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<p>In this paper we consider families of differential equations on some real interval that depend on a parameter <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\varepsilon \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation>, are regular for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\varepsilon \ne 0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>≠</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and have exactly one regular singular point for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\varepsilon =0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. Our aim is to give a formula which describes the asymptotic behavior of the solutions for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\varepsilon \rightarrow 0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. To this end, we use a method that combines matched asymptotic analysis with uniform asymptotic integration. We then apply our results to some typical examples, such as a differential equation with two coalescing regular singular points as well as a singular perturbation of the Bessel equation.</p>

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On the Asymptotic Behavior of the Solutions of Nearly Regular Singular Differential Equations

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  • 发表日期：

    2026年4月24日

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  • Harald Schmid

摘要

In this paper we consider families of differential equations on some real interval that depend on a parameter \(\varepsilon \) ε , are regular for \(\varepsilon \ne 0\) ε ≠ 0 and have exactly one regular singular point for \(\varepsilon =0\) ε = 0 . Our aim is to give a formula which describes the asymptotic behavior of the solutions for \(\varepsilon \rightarrow 0\) ε → 0 . To this end, we use a method that combines matched asymptotic analysis with uniform asymptotic integration. We then apply our results to some typical examples, such as a differential equation with two coalescing regular singular points as well as a singular perturbation of the Bessel equation.