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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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