The Spectral Method for Solving Sine-Gordon Equation Using a New Orthogonal Polynomial

<table>(a) The exact and approximate solutions for <svg height="11.0375" id="M188" style="vertical-align:-0.17555pt" version="1.1" viewbox="0 0 48.4375 11.0375" width="48.4375" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,11.0375)">
<g transform="translate(72,-63.17)">
<text transform="matrix(1,0,0,-1,-71.95,63.4)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑁</tspan>
<tspan style="font-size: 12.50px; " x="14.103384" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="26.143772" y="0">1</tspan>
<tspan style="font-size: 12.50px; " x="32.395271" y="0">6</tspan>
</text>
</g>
</g>
</svg> are shown. (b) The <svg height="14.2375" id="M189" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 21.9375 14.2375" width="21.9375" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.2375)">
<g transform="translate(72,-60.61)">
<text transform="matrix(1,0,0,-1,-71.95,63.79)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝐿</tspan>
</text>
<text transform="matrix(1,0,0,-1,-63.1,60.66)">
<tspan style="font-size: 8.75px; " x="0" y="0">∞</tspan>
</text>
</g>
</g>
</svg> error norm for <svg height="11.0375" id="M190" style="vertical-align:-0.17555pt" version="1.1" viewbox="0 0 48.4375 11.0375" width="48.4375" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,11.0375)">
<g transform="translate(72,-63.17)">
<text transform="matrix(1,0,0,-1,-71.95,63.4)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑁</tspan>
<tspan style="font-size: 12.50px; " x="14.103384" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="26.143772" y="0">1</tspan>
<tspan style="font-size: 12.50px; " x="32.395271" y="0">6</tspan>
</text>
</g>
</g>
</svg> is shown. (c) The <svg height="14.2375" id="M191" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 17.2875 14.2375" width="17.2875" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.2375)">
<g transform="translate(72,-60.61)">
<text transform="matrix(1,0,0,-1,-71.95,63.79)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝐿</tspan>
</text>
<text transform="matrix(1,0,0,-1,-63.1,60.66)">
<tspan style="font-size: 8.75px; " x="0" y="0">2</tspan>
</text>
</g>
</g>
</svg> error norm for different values of <svg height="10.325" id="M192" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 14.8375 10.325" width="14.8375" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.325)">
<g transform="translate(72,-63.74)">
<text transform="matrix(1,0,0,-1,-71.95,63.79)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑁</tspan>
</text>
</g>
</g>
</svg> is shown. (d) The <svg height="14.2375" id="M193" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 21.9375 14.2375" width="21.9375" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.2375)">
<g transform="translate(72,-60.61)">
<text transform="matrix(1,0,0,-1,-71.95,63.79)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝐿</tspan>
</text>
<text transform="matrix(1,0,0,-1,-63.1,60.66)">
<tspan style="font-size: 8.75px; " x="0" y="0">∞</tspan>
</text>
</g>
</g>
</svg> error norm for different values of <svg height="10.325" id="M194" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 14.8375 10.325" width="14.8375" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.325)">
<g transform="translate(72,-63.74)">
<text transform="matrix(1,0,0,-1,-71.95,63.79)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑁</tspan>
</text>
</g>
</g>
</svg> is shown.</table>

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fig2

Figure 2

Figure 2: The Spectral Method for Solving Sine-Gordon Equation Using a New Orthogonal Polynomial 