Homotopy Perturbation Method and Variational Iteration Method for Harmonic Waves Propagation in Nonlinear Magneto-Thermoelasticity with Rotation

<table>Variations of the displacement <svg height="10.925" id="M243" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 13.625 10.925" width="13.625" xmlns="http://www.w3.org/2000/svg">
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<g transform="translate(72,-63.26)">
<text transform="matrix(1,0,0,-1,-71.95,66.44)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑢</tspan>
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<text transform="matrix(1,0,0,-1,-66.02,63.31)">
<tspan style="font-size: 8.75px; " x="0" y="0">2</tspan>
</text>
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</g>
</svg> and temperature <svg height="14.475" id="M244" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 13.7875 14.475" width="13.7875" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.475)">
<g transform="translate(72,-60.42)">
<text transform="matrix(1,0,0,-1,-71.95,63.6)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝜃</tspan>
</text>
<text transform="matrix(1,0,0,-1,-65.9,60.47)">
<tspan style="font-size: 8.75px; " x="0" y="0">2</tspan>
</text>
</g>
</g>
</svg> for various values of the axis <svg height="7.1624999" id="M245" style="vertical-align:-0.11285pt" version="1.1" viewbox="0 0 8.7250004 7.1624999" width="8.7250004" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,7.1625)">
<g transform="translate(72,-66.27)">
<text transform="matrix(1,0,0,-1,-71.95,66.44)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑥</tspan>
</text>
</g>
</g>
</svg> and time <svg height="9.125" id="M246" style="vertical-align:-0.11285pt" version="1.1" viewbox="0 0 5.0124998 9.125" width="5.0124998" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,9.125)">
<g transform="translate(72,-64.7)">
<text transform="matrix(1,0,0,-1,-71.95,64.86)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑡</tspan>
</text>
</g>
</g>
</svg> (<svg height="10.925" id="M247" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 40.424999 10.925" width="40.424999" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.925)">
<g transform="translate(72,-63.26)">
<text transform="matrix(1,0,0,-1,-71.95,66.44)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑢</tspan>
<tspan style="font-size: 12.50px; " x="9.3897533" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="21.430141" y="0">𝑢</tspan>
</text>
<text transform="matrix(1,0,0,-1,-44.59,63.31)">
<tspan style="font-size: 8.75px; " x="0" y="0">2</tspan>
</text>
</g>
</g>
</svg> and <svg height="14.475" id="M248" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 41.362499 14.475" width="41.362499" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.475)">
<g transform="translate(72,-60.42)">
<text transform="matrix(1,0,0,-1,-71.95,63.6)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝜃</tspan>
<tspan style="font-size: 12.50px; " x="10.014903" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="22.055292" y="0">𝜃</tspan>
</text>
<text transform="matrix(1,0,0,-1,-43.84,60.47)">
<tspan style="font-size: 8.75px; " x="0" y="0">2</tspan>
</text>
</g>
</g>
</svg>) when <svg height="14.6" id="M249" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 106.7 14.6" width="106.7" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.6)">
<g transform="translate(72,-60.32)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">Ω</tspan>
<tspan style="font-size: 12.50px; " x="12.765563" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="24.805952" y="0">𝜎</tspan>
</text>
<text transform="matrix(1,0,0,-1,-40.42,60.37)">
<tspan style="font-size: 8.75px; " x="0" y="0">1</tspan>
</text>
<text transform="matrix(1,0,0,-1,-32.08,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="12.040389" y="0">𝜎</tspan>
</text>
<text transform="matrix(1,0,0,-1,-13.32,60.37)">
<tspan style="font-size: 8.75px; " x="0" y="0">2</tspan>
</text>
<text transform="matrix(1,0,0,-1,-4.98,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="12.040389" y="0">0</tspan>
</text>
</g>
</g>
</svg>.</table>

Mathematical Problems in Engineering

fig19

Figure 19

Figure 19: Homotopy Perturbation Method and Variational Iteration Method for Harmonic Waves Propagation in Nonlinear Magneto-Thermoelasticity with Rotation 

Figure 19 | Homotopy Perturbation Method and Variational Iteration Method for Harmonic Waves Propagation in Nonlinear Magneto-Thermoelasticity with Rotation 