Motivated by the wide scope of the applications of the weakly singular Volterra integral equations (VIEs) (see, for instance, [U. J. Choi and R. C. MacCamy, J. Math. Anal. Appl. 139, No. 2, 448–464 (1989; Zbl 0674.45007); K. Diethelm, The analysis of fractional differential equations. An application-oriented exposition using differential operators of Caputo type. Berlin: Springer (2010; Zbl 1215.34001); R. Gorenflo and S. Vessella, Abel integral equations. Analysis and applications. Berlin etc.: Springer-Verlag (1991; Zbl 0717.45002)]), the authors of this paper study the test equation \[ y\left( t \right) = f\left( t \right) + \frac{1}{{\Gamma \left( \alpha \right)}}\int_0^t {{{\left( {t - s} \right)}^{\alpha - 1}}k\left( s \right)y\left( s \right)ds} ,\quad t \geqslant 0,\tag{1} \] in which \(f\left( t \right)\) is the forcing term which is continuous for \(t \geqslant 0\), the parameter \(\alpha \in (0, 1] \), defines the weak singularity, \(k\left( t \right) > 0\) satisfies \[ \frac{1}{{\Gamma \left( \alpha \right)}}\mathop {\sup }\limits_{t \geqslant \bar t} \int_{\bar t}^t {{{\left( {t - s} \right)}^{\alpha - 1}}\left| {k\left( s \right)} \right|ds} = \eta < 1\tag{2} \] for \(\bar t \geqslant 0\). The second and third authors have earlier presented a similar analysis for VIEs with regular kernels in their works [“Stability and convergence of solutions to Volterra integral equations on time scales”, Discrete Dyn. Nat. Soc. 2015, Article ID 612156, 6 p. (2015; doi:10.1155/2015/612156); Appl. Numer. Math. 116, 230–237 (2017; Zbl 1372.65348); Numer. Algorithms 74, No. 4, 1223–1236 (2017; Zbl 1364.65295)], here the analysis covers the Abel-type equations. In yet another previous paper, the second and third authors together with R. Garrappa [Discrete Contin. Dyn. Syst. Ser. B 23, No. 7, 2679–2694 (2018; Zbl 1402.65067)] have studied the stability analysis of (1) relative to (2) for \(\bar t = 0\) using trapezoidal product integration. In the paper under review, the authors extend this approach to the class of convolution quadrature methods (see [Ch. Lubich, IMA J. Numer. Anal. 6, No. 1, 87–101 (1986; Zbl 0587.65090)]) and for any non-negative value of \(\bar t\).
In Section 2, the authors prove
Theorem 2.1. For Problem (1), (2), if the assumptions \(\mathop {\sup }\limits_{t \geqslant 0} \left| {k\left( t \right)} \right| = K < + \infty \) and \(\mathop {\sup }\limits_{t \geqslant 0} \left| {f\left( t \right)} \right| = F < + \infty \) hold, then \(\mathop {\sup }\limits_{t \geqslant 0} \left| {y\left( t \right)} \right| = Y < + \infty \).
Theorem 2.2. Consider (1), (2) and assume that \(\mathop {\sup }\limits_{t \geqslant 0} \left| {k\left( t \right)} \right| = K < + \infty \) holds, furthermore let \(\mathop {\lim }\limits_{t \to + \infty } f\left( t \right) = {f_\infty }\), and \(\mathop {\lim }\limits_{\tau \to + \infty } \mathop {\lim \sup }\limits_{t \to + \infty } \left| {\frac{1} {{\Gamma \left( \alpha \right)}}\int_\tau ^t {{{\left( {t - s} \right)}^{\alpha - 1}}k\left( s \right)ds} - {I_k}} \right| = 0\), for some \(I_k\), then \({\lim _{t \to + \infty }}y\left( t \right) = {y_\infty }\), with \({y_\infty } = {f_\infty }/\left( {1 - {I_k}} \right)\).
By considering the convolution quadrature \({y_n} = {f_n} + {h^\alpha }\sum\limits_{j = 0}^n {{\omega _{n - j}}{k_j}{y_j}}\), \(n \geqslant 0\), \(n = 0,1, \dots\), with \({f_n} = f\left( {{t_n}} \right) + {h^\alpha }\sum\nolimits_{j = - {n_0}}^{ - 1} {{w_{nj}}{y_j}{k_j}} \) and, by imposing certain conditions on the weights, the authors establish the following crucial result of the paper, which links the continuous and discrete problems:
Lemma 3.1. Assume that, for (1), \(\mathop {\sup }\limits_{t \geqslant 0} \left| {k\left( t \right)} \right| = K < + \infty \) holds, furthermore let \(\bar K = {\sup _{t \geqslant \bar t}}\left| {k\left( t \right)} \right|\), \(\bar t \geqslant 0\). Then, we have \[ \mathop {\sup }\limits_{n \geqslant r} {h^\alpha }\sum\limits_{j = r}^n {\left| {{\omega _{n - j}}} \right|} \left| {{k_j}} \right| \leqslant \frac{1} {{\Gamma \left( \alpha \right)}}\mathop {\sup }\limits_{t \geqslant \bar t} \int_{\bar t}^t {{{\left( {t - s} \right)}^{\alpha - 1}}\left| {k\left( s \right)} \right|ds} + {h^\alpha }A,\tag{3} \] for any \(r\geq 0\) and \(h > 0\) such that \(t_r = t_0 + rh = \bar t,\) where \[ A = \frac{1} {{\Gamma \left( \alpha \right)}}\left( {\bar K + \mathop {\sup }\limits_{t \geqslant \bar t} \int_{\bar t}^t {\left| {k'\left( x \right)} \right|dx} } \right) + \left( {\sum\limits_{n = 1}^\infty {\left| {{u_n}} \right|} + \left| {{\omega _0}} \right| + W} \right)\bar K. \]
Two results are proven in Section 4 for the numerical stability of the numerical solutions \(y_n\) of (1) which are analogous to Theorems 2.1 and 2.2. The authors also describe some numerical experiments in Section 5 to support the theory developed by them in the paper and an exemplar application is described in Section 6 of the paper. The reviewer concludes by expressing his affirmation with the authors’ remark, “However, the numerical stability with respect to non-parametric test equations is quite an unexplored subject of study in the context of weakly singular VIEs and the research described here may be considered as a first attempt in this direction.”