TY - JOUR AU - Benyoucef, Abir AU - Alem, Leila AU - Chorfi, Lahcène PY - 2022/08/31 Y2 - 2024/03/29 TI - Uniqueness and stability of parameter identification in elliptic boundary value problem JF - Journal of Innovative Applied Mathematics and Computational Sciences JA - J. Innov. Appl. Math. Comput. Sci VL - 2 IS - 2 SE - Conference paper (ICMA'2021) DO - 10.58205/jiamcs.v2i2.31 UR - https://jiamcs.centre-univ-mila.dz/index.php/jiamcs/article/view/v2i2_31 SP - 30-37 AB - <p>This paper concerns the uniqueness and stability of an inverse problem<br />in PDE. Our problem consists of identifying two parameters <span id="MathJax-Element-1-Frame" class="MathJax" style="position: relative;" tabindex="0" role="presentation" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;"><span id="MathJax-Span-1" class="math" style="width: 2.249em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.789em; height: 0px; font-size: 124%;"><span style="position: absolute; clip: rect(1.443em, 1001.67em, 2.768em, -999.997em); top: -2.359em; left: 0em;"><span id="MathJax-Span-2" class="mrow"><span id="MathJax-Span-3" class="mi" style="font-family: MathJax_Math-italic;">b</span><span id="MathJax-Span-4" class="mo" style="font-family: MathJax_Main;">(</span><span id="MathJax-Span-5" class="mi" style="font-family: MathJax_Math-italic;">x</span><span id="MathJax-Span-6" class="mo" style="font-family: MathJax_Main;">)</span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation">b(x)</span></span> and <span id="MathJax-Element-2-Frame" class="MathJax" style="position: relative;" tabindex="0" role="presentation" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;"><span id="MathJax-Span-7" class="math" style="width: 2.249em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.789em; height: 0px; font-size: 124%;"><span style="position: absolute; clip: rect(1.443em, 1001.67em, 2.768em, -999.997em); top: -2.359em; left: 0em;"><span id="MathJax-Span-8" class="mrow"><span id="MathJax-Span-9" class="mi" style="font-family: MathJax_Math-italic;">c</span><span id="MathJax-Span-10" class="mo" style="font-family: MathJax_Main;">(</span><span id="MathJax-Span-11" class="mi" style="font-family: MathJax_Math-italic;">x</span><span id="MathJax-Span-12" class="mo" style="font-family: MathJax_Main;">)</span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation">c(x)</span></span> in the following boundary-value problem</p><div class="MathJax_Display" style="text-align: center;"><span id="MathJax-Element-3-Frame" class="MathJax" style="text-align: center; position: relative;" tabindex="0" role="presentation" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mtable columnalign=&quot;left left&quot; rowspacing=&quot;.2em&quot; columnspacing=&quot;1em&quot; displaystyle=&quot;false&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;:=&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;&amp;#x2033;&lt;/mo&gt;&lt;/msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;&amp;#x2032;&lt;/mo&gt;&lt;/msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace width=&quot;1em&quot; /&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo fence=&quot;true&quot; stretchy=&quot;true&quot; symmetric=&quot;true&quot;&gt;&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;"><span id="MathJax-Span-13" class="math" style="width: 23.16em; display: inline-block;"><span style="display: inline-block; position: relative; width: 18.666em; height: 0px; font-size: 124%;"><span style="position: absolute; clip: rect(1.731em, 1018.67em, 4.438em, -999.997em); top: -3.338em; left: 0em;"><span id="MathJax-Span-14" class="mrow"><span id="MathJax-Span-15" class="mrow"><span id="MathJax-Span-16" class="mo" style="vertical-align: 0em;"><span style="font-family: MathJax_Size3;">{</span></span><span id="MathJax-Span-17" class="mtable" style="padding-right: 0.176em; padding-left: 0.176em;"><span style="display: inline-block; position: relative; width: 17.572em; height: 0px;"><span style="position: absolute; clip: rect(2.422em, 1017.57em, 5.072em, -999.997em); top: -3.972em; left: 0em;"><span style="display: inline-block; position: relative; width: 17.572em; height: 0px;"><span style="position: absolute; clip: rect(3.056em, 1017.57em, 4.381em, -999.997em); top: -4.548em; left: 0em;"><span id="MathJax-Span-18" class="mtd"><span id="MathJax-Span-19" class="mrow"><span id="MathJax-Span-20" class="mi" style="font-family: MathJax_Math-italic;">L</span><span id="MathJax-Span-21" class="mi" style="font-family: MathJax_Math-italic;">u</span><span id="MathJax-Span-22" class="mo" style="font-family: MathJax_Main; padding-left: 0.291em;">:<span style="font-family: MathJax_Main;">=</span></span><span id="MathJax-Span-23" class="mo" style="font-family: MathJax_Main; padding-left: 0.291em;">−</span><span id="MathJax-Span-24" class="mi" style="font-family: MathJax_Math-italic;">b</span><span id="MathJax-Span-25" class="mo" style="font-family: MathJax_Main;">(</span><span