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

<table>Variations of the displacement <svg height="7.1624999" id="M121" style="vertical-align:-0.11285pt" version="1.1" viewbox="0 0 7.5374999 7.1624999" width="7.5374999" 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">𝑢</tspan>
</text>
</g>
</g>
</svg> and temperature <svg height="10.7375" id="M122" style="vertical-align:-0.13794pt" version="1.1" viewbox="0 0 8.3125 10.7375" width="8.3125" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.7375)">
<g transform="translate(72,-63.41)">
<text transform="matrix(1,0,0,-1,-71.95,63.6)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝜃</tspan>
</text>
</g>
</g>
</svg> for various values of the axis <svg height="7.1624999" id="M123" 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="M124" 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="14.3875" id="M125" style="vertical-align:-3.25793pt" version="1.1" viewbox="0 0 107.4625 14.3875" width="107.4625" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.3875)">
<g transform="translate(72,-60.49)">
<text transform="matrix(1,0,0,-1,-71.95,63.79)">
<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">𝑉</tspan>
</text>
<text transform="matrix(1,0,0,-1,-43.32,60.66)">
<tspan style="font-size: 8.75px; " x="0" y="0">0</tspan>
</text>
<text transform="matrix(1,0,0,-1,-35.67,63.79)">
<tspan style="font-size: 12.50px; " x="0" y="0">+</tspan>
<tspan style="font-size: 12.50px; " x="11.340221" y="0">𝑉</tspan>
</text>
<text transform="matrix(1,0,0,-1,-17.14,60.66)">
<tspan style="font-size: 8.75px; " x="0" y="0">1</tspan>
</text>
<text transform="matrix(1,0,0,-1,-9.49,63.79)">
<tspan style="font-size: 12.50px; " x="0" y="0">+</tspan>
<tspan style="font-size: 12.50px; " x="11.340221" y="0">𝑉</tspan>
</text>
<text transform="matrix(1,0,0,-1,9.05,60.66)">
<tspan style="font-size: 8.75px; " x="0" y="0">2</tspan>
</text>
</g>
</g>
</svg> and <svg height="14.75" id="M126" style="vertical-align:-3.25793pt" version="1.1" viewbox="0 0 115.1375 14.75" width="115.1375" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.75)">
<g transform="translate(72,-60.2)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<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,-40.86,60.37)">
<tspan style="font-size: 8.75px; " x="0" y="0">0</tspan>
</text>
<text transform="matrix(1,0,0,-1,-33.21,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">+</tspan>
<tspan style="font-size: 12.50px; " x="11.340221" y="0">Θ</tspan>
</text>
<text transform="matrix(1,0,0,-1,-12.84,60.37)">
<tspan style="font-size: 8.75px; " x="0" y="0">1</tspan>
</text>
<text transform="matrix(1,0,0,-1,-5.18,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">+</tspan>
<tspan style="font-size: 12.50px; " x="11.327718" y="0">Θ</tspan>
</text>
<text transform="matrix(1,0,0,-1,15.19,60.37)">
<tspan style="font-size: 8.75px; " x="0" y="0">2</tspan>
</text>
</g>
</g>
</svg>) when <svg height="14.6" id="M127" 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>
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<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

fig9

Figure 9

Figure 9: Homotopy Perturbation Method and Variational Iteration Method for Harmonic Waves Propagation in Nonlinear Magneto-Thermoelasticity with Rotation 

Figure 9 | Homotopy Perturbation Method and Variational Iteration Method for Harmonic Waves Propagation in Nonlinear Magneto-Thermoelasticity with Rotation 