Abstract <p> We consider the equation <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11232_2025_2635_Article_IEq1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="191" /> </InlineMediaObject> <EquationSource Format="TEX">\(\int_{0}^{r}T(|x-t|)f(t)\,dt =g(x)\)</EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11232_2025_2635_Article_IEq2.gif" Format="GIF" Height="11" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">\(r<\infty\)</EquationSource> </InlineEquation> and the functions <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11232_2025_2635_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(T\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11232_2025_2635_Article_IEq4.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(g\)</EquationSource> </InlineEquation> and their first derivatives are absolutely continuous on <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11232_2025_2635_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="34" /> </InlineMediaObject> <EquationSource Format="TEX">\([0,r]\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11232_2025_2635_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="68" /> </InlineMediaObject> <EquationSource Format="TEX">\(T'(0)\ne 0\)</EquationSource> </InlineEquation>. An arbitrary term <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11232_2025_2635_Article_IEq7.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">\(ax+b\)</EquationSource> </InlineEquation> is added to the right-hand side of the equation. The obtained family of equations is reduced by double differentiation to an equation of the second kind. In the case of its unique solvability in <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11232_2025_2635_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">\(L_{1}(0,r)\)</EquationSource> </InlineEquation>, the solution <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11232_2025_2635_Article_IEq9.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\tilde{f}\)</EquationSource> </InlineEquation> is called a <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11232_2025_2635_Article_IEq10.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(D^{2}\)</EquationSource> </InlineEquation>-pseudosolution of the original equation. We introduce the partial regularization of the equation and present some cases of the existence of a <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11232_2025_2635_Article_IEq10.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(D^{2}\)</EquationSource> </InlineEquation>-pseudosolution. We propose a criterion for the suitability of <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11232_2025_2635_Article_IEq9.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\tilde{f}\)</EquationSource> </InlineEquation> as an approximate solution. The problem of constructing a pseudosolution of an equation on the half-line is discussed. </p>

Pseudosolution of an integral convolution equation of the first kind

期刊论文
发表日期：

2025年5月26日

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N. B. Yengibaryan

摘要

Abstract

We consider the equation \(\int_{0}^{r}T(|x-t|)f(t)\,dt =g(x)\) , where \(r<\infty\) and the functions \(T\) and \(g\) and their first derivatives are absolutely continuous on \([0,r]\) and \(T'(0)\ne 0\) . An arbitrary term \(ax+b\) is added to the right-hand side of the equation. The obtained family of equations is reduced by double differentiation to an equation of the second kind. In the case of its unique solvability in \(L_{1}(0,r)\) , the solution \(\tilde{f}\) is called a \(D^{2}\) -pseudosolution of the original equation. We introduce the partial regularization of the equation and present some cases of the existence of a \(D^{2}\) -pseudosolution. We propose a criterion for the suitability of \(\tilde{f}\) as an approximate solution. The problem of constructing a pseudosolution of an equation on the half-line is discussed.