id="MathJax-Span-26" class="mi" style="font-family: MathJax_Math-italic;">x</span><span id="MathJax-Span-27" class="mo" style="font-family: MathJax_Main;">)</span><span id="MathJax-Span-28" class="msup"><span style="display: inline-block; position: relative; width: 1.04em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.58em, 4.15em, -999.997em); top: -3.972em; left: 0em;"><span id="MathJax-Span-29" class="mi" style="font-family: MathJax_Math-italic;">u</span></span><span style="position: absolute; top: -4.317em; left: 0.579em;"><span id="MathJax-Span-30" class="mo" style="font-size: 70.7%; font-family: MathJax_Main;">′′</span></span></span></span><span id="MathJax-Span-31" class="mo" style="font-family: MathJax_Main;">(</span><span id="MathJax-Span-32" class="mi" style="font-family: MathJax_Math-italic;">x</span><span id="MathJax-Span-33" class="mo" style="font-family: MathJax_Main;">)</span><span id="MathJax-Span-34" class="mo" style="font-family: MathJax_Main; padding-left: 0.233em;">+</span><span id="MathJax-Span-35" class="mi" style="font-family: MathJax_Math-italic; padding-left: 0.233em;">c</span><span id="MathJax-Span-36" class="mo" style="font-family: MathJax_Main;">(</span><span id="MathJax-Span-37" class="mi" style="font-family: MathJax_Math-italic;">x</span><span id="MathJax-Span-38" class="mo" style="font-family: MathJax_Main;">)</span><span id="MathJax-Span-39" class="msup"><span style="display: inline-block; position: relative; width: 0.867em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.58em, 4.15em, -999.997em); top: -3.972em; left: 0em;"><span id="MathJax-Span-40" class="mi" style="font-family: MathJax_Math-italic;">u</span></span><span style="position: absolute; top: -4.317em; left: 0.579em;"><span id="MathJax-Span-41" class="mo" style="font-size: 70.7%; font-family: MathJax_Main;">′</span></span></span></span><span id="MathJax-Span-42" class="mo" style="font-family: MathJax_Main;">(</span><span id="MathJax-Span-43" class="mi" style="font-family: MathJax_Math-italic;">x</span><span id="MathJax-Span-44" class="mo" style="font-family: MathJax_Main;">)</span><span id="MathJax-Span-45" class="mo" style="font-family: MathJax_Main; padding-left: 0.291em;">=</span><span id="MathJax-Span-46" class="mi" style="font-family: MathJax_Math-italic; padding-left: 0.291em;">f</span><span id="MathJax-Span-47" class="mo" style="font-family: MathJax_Main;">(</span><span id="MathJax-Span-48" class="mi" style="font-family: MathJax_Math-italic;">x</span><span id="MathJax-Span-49" class="mo" style="font-family: MathJax_Main;">)</span><span id="MathJax-Span-50" class="mo" style="font-family: MathJax_Main;">,</span><span id="MathJax-Span-51" class="mspace" style="height: 0em; vertical-align: 0em; width: 0.982em; display: inline-block; overflow: hidden;"></span></span></span></span><span style="position: absolute; clip: rect(3.056em, 1007.15em, 4.381em, -999.997em); top: -3.338em; left: 0em;"><span id="MathJax-Span-52" class="mtd"><span id="MathJax-Span-53" class="mrow"><span id="MathJax-Span-54" class="mi" style="font-family: MathJax_Math-italic;">u</span><span id="MathJax-Span-55" class="mo" style="font-family: MathJax_Main;">(</span><span id="MathJax-Span-56" class="mn" style="font-family: MathJax_Main;">0</span><span id="MathJax-Span-57" class="mo" style="font-family: MathJax_Main;">)</span><span id="MathJax-Span-58" class="mo" style="font-family: MathJax_Main; padding-left: 0.291em;">=</span><span id="MathJax-Span-59" class="mi" style="font-family: MathJax_Math-italic; padding-left: 0.291em;">u</span><span id="MathJax-Span-60" class="mo" style="font-family: MathJax_Main;">(</span><span id="MathJax-Span-61" class="mn" style="font-family: MathJax_Main;">1</span><span id="MathJax-Span-62" class="mo" style="font-family: MathJax_Main;">)</span><span id="MathJax-Span-63" class="mo" style="font-family: MathJax_Main; padding-left: 0.291em;">=</span><span id="MathJax-Span-64" class="mn" style="font-family: MathJax_Main; padding-left: 0.291em;">0</span><span id="MathJax-Span-65" class="mo" style="font-family: MathJax_Main;">,</span></span></span></span></span></span></span></span><span id="MathJax-Span-66" class="mo"></span></span></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation">{Lu:=−b(x)u″(x)+c(x)u′(x)=f(x),u(0)=u(1)=0,</span></span></div><p><br />from distributed observations <span id="MathJax-Element-4-Frame" class="MathJax" style="position: relative;" tabindex="0" role="presentation" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;"><span id="MathJax-Span-67" class="math" style="width: 1.213em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.982em; height: 0px; font-size: 124%;"><span style="position: absolute; clip: rect(1.731em, 1000.98em, 2.71em, -999.997em); top: -2.359em; left: 0em;"><span id="MathJax-Span-68" class="mrow"><span id="MathJax-Span-69" class="msubsup"><span style="display: inline-block; position: relative; width: 