Solving Linear Coupled Fractional Differential Equations by Direct Operational Method and Some Applications

<table>Variation of waveform characteristics for <svg height="16.275" id="M250" style="vertical-align:-3.276pt" version="1.1" viewbox="0 0 33.525002 16.275" width="33.525002" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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<g transform="matrix(.017,-0,0,-.017,15.9,12.137)"><use xlink:href="#x28"></use></g><g transform="matrix(.017,-0,0,-.017,21.782,12.137)"><path d="M324 430l-26 -36l-112 -4l-55 -265q-13 -66 7 -66q13 0 44.5 20t50.5 40l17 -24q-38 -40 -85.5 -73.5t-87.5 -33.5q-50 0 -21 138l55 262h-80l-2 8l25 34h66l25 99l78 63l10 -9l-37 -153h128z" id="x1D461"></path></g><g transform="matrix(.017,-0,0,-.017,27.578,12.137)"><use xlink:href="#x29"></use></g>
</svg>. Parameters: <svg height="16.275" id="M251" style="vertical-align:-3.276pt" version="1.1" viewbox="0 0 152.8125 16.275" width="152.8125" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.137)"><use xlink:href="#x1D465"></use></g>
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<g transform="matrix(.012,-0,0,-.012,71.725,16.212)"><use xlink:href="#x2217"></use></g><g transform="matrix(.012,-0,0,-.012,78.71,16.212)"><use xlink:href="#x32"></use></g>
<g transform="matrix(.017,-0,0,-.017,85.05,12.137)"><use xlink:href="#x28"></use></g><g transform="matrix(.017,-0,0,-.017,90.932,12.137)"><use xlink:href="#x30"></use></g><g transform="matrix(.017,-0,0,-.017,99.091,12.137)"><use xlink:href="#x29"></use></g><g transform="matrix(.017,-0,0,-.017,109.681,12.137)"><use xlink:href="#x3D"></use></g><g transform="matrix(.017,-0,0,-.017,124.385,12.137)"><use xlink:href="#x30"></use></g><g transform="matrix(.017,-0,0,-.017,132.545,12.137)"><use xlink:href="#x2E"></use></g><g transform="matrix(.017,-0,0,-.017,136.42,12.137)"><use xlink:href="#x30"></use></g><g transform="matrix(.017,-0,0,-.017,144.58,12.137)"><use xlink:href="#x35"></use></g>
</svg>, (a) Capacitor order affects both frequency and amplitude. Legend: <svg height="11.125" id="M252" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 49.400002 11.125" width="49.400002" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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</svg> (solid), 0.5 (dashed), 0.7 (dash-dotted), 1.0 (dotted) and (b) Time constant affects frequency while amplitude remains constant. Legend: <svg height="11.75" id="M253" style="vertical-align:-0.23206pt" version="1.1" viewbox="0 0 62.674999 11.75" width="62.674999" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><path d="M627 18l-10 -26q-79 6 -116 27t-69 76q-41 71 -71 138q-13 29 -27.5 39t-42.5 10h-46l-27 -145q-13 -74 -2.5 -88.5t78.5 -20.5l-6 -28h-271l5 28q66 6 82.5 21.5t30.5 87.5l71 387q12 66 2 78.5t-77 19.5l8 28h233q102 0 147 -29q65 -43 65 -129q0 -69 -45.5 -117
t-115.5 -72q40 -86 66 -133q39 -68 65 -101q28 -37 73 -51zM491 483q0 67 -33.5 101t-91.5 34q-35 0 -51 -10q-13 -8 -20 -48l-45 -245h49q71 0 113 28q79 52 79 140z" id="x1D445"></path></g><g transform="matrix(.017,-0,0,-.017,11.01,11.4)"><path d="M682 629q-1 -16 -1.5 -72t-2.5 -86l-31 -4q-5 92 -51 129t-139 37q-100 0 -177 -49t-116 -125t-39 -162q0 -122 66 -201t182 -79q83 0 137 42.5t112 128.5l26 -15q-12 -31 -42.5 -88t-45.5 -75q-139 -27 -199 -27q-148 0 -243 81.5t-95 226.5q0 173 129.5 274.5
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t98.5 -62.5q61 48 61 124q0 59 -29.5 90t-69.5 31zM248 20q46 0 76.5 33.5t30.5 89.5q0 50 -39 85.5t-110 71.5q-81 -54 -81 -137q0 -67 35.5 -105t87.5 -38z" id="x38"></path></g>
</svg> (solid), 0.9 (dashed), and 1.0 (dash-dotted), and 1.1 (dotted).</table>

Mathematical Problems in Engineering

fig4

Figure 4

Figure 4: Solving Linear Coupled Fractional Differential Equations by Direct Operational Method and Some Applications 