<p>An exhaustive study is presented in this work to solve a chemical clock reaction model, which has a vital role in chemistry. The non-integer order chemical clock reaction framework in terms of the Caputo operator is discussed in this paper. In this research work, fractional-order chemical clock reaction equations are addressed with the assistance of the Haar wavelet approach. To check that the obtained solutions are correct, the Adams–Bashforth–Moulton method is used. Also, we conducted a comparative study of the outcomes of the chemical clock reaction model with the spectral collocation technique. Further, the Haar wavelet operational matrix is derived to convert the set of differential equation transforms into a set of algebraic equations. This set of complex nonlinear equations is resolved by utilizing MATLAB (2023a). Moreover, the focus lies on the convergent analysis, stability analysis, and the existence and uniqueness of the obtained outcomes. Furthermore, error analysis by contrasting the Haar wavelet technique and the spectral collocation technique is also discussed. This work not only shows the efficiency of the Haar wavelet technique in exactly calculating the dynamics of the chemical clock reaction model but also provides some examination of the chemical clock reaction system. Convergence analysis tells us that <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10910_2025_1719_Article_IEq1.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="137" /> </InlineMediaObject> <EquationSource Format="TEX">$\left\Vert e_\mathfrak {M}(t) \right\Vert _2 = o\left( \frac{1}{\mathfrak {M}}\right) .$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mfenced close="∥" open="∥"> <msub> <mi>e</mi> <mi mathvariant="fraktur">M</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mfenced> <mn>2</mn> </msub> <mo>=</mo> <mi>o</mi> <mfenced close=")" open="("> <mfrac> <mn>1</mn> <mi mathvariant="fraktur">M</mi> </mfrac> </mfenced> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> This implies that as <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10910_2025_1719_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$ \mathfrak {M} $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">M</mi> </math></EquationSource> </InlineEquation> increases, the error decreases. Specifically, for <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10910_2025_1719_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">$ \mathfrak {M} = 8 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="fraktur">M</mi> <mo>=</mo> <mn>8</mn> </mrow> </math></EquationSource> </InlineEquation>, the absolute error is approximately <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10910_2025_1719_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">$ 0.125 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0.125</mn> </mrow> </math></EquationSource> </InlineEquation>, while for <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10910_2025_1719_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">$ \mathfrak {M} = 16 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="fraktur">M</mi> <mo>=</mo> <mn>16</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10910_2025_1719_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">$ \mathfrak {M} = 32 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="fraktur">M</mi> <mo>=</mo> <mn>32</mn> </mrow> </math></EquationSource> </InlineEquation>, the errors reduce to <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10910_2025_1719_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">$ 0.0625 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0.0625</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10910_2025_1719_Article_IEq8.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">$ 0.03125 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0.03125</mn> </mrow> </math></EquationSource> </InlineEquation>, respectively. The error analysis shows that the error between Haar wavelet method and Adams–Bashforth–Moulton method maintain a low error rate, often in the range of <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10910_2025_1719_Article_IEq9.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="TEX">$ \mathbf {10^{-4}} $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn mathvariant="bold">10</mn> <mrow> <mo>-</mo> <mn mathvariant="bold">4</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10910_2025_1719_Article_IEq10.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="TEX">$ \mathbf {10^{-1}} $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn mathvariant="bold">10</mn> <mrow> <mo>-</mo> <mn mathvariant="bold">1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>, whereas the error between Spectral Collocation method and the Adams–Bashforth–Moulton method exhibit higher absolute errors, highlighting accuracy of the Haar wavelet approach. Additionally, the stability of the proposed method is theoretically established, ensuring that the solutions remain bounded within a well-defined range.</p>

Numerical study of a chemical clock reaction framework utilizing the Haar wavelet approach

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发表日期：

2025年3月20日

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Jasinth Sylvia,
Surath Ghosh

摘要

An exhaustive study is presented in this work to solve a chemical clock reaction model, which has a vital role in chemistry. The non-integer order chemical clock reaction framework in terms of the Caputo operator is discussed in this paper. In this research work, fractional-order chemical clock reaction equations are addressed with the assistance of the Haar wavelet approach. To check that the obtained solutions are correct, the Adams–Bashforth–Moulton method is used. Also, we conducted a comparative study of the outcomes of the chemical clock reaction model with the spectral collocation technique. Further, the Haar wavelet operational matrix is derived to convert the set of differential equation transforms into a set of algebraic equations. This set of complex nonlinear equations is resolved by utilizing MATLAB (2023a). Moreover, the focus lies on the convergent analysis, stability analysis, and the existence and uniqueness of the obtained outcomes. Furthermore, error analysis by contrasting the Haar wavelet technique and the spectral collocation technique is also discussed. This work not only shows the efficiency of the Haar wavelet technique in exactly calculating the dynamics of the chemical clock reaction model but also provides some examination of the chemical clock reaction system. Convergence analysis tells us that $\left\Vert e_\mathfrak {M}(t) \right\Vert _2 = o\left( \frac{1}{\mathfrak {M}}\right) .$ ${∥e_{M}, (t)∥}_{2} = o (\frac{1}{M}) .$ This implies that as $ \mathfrak {M} $ $M$ increases, the error decreases. Specifically, for $ \mathfrak {M} = 8 $ $M = 8$ , the absolute error is approximately $ 0.125 $ $0.125$ , while for $ \mathfrak {M} = 16 $ $M = 16$ and $ \mathfrak {M} = 32 $ $M = 32$ , the errors reduce to $ 0.0625 $ $0.0625$ and $ 0.03125 $ $0.03125$ , respectively. The error analysis shows that the error between Haar wavelet method and Adams–Bashforth–Moulton method maintain a low error rate, often in the range of $ \mathbf {10^{-4}} $ $10^{- 4}$ to $ \mathbf {10^{-1}} $ $10^{- 1}$ , whereas the error between Spectral Collocation method and the Adams–Bashforth–Moulton method exhibit higher absolute errors, highlighting accuracy of the Haar wavelet approach. Additionally, the stability of the proposed method is theoretically established, ensuring that the solutions remain bounded within a well-defined range.