A Three-Stage Fifth-Order Runge-Kutta Method for Directly Solving Special Third-Order Differential Equation with Application to Thin Film Flow Problem

<table>Plot of the solution <svg height="12.9125" id="M236" style="vertical-align:-4.15506pt" version="1.1" viewbox="0 0 13.8375 12.9125" width="13.8375" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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<g transform="matrix(.012,-0,0,-.012,9.9,12.713)"><path d="M244 607q0 -25 -15.5 -43t-37.5 -18q-19 0 -32 13t-13 35q0 21 15 41t39 20q20 0 32 -14t12 -34zM222 91q-29 -33 -79 -68t-75 -35q-13 0 -19 7.5t-6 31t10 65.5l62 253q5 26 -1 26q-21 0 -72 -43l-13 24q43 40 91 68t71 28q30 0 10 -78l-71 -274q-8 -30 3 -30
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</svg> for problem (<a href="https://static.hindawi.com/articles/mpe/volume-2013/795397/figures/#EEq41">37</a>) for <svg height="12.4375" id="M237" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 36.262501 12.4375" width="36.262501" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.162)"><path d="M480 416q0 -21 -18 -41q-9 -11 -17 -7q-20 9 -42 9q-62 0 -140 -78q23 -69 88 -192q17 -31 27 -42t20 -11q16 0 62 46l17 -20q-64 -92 -119 -92q-35 0 -70 66q-41 73 -84 187q-36 -30 -62 -61q-27 -115 -35 -172q-41 -8 -78 -20l-6 6l140 612q7 28 0.5 34t-37.5 7l-34 1
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</svg>, <svg height="12.4375" id="M238" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 56.674999 12.4375" width="56.674999" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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</svg>.</table>

Mathematical Problems in Engineering

fig3

Figure 3

Figure 3: A Three-Stage Fifth-Order Runge-Kutta Method for Directly Solving Special Third-Order Differential Equation with Application to Thin Film Flow Problem 

Figure 3 | A Three-Stage Fifth-Order Runge-Kutta Method for Directly Solving Special Third-Order Differential Equation with Application to Thin Film Flow Problem 