<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>This study aims to investigate the problem of determining the unknown initial temperature in a variable coefficient heat equation. We obtain the existence and uniqueness of the solution of the optimal control problem considered under some conditions. Using the adjoint problem approach, we get the Frechet differential of the cost functional. We construct a minimizing sequence and give the convergence rate of this sequence. Also, we test the theoretical results in a numerical example by using the MAPLE<sup>®</sup> program.</p>
</abstract>

This study aims to investigate the problem of determining the unknown initial temperature in a variable coefficient heat equation. We obtain the existence and uniqueness of the solution of the optimal control problem considered under some conditions. Using the adjoint problem approach, we get the Frechet differential of the cost functional. We construct a minimizing sequence and give the convergence rate of this sequence. Also, we test the theoretical results in a numerical example by using the MAPLE® program.

Identification of the initial temperature from the given temperature data at the left end of a rod

Science Faculty, Department of Mathematics

Applied Mathematics and Nonlinear Sciences

This work is licensed under the Creative Commons Attribution 4.0 Public License.

This study aims to investigate the problem of determining the unknown initial temperature in a variable coefficient heat equation. We obtain the existence and...

α	$\begin{matrix} {∥u (0, t;; v) - e^{t}∥}_{L_{2} (0, 1)}^{2} \end{matrix}$ $\begin{array}{} \displaystyle {\left\\|{\mathbf u}\left({\mathbf 0},{\mathbf t};;{\mathbf v}\right){\mathbf -}{{\mathbf e}}^{{\mathbf t}}\right\\|}^{{\mathbf \, }{\mathbf 2}}_{{{\mathbf L}}_{{\mathbf 2}}\left({\mathbf 0},{\mathbf 1}\right)} \end{array}$	$\begin{matrix} {∥v∥}_{L_{2} (0, 1)}^{2} \end{matrix}$ $\begin{array}{} \displaystyle {\left\\|{\mathbf v}\right\\|}^{{\mathbf 2}}_{{{\mathbf L}}_{{\mathbf 2}}\left({\mathbf 0},{\mathbf 1}\right)} \end{array}$
0.1	0.02703	0.20904
0.2	0.03356	0.18194
0.3	0.04015	0.16205
0.4	0.04673	0.14153
0.5	0.05348	0.12598
1.0	0.08095	0.07398
2.0	0.11747	0.03282

Identification of the initial temperature from the given temperature data at the left end of a rod

Published Online: Dec 18, 2019

Page range: 469 - 474

Received: May 08, 2019

Accepted: Jul 02, 2019

DOI: https://doi.org/10.2478/AMNS.2019.2.00044

Keywords
Optimal Control, Frechet Differential, Adjoint Approach

© 2019 Yesim Sarac and S. Sule Sener, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Some α, u0,t;v100−etL20,12 and v100L20,12 $\begin{array}{} \displaystyle {\left\|u\left(0,t;v.{100}\right)-e^t\right\|}^{\ 2}.{L.2\left(0,1\right)} and {\left\|v.{100}\right\|}^2.{L.2\left(0,1\right)} \end{array}$ values

Identification of the initial temperature from the given temperature data at the left end of a rod

Published Online: Dec 18, 2019

Page range: 469 - 474

Received: May 08, 2019

Accepted: Jul 02, 2019

DOI: https://doi.org/10.2478/AMNS.2019.2.00044

KeywordsOptimal Control, Frechet Differential, Adjoint Approach

© 2019 Yesim Sarac and S. Sule Sener, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Some α, u0,t;v100−etL20,12 and v100L20,12 $\begin{array}{} \displaystyle {\left\|u\left(0,t;v.{100}\right)-e^t\right\|}^{\ 2}.{L.2\left(0,1\right)}~~ and ~~{\left\|v.{100}\right\|}^2.{L.2\left(0,1\right)} \end{array}$ values

Keywords
Optimal Control, Frechet Differential, Adjoint Approach

Some α, u0,t;v100−etL20,12 and v100L20,12 $\begin{array}{} \displaystyle {\left\|u\left(0,t;v.{100}\right)-e^t\right\|}^{\ 2}.{L.2\left(0,1\right)} and {\left\|v.{100}\right\|}^2.{L.2\left(0,1\right)} \end{array}$ values