Comparison between Taylor and perturbed method for Volterra integral equation of the first kind

Abstract

<p style='text-indent:20px;'>As it is known the equation <inline-formula><tex-math id="M1">\begin{document}$ A\varphi = f $\end{document}</tex-math></inline-formula> with injective compact operator has a unique solution for all <inline-formula><tex-math id="M2">\begin{document}$ f $\end{document}</tex-math></inline-formula> in the range <inline-formula><tex-math id="M3">\begin{document}$ R(A). $\end{document}</tex-math></inline-formula>Unfortunately, the right-hand side <inline-formula><tex-math id="M4">\begin{document}$ f $\end{document}</tex-math></inline-formula> is never known exactly, so we can take an approximate data <inline-formula><tex-math id="M5">\begin{document}$ f_{\delta } $\end{document}</tex-math></inline-formula> and used the perturbed problem <inline-formula><tex-math id="M6">\begin{document}$ \alpha \varphi +A\varphi = f_{\delta } $\end{document}</tex-math></inline-formula> where the solution <inline-formula><tex-math id="M7">\begin{document}$ \varphi _{\alpha \delta } $\end{document}</tex-math></inline-formula> depends continuously on the data <inline-formula><tex-math id="M8">\begin{document}$ f_{\delta }, $\end{document}</tex-math></inline-formula> and the bounded inverse operator <inline-formula><tex-math id="M9">\begin{document}$ \left( \alpha I+A \right) ^{-1} $\end{document}</tex-math></inline-formula> approximates the unbounded operator <inline-formula><tex-math id="M10">\begin{document}$ A^{-1} $\end{document}</tex-math></inline-formula> but not stable. In this work we obtain the convergence of the approximate solution of <inline-formula><tex-math id="M11">\begin{document}$ \varphi _{\alpha \delta } $\end{document}</tex-math></inline-formula> of the perturbed equation to the exact solution <inline-formula><tex-math id="M12">\begin{document}$ \varphi $\end{document}</tex-math></inline-formula> of initial equation provided <inline-formula><tex-math id="M13">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula> tends to zero with <inline-formula><tex-math id="M14">\begin{document}$ \dfrac{\delta }{\sqrt{\alpha }}. $\end{document}</tex-math></inline-formula></p>