displaystyle\sin\frac{\pi}{2}-\sin\pi=1\neq 0$.

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$\displaystyle r_{0}(t)=t$. (4.8):

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. , :

[TABLE]

5

• . , ( ) . - .

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