Blow-up behavior of collocation solutions to Hammerstein-type Volterra integral equations (Q2855107)

The following Volterra Integral Equation (VIE) arises as a mathematical model for the formation of shear bands in steel that is subjected to very high strain rates: NEWLINE\[NEWLINE u(t)=\gamma \int_{0}^{t}( \pi(t-s))^{-1/2}(1+s)^{q}[u(s)+1]^{p}ds,\tag{*} NEWLINE\]NEWLINE where \( \gamma>0 \) and \( p\geq 0 \), \( q\geq 0\) are material parameters related to the constitutive law for plastic straining (see [\textit{C. A. Roberts} et al., J. Integral Equations Appl. 5, No. 4, 531--546 (1993; Zbl 0804.45002)]). Equation (*) can be rewritten in the form NEWLINE\[NEWLINE u(t)=\varphi(t)+\int_{0}^{t}k(t-s)G(s,u(s))ds, t\in [0,T],\tag{**} NEWLINE\]NEWLINE where \( \varphi: \mathbb{R}\rightarrow \mathbb{R}_{+} \) and \( G: \mathbb{R}\times \mathbb{R}\rightarrow \mathbb{R}_{+} \) are continuously differentiable, and the kernel \(k:(0,\infty)\rightarrow (0,\infty)\) is locally integrable. This paper is devoted to a systematic study of the numerical solution of nonlinear VIE (**) by a class of(one parameter)collocation methods. One of the key results in this paper is that these numerical methods could be used to detect finite-time blow-up( an important aspect since in many practical applications it is not known a priori whether or not the given model VIE will exhibit finite-time blow-up). The authors have shown, for the blow-up case, the convergence of the numerical blow-up time to the exact one. They have defined an adaptive step-size strategy for (**) so that the collocation solutions of implicit methods exist uniquely at each time level. It is shown that the asymptotic behavior of the collocation solutions with adaptive step-size is the same as for the exact ones, regardless of whether or not the exact solutions blow-up in finite time.