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<p>The study examines a generalized Bernoulli differential equation with piecewise constant arguments, which includes two terms with powers of the unknown. It presents an efficient approach to determining an approximate solution to the equation. A differential equation with piecewise constant arguments corresponding to the considered initial value problem, which depends on a positive integer <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40819_2025_1855_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(N\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>N</mi> </math></EquationSource> </InlineEquation>, is introduced. It is proven that this equation has a unique piecewise-smooth solution, which serves as an approximate solution to the initial value problem for large <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40819_2025_1855_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(N\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>N</mi> </math></EquationSource> </InlineEquation>. Numerical results are provided through examples, demonstrating the efficiency and high accuracy of the proposed method.</p>

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On the Approximation Algorithms for Solving Generalized Bernoulli Differential Equations Using Piecewise Constant Argument Method

  • 期刊论文

  • 发表日期：

    2025年2月25日

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  • Mukhiddin I. Muminov,
  • Zafar Z. Jumaev

摘要

The study examines a generalized Bernoulli differential equation with piecewise constant arguments, which includes two terms with powers of the unknown. It presents an efficient approach to determining an approximate solution to the equation. A differential equation with piecewise constant arguments corresponding to the considered initial value problem, which depends on a positive integer \(N\) N , is introduced. It is proven that this equation has a unique piecewise-smooth solution, which serves as an approximate solution to the initial value problem for large \(N\) N . Numerical results are provided through examples, demonstrating the efficiency and high accuracy of the proposed method.