Analytical Solution of the Hyperbolic Heat Conduction Equation for Moving Semi-Infinite Medium under the Effect of Time-Dependent Laser Heat Source

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<tspan style="font-size: 12.50px; " x="0" y="0">𝜂</tspan>
<tspan style="font-size: 12.50px; " x="9.7148314" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="21.75522" y="0">1</tspan>
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</svg> for the instantaneous heat source and various values of <svg height="13.425" id="M96" style="vertical-align:-2.29482pt" version="1.1" viewbox="0 0 8.8500004 13.425" width="8.8500004" xmlns="http://www.w3.org/2000/svg">
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<tspan style="font-size: 12.50px; " x="0" y="0">𝛽</tspan>
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</svg>; <svg height="14.8625" id="M97" style="vertical-align:-3.25793pt" version="1.1" viewbox="0 0 233.125 14.8625" width="233.125" xmlns="http://www.w3.org/2000/svg">
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<text transform="matrix(1,0,0,-1,-71.95,63.41)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝜙</tspan>
<tspan style="font-size: 12.50px; " x="8.064435" y="0">(</tspan>
<tspan style="font-size: 12.50px; " x="12.227934" y="0">𝜂</tspan>
<tspan style="font-size: 12.50px; " x="18.46693" y="0">)</tspan>
<tspan style="font-size: 12.50px; " x="26.106264" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="38.146652" y="0">𝛿</tspan>
<tspan style="font-size: 12.50px; " x="44.435661" y="0">(</tspan>
<tspan style="font-size: 12.50px; " x="48.599159" y="0">𝜂</tspan>
<tspan style="font-size: 12.50px; " x="54.850662" y="0">)</tspan>
<tspan style="font-size: 12.50px; " x="59.01416" y="0">,</tspan>
<tspan style="font-size: 12.50px; " x="72.554909" y="0">𝜓</tspan>
</text>
<text transform="matrix(1,0,0,-1,10.01,60.28)">
<tspan style="font-size: 8.75px; " x="0" y="0">0</tspan>
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<text transform="matrix(1,0,0,-1,18.35,63.41)">
<tspan style="font-size: 12.50px; " x="0" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="12.040389" y="0">1</tspan>
<tspan style="font-size: 12.50px; " x="18.291889" y="0">,</tspan>
<tspan style="font-size: 12.50px; " x="27.66914" y="0">a</tspan>
<tspan style="font-size: 12.50px; " x="33.22047" y="0">n</tspan>
<tspan style="font-size: 12.50px; " x="39.47197" y="0">d</tspan>
<tspan style="font-size: 12.50px; " x="53.012718" y="0">𝑈</tspan>
<tspan style="font-size: 12.50px; " x="65.303169" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="77.343559" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="83.595055" y="0">.</tspan>
<tspan style="font-size: 12.50px; " x="86.72081" y="0">1</tspan>
<tspan style="font-size: 12.50px; " x="92.972305" y="0">.</tspan>
</text>
</g>
</g>
</svg></table>

Journal of Applied Mathematics

fig7

Figure 7

Figure 7: Analytical Solution of the Hyperbolic Heat Conduction Equation for Moving Semi-Infinite Medium under the Effect of Time-Dependent Laser Heat Source 

Figure 7 | Analytical Solution of the Hyperbolic Heat Conduction Equation for Moving Semi-Infinite Medium under the Effect of Time-Dependent Laser Heat Source 