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<p>The study presents a novel algorithm for solving third-order non-linear equations (Emden–Fowler type), which can be applied to various physical models. The algorithm uses a quintic trigonometric B-spline collocation method and a quasilinearization technique to avoid the non-linearity term in the equation. The study established a comprehensive error analysis for the proposed algorithm and proved that it has fourth order, i.e., <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="500_2025_10431_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">\((\mathscr {O}(h^4))\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">O</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mi>h</mi> <mn>4</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> convergent. The algorithm’s ability to handle singular behavior at the point <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="500_2025_10431_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">\(x=0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and its faster rate of convergence exhibit a promising approach to solving such problems. The study also validates the theoretical results through numerical experiments and shows that the proposed algorithm has a faster rate of convergence in comparison to the existing methods.</p>

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An efficient collocation algorithm for third order non-linear Emden–Fowler equation

  • 期刊论文

  • 发表日期：

    2025年2月15日

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  • Mohammad Prawesh Alam,
  • Arshad Khan

摘要

The study presents a novel algorithm for solving third-order non-linear equations (Emden–Fowler type), which can be applied to various physical models. The algorithm uses a quintic trigonometric B-spline collocation method and a quasilinearization technique to avoid the non-linearity term in the equation. The study established a comprehensive error analysis for the proposed algorithm and proved that it has fourth order, i.e., \((\mathscr {O}(h^4))\) ( O ( h 4 ) ) convergent. The algorithm’s ability to handle singular behavior at the point \(x=0\) x = 0 and its faster rate of convergence exhibit a promising approach to solving such problems. The study also validates the theoretical results through numerical experiments and shows that the proposed algorithm has a faster rate of convergence in comparison to the existing methods.