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Analysis of collocation solutions for nonstandard Volterra integral equations. (English) Zbl 1259.65214
The following nonlinear Volterra integral equation with respect to the unknown function \(u(t)\) is considered \[ u(t)= g(t)+ \int^t_0 K(t,s,u(t),u(s))ds,\quad t \in I:=[0,T]. \tag{1} \]
Such equations have numerous applications in modeling the physical and biological processes and are nonstandard due to the function \(u(t)\) in the integral.
The authors investigate the existence, uniqueness and regularity properties of solutions for (1). For this purpose, the general Banach fixed point theorem in small subintervals is applied. It is particularly shown that under certain smoothness conditions for \(g\) and \(K\), the exact solution \(u(t)\) is in \(C^m(I)\). They also present a collocation method to solve this equation, and analyse the convergence and superconvergence of piecewise polynomial collocation approximations.
There are also adduced two model equations with extensive numerical experiments showing that the collocation method behaves according to the theoretical results.

MSC:
65R20 Numerical methods for integral equations
45D05 Volterra integral equations
45G10 Other nonlinear integral equations
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