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<p>Wavelets have numerous applications in almost every field as they are powerful mathematical tools. They are becoming common for solving boundary value problems. One of the reasons wavelets are efficient is because they are capable of approximating the function exceptionally well. In this paper, a new wavelet, called the vertex covering wavelet, is introduced and originates from the vertex covering polynomial of the path graph <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40435_2025_1732_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\({P}_{n}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>. The proposed wavelet is applied together with collocation method, and singular boundary value problems are tackled by deriving an operational matrix of integration. The powerfulness of the method is proved through numerical tests where results are compared to other existing techniques in terms of accuracy and computational efficiency. The results presented in graphs and tables demonstrate the effectiveness of the vertex covering wavelet collocation method in providing the high-accuracy numerical solutions. This work introduces the notion of graph-based wavelets as an innovative approach to solve difficult problems in numerics and offers a new direction in wavelet-based numerical analysis.</p>

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An application of novel vertex covering wavelet (of path graph) for solving singular boundary value problems

  • 期刊论文

  • 发表日期：

    2025年5月28日

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  • K. J. Gowtham,
  • B. J. Gireesha

摘要

Wavelets have numerous applications in almost every field as they are powerful mathematical tools. They are becoming common for solving boundary value problems. One of the reasons wavelets are efficient is because they are capable of approximating the function exceptionally well. In this paper, a new wavelet, called the vertex covering wavelet, is introduced and originates from the vertex covering polynomial of the path graph \({P}_{n}\) P n . The proposed wavelet is applied together with collocation method, and singular boundary value problems are tackled by deriving an operational matrix of integration. The powerfulness of the method is proved through numerical tests where results are compared to other existing techniques in terms of accuracy and computational efficiency. The results presented in graphs and tables demonstrate the effectiveness of the vertex covering wavelet collocation method in providing the high-accuracy numerical solutions. This work introduces the notion of graph-based wavelets as an innovative approach to solve difficult problems in numerics and offers a new direction in wavelet-based numerical analysis.