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Spectral collocation methods for Volterra-integro differential equations with noncompact kernels. (English) Zbl 1263.65134
The paper is devoted to a study of a numerical method for the linear Volterra integro-differential equation \(y'(t) + a(t) y(t) = \int_0^t s^{\mu-1}t^{-\mu} k(t,s) y(s) ds + g(t)\), \(t \in [0,T]\), subject to the initial condition \(y(0) = y_0\), with sufficiently smooth functions \(a\), \(g\) and \(k\) and \(\mu > 0\). The integral operator in this equation is not compact; this property makes the investigation more difficult.
The authors begin by providing sufficient conditions for the existence and uniqueness of solutions and prove regularity properties of these solutions. Then, a spectral collocation method based on Chebyshev polynomials is developed and its convergence is analyzed.

MSC:
65R20 Numerical methods for integral equations
45J05 Integro-ordinary differential equations
45A05 Linear integral equations
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