0.982em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.58em, 4.15em, -999.997em); top: -3.972em; left: 0em;"><span id="MathJax-Span-70" class="mi" style="font-family: MathJax_Math-italic;">u</span></span><span style="position: absolute; top: -3.799em; left: 0.579em;"><span id="MathJax-Span-71" class="mn" style="font-size: 70.7%; font-family: MathJax_Main;">1</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation">u1</span></span> (resp. <span id="MathJax-Element-5-Frame" class="MathJax" style="position: relative;" tabindex="0" role="presentation" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;"><span id="MathJax-Span-72" class="math" style="width: 1.213em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.982em; height: 0px; font-size: 124%;"><span style="position: absolute; clip: rect(1.731em, 1000.98em, 2.71em, -999.997em); top: -2.359em; left: 0em;"><span id="MathJax-Span-73" class="mrow"><span id="MathJax-Span-74" class="msubsup"><span style="display: inline-block; position: relative; width: 0.982em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.58em, 4.15em, -999.997em); top: -3.972em; left: 0em;"><span id="MathJax-Span-75" class="mi" style="font-family: MathJax_Math-italic;">u</span></span><span style="position: absolute; top: -3.799em; left: 0.579em;"><span id="MathJax-Span-76" class="mn" style="font-size: 70.7%; font-family: MathJax_Main;">2</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation">u2</span></span>) associated with the source <span id="MathJax-Element-6-Frame" class="MathJax" style="position: relative;" tabindex="0" role="presentation" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;"><span id="MathJax-Span-77" class="math" style="width: 1.155em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px; font-size: 124%;"><span style="position: absolute; clip: rect(1.501em, 1000.92em, 2.768em, -999.997em); top: -2.359em; left: 0em;"><span id="MathJax-Span-78" class="mrow"><span id="MathJax-Span-79" class="msubsup"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.113em, 1000.58em, 4.381em, -999.997em); top: -3.972em; left: 0em;"><span id="MathJax-Span-80" class="mi" style="font-family: MathJax_Math-italic;">f</span></span><span style="position: absolute; top: -3.799em; left: 0.521em;"><span id="MathJax-Span-81" class="mn" style="font-size: 70.7%; font-family: MathJax_Main;">1</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation">f1</span></span> (resp. <span id="MathJax-Element-7-Frame" class="MathJax" style="position: relative;" tabindex="0" role="presentation" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt;&lt;/math&gt;"><span id="MathJax-Span-82" class="math" style="width: 1.155em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px; font-size: 124%;"><span style="position: absolute; clip: rect(1.501em, 1000.92em, 2.768em, -999.997em); top: -2.359em; left: 0em;"><span id="MathJax-Span-83" class="mrow"><span id="MathJax-Span-84" class="msubsup"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.113em, 1000.58em, 4.381em, -999.997em); top: -3.972em; left: 0em;"><span id="MathJax-Span-85" class="mi" style="font-family: MathJax_Math-italic;">f</span></span><span style="position: absolute; top: -3.799em; left: 0.521em;"><span id="MathJax-Span-86" class="mn" style="font-size: 70.7%; font-family: MathJax_Main;">2</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation">f2</span></span>). For one observation, the solution is not unique. However, we prove, under some conditions, the uniqueness of the solution <span id="MathJax-Element-8-Frame" class="MathJax" style="position: relative;" tabindex="0" role="presentation" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;"><span id="MathJax-Span-87" class="math" style="width: 4.957em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.978em; height: 0px; font-size: 124%;"><span style="position: absolute; clip: rect(1.443em, 1003.86em, 2.768em, -999.997em); top: -2.359em; left: 0em;"><span id="MathJax-Span-88" class="mrow"><span id="MathJax-Span-89" class="mi" style="font-family: MathJax_Math-italic;">p</span><span id="MathJax-Span-90" class="mo" style="font-family: MathJax_Main; padding-left: 0.291em;">=</span><span id="MathJax-Span-91" class="mo" style="font-family: MathJax_Main; padding-left: 0.291em;">(</span><span id="MathJax-Span-92" class="mi" style="font-family: MathJax_Math-italic;">b</span><span id="MathJax-Span-93" class="mo" style="font-family: MathJax_Main;">,</span><span id="MathJax-Span-94" class="mi" style="font-family: MathJax_Math-italic; padding-left: 0.176em;">c</span><span id="MathJax-Span-95" class="mo" style="font-family: MathJax_Main;">)</span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation">p=(b,c)</span></span> in the case of two observations. Furthermore, we derive a H\"older-type stability result. The algorithm of reconstruction uses the least squares method. Finally, we present some numerical examples with exact and noisy data to illustrate our method.</p> ER -