A Reliable Treatment of Homotopy Perturbation Method for Solving the Nonlinear Klein-Gordon Equation of Arbitrary (Fractional) Orders

<table><svg height="13.6375" id="M100" style="vertical-align:-2.21957pt" version="1.1" viewbox="0 0 52.587502 13.6375" width="52.587502" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,13.6375)">
<g transform="translate(72,-61.09)">
<text transform="matrix(1,0,0,-1,-71.95,63.35)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑢</tspan>
<tspan style="font-size: 12.50px; " x="5.9264221" y="0">(</tspan>
<tspan style="font-size: 12.50px; " x="10.089921" y="0">𝑥</tspan>
<tspan style="font-size: 12.50px; " x="16.966572" y="0">,</tspan>
<tspan style="font-size: 12.50px; " x="22.167818" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="28.419319" y="0">.</tspan>
<tspan style="font-size: 12.50px; " x="31.545069" y="0">5</tspan>
<tspan style="font-size: 12.50px; " x="37.79657" y="0">)</tspan>
</text>
</g>
</g>
</svg> of Example <a href="https://static.hindawi.com/articles/jam/volume-2012/581481/figures/#ex1">4.1</a> Case <a href="https://static.hindawi.com/articles/jam/volume-2012/581481/figures/#casee2">2</a> for 6th-order HPM approximation as parameterized by <svg height="13.425" id="M101" 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">
<g transform="matrix(1.25,0,0,-1.25,0,13.425)">
<g transform="translate(72,-61.26)">
<text transform="matrix(1,0,0,-1,-71.95,63.6)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝛽</tspan>
</text>
</g>
</g>
</svg>.</table>

Journal of Applied Mathematics

fig2

Figure 2

Figure 2: A Reliable Treatment of Homotopy Perturbation Method for Solving the Nonlinear Klein-Gordon Equation of Arbitrary (Fractional) Orders 