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<p>Bone mineralization is a crucial physiological process that ensures skeletal strength and integrity, playing a vital role in bone development, repair, and overall health. Using the mathematical framework developed by Komarova in 2015, this manuscript attempts to investigate the practical implications of combining Fractal–Fractional (<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2472_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal{F}\mathcal{F}\)</EquationSource> </InlineEquation>) operators in mathematical modeling, with a particular focus on the setting of bone mineralization. Through investigating the potential advantages of fractal operators, we develop a more precise and sophisticated mathematical model, evaluate the existence and uniqueness of solutions, Ulam–Hyers (UH) stability, and present numerical results that highlight the improved performance offered by this innovative approach. Lagrange’s two-step method has been used to estimate solutions for the <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2472_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal{F}\mathcal{F}\)</EquationSource> </InlineEquation> order bone mineralization model. MATLAB software is used to perform the numerical simulations, which provide insightful information on the dynamics of the model. Our simulations suggest early detection of the mineralization.</p>

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Numerical and mathematical analysis of the novel fractal–fractional Komarova’s coupled system for the bone mineralization: existence and uniqueness of solution

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年4月10日

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  • Ritu Agarwal,
  • Pooja Airan,
  • Haci Mehmet Baskonus

摘要

Bone mineralization is a crucial physiological process that ensures skeletal strength and integrity, playing a vital role in bone development, repair, and overall health. Using the mathematical framework developed by Komarova in 2015, this manuscript attempts to investigate the practical implications of combining Fractal–Fractional ( \(\mathcal{F}\mathcal{F}\) ) operators in mathematical modeling, with a particular focus on the setting of bone mineralization. Through investigating the potential advantages of fractal operators, we develop a more precise and sophisticated mathematical model, evaluate the existence and uniqueness of solutions, Ulam–Hyers (UH) stability, and present numerical results that highlight the improved performance offered by this innovative approach. Lagrange’s two-step method has been used to estimate solutions for the \(\mathcal{F}\mathcal{F}\) order bone mineralization model. MATLAB software is used to perform the numerical simulations, which provide insightful information on the dynamics of the model. Our simulations suggest early detection of the mineralization.