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

<table>Dimensionless temperature distributions at <svg height="13.55" id="M99" style="vertical-align:-2.29482pt" version="1.1" viewbox="0 0 35.137501 13.55" width="35.137501" xmlns="http://www.w3.org/2000/svg">
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<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<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 exponential heat source and various values of <svg height="13.425" id="M100" 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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<text transform="matrix(1,0,0,-1,-71.95,63.6)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝛽</tspan>
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</svg>; <svg height="14.8625" id="M101" style="vertical-align:-3.25793pt" version="1.1" viewbox="0 0 274.16251 14.8625" width="274.16251" xmlns="http://www.w3.org/2000/svg">
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<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">e</tspan>
<tspan style="font-size: 12.50px; " x="43.697987" y="0">x</tspan>
<tspan style="font-size: 12.50px; " x="49.949486" y="0">p</tspan>
<tspan style="font-size: 12.50px; " x="60.364483" y="0">(</tspan>
<tspan style="font-size: 12.50px; " x="64.527985" y="0">−</tspan>
<tspan style="font-size: 12.50px; " x="73.092537" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="79.34404" y="0">.</tspan>
<tspan style="font-size: 12.50px; " x="82.469788" y="0">4</tspan>
<tspan style="font-size: 12.50px; " x="88.721291" y="0">𝜂</tspan>
<tspan style="font-size: 12.50px; " x="94.972786" y="0">)</tspan>
<tspan style="font-size: 12.50px; " x="99.136284" y="0">,</tspan>
<tspan style="font-size: 12.50px; " x="108.51353" y="0">𝜓</tspan>
</text>
<text transform="matrix(1,0,0,-1,45.97,60.28)">
<tspan style="font-size: 8.75px; " x="0" y="0">0</tspan>
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<text transform="matrix(1,0,0,-1,54.31,63.41)">
<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="49.886971" y="0">𝑈</tspan>
<tspan style="font-size: 12.50px; " x="62.177418" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="74.217812" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="80.469307" y="0">.</tspan>
<tspan style="font-size: 12.50px; " x="83.595055" y="0">1</tspan>
<tspan style="font-size: 12.50px; " x="89.846558" y="0">.</tspan>
</text>
</g>
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</svg></table>

Journal of Applied Mathematics

fig9

Figure 9

Figure 9: Analytical Solution of the Hyperbolic Heat Conduction Equation for Moving Semi-Infinite Medium under the Effect of Time-Dependent Laser Heat Source 

Figure 9 | Analytical Solution of the Hyperbolic Heat Conduction Equation for Moving Semi-Infinite Medium under the Effect of Time-Dependent Laser Heat Source 